DISS. ETH Nr. 12794
NUMERICAL METHODS
FOR THE THERMODYNAMICS
OF LOW-DIMENSIONAL
FERMION SYSTEMS
A dissertation submitted to the
SWISS FEDERAL INSTITUTE
OF TECHNOLOGY ZURICH
for the
degree
of
Doctor of Natural Sciences
Presented
by
Beat J. AMMON
Dipl. Phys.
born
August 27th,
citizen of
accepted
on
E.T.H.
1969
Herzogenbuchsee,
Berne
the recommendation of
Prof. Dr. T.M. Rice, examiner
Prof. Dr. M.
PD Dr. D.
Sigrist,
Wiirtz,
1998
co-examiner
co-examiner
Acknowledgements
I wish to thank T.M. Rice and D. Wiirtz for
giving
work,
I whish to thank the CSCS and the Institut
and for very instructive conversations.
fvir theoretische
gratitute
Troyer
to M.
support, I
to
Physik
am
also very
acknowledge helpfull
D.R.
Fokkema,
P. de
for
for
providing
giving
grateful
me
to M.
excellent
a
the opportunity to carry out this
me
working
conditions.
I
owe
a
special
debt of
first introduction into the field and for his constant
Sigrist
Agterberg,
discussions with D.
Forcrand, A. Friedli,
for many instructive conversations. I would like
B.
F.F.
Frischmuth,
Assad,
A.
G.
Blatter,
H.G.
Evertz,
Furusaki, A. Galli, S. Haas, E.
Heeb, N. Kawashima, B. Loepfe, T.M. Miiller, N. Shibata, H. Tsunetsugu, M. Vollmer, and
Wang. Finally
X.
B.
Lienhart,
putting
N.
up with
This work
I wish to thank my office
Jain, J.
Majmudar,
and R.
Cray
Schnidrig
at the SCSC M.
for many
Hanf, H.Labermeier,
stimulating
conversations and
me.
was
carried out under the support of
an
ETH-internal grant No.
performed
on
the Intel
2511.5. Most of the calculations have been
the
colleagues
J90 of ETH
Zurich, and
on
the DEC 8400
5/300
9452/41-
Paragon XP/S-22 MP,
on
of the C4-cluster at ETH Zurich.
ii
iii
Abstract
In this PhD work
we
algorithms
investigation
for the
have
improved
correlated fermions. We have
and
developed improved
improved
of
larger systems
limit of continuous
error
spin S
three
study
allows the
DMRG
of random
investigation
Examples
a
algorithm
as
length
to
a
we
is
given by
overcome
show that the
we
which eliminates the
are
randomly depleted
arguments that predict
Curie-like susceptibility
specific
anomalies in the
SrsCuPtj-^Ir^Og,
heat.
which
matrix renormalization group
the combination of the zero-temperature
algorithm.
algorithm
does not suf¬
exact results down to very low
temperatures
This
By the incorporation of
numerical instabilities of this
implementation
possible for the first
at low
of this
time to
new
of the
algorithm
a
re-
and ob¬
method.
investigate accurately
the
temperatures. Ladder systems
they allow the study of the transition from chains
groundstate
own
realistic model of
without any approximation.
stable
it is
because of the spin-liquid
orders
results.
properties of the t-J ladder model
interest
separated by
are
negative sign problem and gives
numerically
by
ferromagnetic-antiferromagnetic
each with its
with the virtual transfer-matrix
biorthogonalization algorithm
With this
in combination with
decomposition.
of systems of this type
finite-temperature density
algorithm
model,
We confirm theoretical
generic model
is the
This
for systems of infinite
tain the first
regimes
experimental
algorithm
algorithm
fer from the
to
loop algorithm
of the Trotter-Suzuki
SrzCviPt\-xlrxO§.
The different
comparison
(DMRG) algorithm.
the
time-steps
chains.
the differences of
The second
namic
slowing
Our results
sign problem.
a
down" and the simulation times
clearly distinguishable temperature regimes,
and Curie constant.
We
of the
strongly
investigation of the t-J model.
time is well defined for the the t-J
is used for the
ladders and
numerical
loop algorithm from the
temperatures than before. Further
at much lower
1/2 Heisenberg
Heisenberg
powerful
previous local updating algorithms. This allows the investigation
to
imaginary
algorithm
=
which allows the
estimators for simulations with
due to the finite size of the
This
model,
estimators reduces the "critical
magnitude compared
of much
the quantum Monte Carlo
generalized
global updating procedure
demonstrate that the
two very
of the thermodynamics of low-dimensional systems of
six-vertex model to the fifteen-vertex
We have
efficiently implemented
to two-dimensional
undoped system.
First results
magnetic susceptibility, the entropy, and the specific heat. By taking
are
thermody¬
are
of
special
systems and
presented for
into account the
lowest
lying
entropy
s
excitations of the t-J ladder
in the limit of strong
couplings
we can
on
describe the
the rungs.
magnetic susceptibility
and the
V
Zusammenfassung
Zwei numerische
Algorithmen
Quanten
Modell
Monte Carlo
zur
Untersuchung
Systeme
korrelierter fermionischer
Loop-Algorithmus
verallgemeinert worden,
der
thermodynamischen Eigenschaften stark
sind verbessert und effizient
was
die
ist
vom
den. Durch die Kombination der
Der
sechs Vertex Modell auf das funfzehn Vertex
des t-J Modells erlaubt. Die Methode
Untersuchung
globalen
Loop-Algorithmus
Methode des
slowing
estimators" werden die Simulationszeiten und das "critical
konventionellen lokalen
gezeigt
Algorithmen
viel grosserer
tersuchung
um
Grossenordnungen
bei markant tieferen
Systeme
dass der Grenzfall kontinuierlicher
imaginarer
durch die endlichen Zeitschritte der Trotter-Suzuki
Mit diesem
Algorithmus
ferromagnetischen
der
solcher
Systeme
bestatigen
und
sind die
Regimes
wird die
Spin
5
=
entleerte
Vorhersagen
Suszeptibilitat
reduziert.
Zeit wohldefiniert
ist,
SraCuPti-^Ir^Oe
Der zweite
Leiter und
Algorithmus
eigenen
extrem tiefen
Beispiele
Wir
Temperaturskalen,
mit
Warme. Der
Vergleich
Ex¬
zum
eines realistischen Modells
Quantentransfermatrixmethode
Dieser
Untersuchung
Algorithmus
unendlich
hat kein
mit
negatives
langer Systeme
bis
zu
Temperaturen ohne jegliche Naherungen. Mit einer Rebiorthogonalisierungs-
methode konnen die numerischen Instabilitaten des
bisherigen Algorithmus
den und wir erhalten die erste numerisch stabile Version dieser
Mit diesem
Algorithmus
ist
es
zum
ersten mal
schaften der t-J Leiter bei tiefen Temperaturen
besonderem
untersucht.
Modell.
ist eine Kombination der
und erlaubt die exakte
die Fehler
Curie-Konstanten. Die verschiedenen
spezifischen
Dichtematrix-Renormalisierungsgruppenmethode.
Vorzeichenproblem
was
SrsCuPti-^Ir^Oe-
dreier klar unterscheidbarer
generischen
Ferner wird
eliminiert.
periment wird ermoglicht durch die Untersuchung der Unterschiede
von
zu
1/2 Heisenberg-Kette mit zufalliger Verteilung
sind getrennt durch Anomalien in der
mit einem
"improved
Vergleich
zuvor.
Zerlegung
wor¬
Dies erlaubt die Un¬
Temperaturen als
Heisenberg
und
mit den
down" im
antiferromagnetischen Wechselwirkungen
zufallig
die theoretischen
einem Curie-Gesetz der
der
worden.
"improved estimators" ist auf Modelle mit einem Vorzeichenproblem verallgemeinert
der
je
implementiert
Interesse,
da hier der
Ubergang
von
die
zu
Systeme
wer¬
Methode.
thermodynamischen Eigen¬
untersuchen.
Ketten
tersucht werden kann und da sich die undotierten
stand befinden. Erste Resultate werden fur die
moglich
zu
neuen
iiberwunden
Leitersysteme
zweidimensionalen
in einem
Systemen
"spin-liquid"
magnetische Suszeptibilitat,
die
sind
von
un¬
Grundzu-
Entropie
und
VI
die
spezinsche
Warme
gezeigt.
Durch die
Betrachtung
gen der t-J Leiter kann die
magnetische Suszeptibilitat
Wechselwirkungen auf
Sprossen beschrieben
den
der
niedrigst-energetischen Anregun-
und die
werden.
Entropie
im Grenzfall starker
vii
Contents
1
Introduction
1
2
Quantum Monte Carlo Loop Algorithm
7
2.1
Introduction
2.2
Background
materials
Worldline representation
2.2.2
Local worldline
algorithms
Loop algorithm
for the
Loop algorithm
2.4
Improved estimators
Simulations without
2.4.2
Simulations with
2.4.3
Improved
Loop algorithm
2.6
Results
12
Heisenberg
model
14
18
23
2.4.1
2.5
9
for the t-J model
2.3
2.7
9
2.2.1
2.2.3
3
7
a
sign problem
24
sign problem
25
estimators for correlation functions
in continuous
imaginary
28
time
31
33
2.6.1
Autocorrelation times
33
2.6.2
Improved
35
2.6.3
Two-leg
2.6.4
Three-leg
estimators
t-J ladder
39
t-J ladder
41
Discussion
Random
ferromagnetic-antiferromagnetic Heisenberg
44
chains
47
3.1
Introduction
47
3.2
Numerical methods
50
3.3
Results
51
viii
3.4
4
5
Susceptibility
52
3.3.2
Specific
58
heat
61
Discussion
Finite
Temperature Density Matrix Renormalization Group Method
4.1
Introduction
4.2
Background
63
63
65
materials
4.2.1
Zero temperature DMRG
65
4.2.2
Transfer matrix method
71
77
Temperature DMRG
4.3
Finite
4.4
Breakdown of the
4.5
The one-dimensional t-J model
4.6
Discussion
82
algorithm
as a
test-case
86
90
Thermodynamic properties
of the t-J ladder
93
5.1
Introduction
93
5.2
Numerical methods
95
5.3
Results
96
5.4
6
3.3.1
properties
98
5.3.1
Low temperature
5.3.2
Susceptibility
100
5.3.3
Entropy and specific heat
102
Discussion
Conclusions
104
107
1
Introduction
1.
The
of strongly correlated fermion systems has become the most
investigation
field in solid state
[1]
than
more
physics
since the
decade ago by
a
controversial and
we
discovery of high-Tc superconductivity
Despite these intensive
now.
still lack of
consistent
a
efforts,
Especially
picture.
electron-electron interactions in the Cu02-planes turned out to be
Despite
the
large variety
family
can
be attributed to the dominating role of the electrons in the
ducting
the
high-Tc superconductors,
corner
building block
atoms
(Ba, O, La, Hg,...)
planes
is
weak,
correlations
are
by
are
now
responsible
among the
essentially
for the
electrons,
serve as
challenge.
Cu02-planes, where
the
separated by
are
strongly
high-Tc superconductivity.
the
con¬
highly anisotropic,
which
various
reservoirs. The
undoped parent materials
high-Tc superconductivity
is
CuC>4-squares,
the
are
CuCvplanes
rather universal. This
superconductors
charge
the best examples of planar
as
investigation of the
very difficult
are
responsible for the unusual strange metal behavior
considered
models of
that
are
planes
These
but the electrons in the
believed to be
repulsion
of all these materials
sharing CuC>2-planes.
properties
their electronic
electrons reside. The structure of the cuprate
common
in all
topics remain
among the structures of the unit cells of the different members of
the
of
studied
in the cuprates
many
the
a
actively
are
arranged
layers of other
coupling
correlated.
among the
These strong
in the normal state and
Due to the strong Coulomb
are
Mott-insulators that
are
spin-1/2 antiferromagnets. Microscopic
must therefore
capture the essential physics of these
extraordinary Cu02-planes.
It has early been
serve
[3]
too
as
the
recognized,
that the
groundstate
of the
doped Mott-insulator
should
starting point for theoretical investigations [2]. The multi-band Emery model
could be used
complicated
as
to
it has been shown
the generic
be
description
of the
CuC>2-planes,
investigated exactly by analytical
by Zhang
and Rice that this system
effective model, the t-J model
[4].
There
are
by
now
or
but its Hamiltonian is far
numerical methods.
be reduced to
However,
a
single
band
many indications that
single
band
can
2
models such
the t-J
as
or
Hubbard-model
are
able to capture the essential features of the
doped antiferromagnets.
models, though simple in
These
appropriate analytic
applied
be
tools to handle them
theories fail because of the
coupling
weak
there is
as
because of the low
limit to
no
dimensionality
important questions still remain
high-Tc superconductivity
Which
can
are
they
play
of
new
mean
liquid
theories cannot
field theories
Are strong correlations
to elucidate these
are
enough
unreliable
describe
to
in which
algorithm [5],
computing
and with the
sality class of Luttinger- [6]
difficult
as
and
renormalization group
investigation
of
are
confirm
can
or
cuprates?
quantities
help
are a
of conformal field
dimensional
one
not known.
on
(DMRG) algorithm [8],
In
The
a
own
and
are
groundstate
theory, these findings and
fast with
powerful
the
can
to the univer¬
the situation
system size and the
with the zero-temperature
there exists
mean-
diagonalization by
higher dimensions,
exponentially
However,
its
(ID) systems belonging
Luther-Emery-liquids [7].
inves¬
can
Numerical methods
difficult problem
be calculated with exact
they
as
reject existing
super-computing applications.
can
the number of states grows
relations
scaling
or
questions,
directions for theoretical models.
correlated fermions
be scaled to infinite system size for
exact
Standard Landau-Fermi
doped antiferromagnets and
important role
an
strongly
the forefront of scientific
more
and
investigate and
strong correlation effects of the problem. Several
open therefore.
and excitation spectrum of small systems
is
to
strong correlations, perturbation
properties of microscopic models exactly, they
investigation
Lanczos
missing.
still
perturb around,
and the
extremely difficult
and the strange metal behavior of the normal state of the
field based theories and indicate
on
are
be observed? Is the t-J model sufficient and if not, what else is needed?
the
for the
are
the relevant energy scales of the
Numerical methods
tigate
appearance,
density
matrix
tool that allows the
markedly larger systems.
Nevertheless, the investigation of thermodynamic quantities is difficult with these methods
and
that work at finite temperatures
algorithms
known
example
of this type
the Trotter-Suzuki
with Lanczos
more
than
or
one
sign problem,
are
algorithms
dimension
or
(QMC)
of the partition function.
can
be
investigated
for frustrated systems,
that limits their
better suited for this purpose.
the quantum Monte Carlo
decomposition [9]
DMRG
are
applicability
to
simulations based
Much
larger systems
with this method.
QMC
on
than
Unfortunately,
methods suffer from the
high temperatures
A well
in
negative
and small system sizes.
3
chapter
second
these
of this method
problem
Another
discuss the quantum Monte Carlo
we
Initially
model
[10]
application
and its
Starting
applied successfully
has been
algorithm
for the Hubbard model is
model,
from the six-vertex
due to
a
huge
reduced
improved
orders of
by
the
use
of
more
improved
decomposition
of the
steps by the Trotter-Suzuki decomposition.
get
are
rid of this
Heisenberg
needed in order to
systematic
model
error.
by working
in continuous
Ref.
function into
infinitely
weight
global updates
a
can
the concept
further reduction
source
of
prolonged
numbers of
varying
many time steps and to
recently
for the
case
of the
We show that this
factors and transition
chapter
updates
finite number of time
directly [18].
time
The results of this
case.
a
to
have been
probabilities
published
in
[19].
With this
algorithm
atures than before.
some
change
we are
extent in
the
able to
much
investigate
larger systems
Quantum mechanical systems with disorder
the simulation of
require
to
for this
presented explicitly
Another
has been eliminated
imaginary
generalized
generalize
estimators. We
partition
limit is well denned for the t-J model too, and the
are
advantages
to the limit of
problem
[11-16],
We find that the
of
Different simulations with
extrapolate
This
model
to conventional local
sign problem, giving
a
be
can
t-J model.
and of the simulation times therefore.
errors
simulation times is the
time steps
complicated
estimators to quantum systems with
of the statistical
Heisenberg
algorithm
magnitude compared
called
so
global updating procedure.
possible [17].
reduction of the autocorrelation times. The
leveraged by
further be
of
are
which reduces both of
and Marcu for the six-vertex
for the
show how the
we
the fifteen-vertex model and thus to the
simulation times
Lana,
has been developed by Evertz,
loop algorithm
and the
a
In the
simulation times.
loop algorithm,
updates by
the conventional local
problems by replacing
the
extensively long
sometimes
are
vast number of
a
sufficiently large
physical properties drastically,
systems. Disorder
drive the system to
or even
recently by Westerberg, Furusaki, Sigrist,
antiferromagnetic
among the
temper¬
problems
always
that
occurs
reality, and especially in low dimensional systems already little disorder
class than the pure system. A particular example of such
discovered
are
at much lower
random
low temperatures, but
drives the system to
spin
already
a new
S
=
1/2 Heisenberg
the admixture of
regime,
the
a
a new
and Lee
[20].
a
class has been
The one-dimensional
a
dimerization
small percentage of
large spin scaling regime
different universality
universality
chain scales to
can
regime
ferromagnetic
with universal
(ID)
at
bonds
scaling
con-
4
Examples
stants.
Heisenberg
ladder
of systems
or
romagnetic spin S
the
1/2
=
of this type have been
in the
belonging
new
alloy SrsCuPti-^Ir^Oe [21],
bonds
arguments predict
and random
investigated analytically
couplings
formed
by
effective and material
an
confirming
algorithm
clearly
It is
be
an
iments.
the
this
and
seen
theory.
we
chapter
whether these
spins
regime
specific
of effective
ferromagnetic spins
are
heat
spins
still exists and
x and
properties
not
as
formed
give
we
by
theoretical
magnitude
of the spins
regime
we
Despite
ties of
the
strongly
problem is still
tems in
on
more
significant improvements
as
segments
one
the Trotter-Suzuki
method
[26].
dimension to
investigation
application
of the
purely antiferromagnetic
of the
can
be obtained for systems of infinite
not suffer from the
of the Hilbert space
ate and
negative sign problem.
by lowering
high temperatures.
In
the
The results
thermodynamic
the
loop algorithm
proper¬
negative sign
for fermion sys¬
Another method which is also based
the partition function is calculated
in this case, and in combination with the non-hermitian Lanczos
sults
regime.
partition function is the virtual transfer-matrix
QMC algorithms,
In contrast to the
generic model. Nev¬
loop algorithm,
of the
high temperatures.
decomposition
Curie-regime
[25].
correlated fermion systems with the QMC
than
actual exper¬
show that in contrast to
for the
of
can
heat Cy-
upper and lower bounds for the value of the
in the
not solved and limits the
by
in the real system and the
the
QMC loop
specific
also be observed
x
exact results
are no
is well denned and
effective Curie constant of this system in the intermediate temperature
of this chapter have been reported in Ref.
antiferromagnetic
magnetic susceptibility
this system with the
pronounced
by
or
anomalies in the
can
and
scale,
but there
SrsCuPti-^Ir^Os,
overlapping
are
high temperatures
at
purely ferromagnetic
[22],
and fer¬
Generic systems
chain.
with random
investigate
we
realistic model of
the energy scales
and the anomalies of the
or
a
single
numerically
Curie constant
magnetic susceptibility
By investigating
ertheless,
on a
randomly depleted
antiferromagnetic
Curie-like behavior of the
dependent
In the third
interesting question
the
a
where
the
are
intermediate temperature
an
the segments of
to
class
show that this intermediate temperature
in the
generic model,
and
of effective
interesting regime
spins. These effective spins give rise
with
arranged randomly
are
low-temperature limit [20,22-24]. On
an
universality
to this
length [27].
Most
algorithm,
importantly,
very
exactly
precise
re¬
this method does
Due to the exponential increase of the dimension
temperature, this method is still restricted
chapter four
we
show how this
algorithm
can
to intermedi¬
be extended to
5
low temperatures in combination with the DMRG method. This finite temperature DMRG
algorithm
has been
lattice model
applied
recently [28-30].
instabilities hinder the
of the left- and
instabilities,
for the
doped
a
new
method. We
the first
density
away from half filling, numerical
identify the loss
matrix
as
the
numerically stable implementation of this algo¬
applicability
is demonstrated
able for the first time to calculate the
thermodynamic
re-biorthogonalization
algorithm
new
we
are
of the t-J ladder system
especially interesting
single chains,
for
such
method. Its
(two coupled chains) exactly
theoreticians,
finite
a
as
superconducting pairing
ble for
spin
and without any
can
section,
excitations not
creation of
tigate
by
the
the
finite-temperature
of excitations
of this
are
chapter
methods,
programming
just
a
carrying
DMRG
case
algorithm.
of strong
be found in Ref.
can
As
high-Tc superconductivity
doping,
are now
but
possible by
spin. In the last chapter
we
in
new
the
inves¬
specific heat of two-leg t-J -ladders
We discuss the different types of excitations of
be
on
explained by
the rungs,
an
as
intuitive
the different types
picture. The results
[32].
this PhD work
are
much
complicated than
more
the size of their code often exceeds lO'OOO lines. Careful and
is therefore
responsi¬
(RVB) picture.
for
gap remains upon
and
exchange couplings
algorithms discussed in
single problem.
important if
We have made
programs have been written in C++.
parallelization
charge
both
the entropy, and the
best visible then and
can
The numerical
older
holes
magnetic susceptibility,
the t-J -ladders for the
spin
gap is
its
present in the undoped Heisenberg ladder
separated single
present
undoped Heisenberg-ladder,
valence bond
starting point
Anderson. The
spin
The finite
correlations in the
exactly
doped antiferromagnets proposed by
magnetic
doped system.
the
not
are
undoped parent system [31] and d-wave
by the short-range resonating
this is
approxi¬
Ladder systems
interesting properties
show many
gap in the
correlations in the
be described
stated earlier in this
they
as
exponentially decaying spin-spin
groundstate
of the numerical
source
mation down to very low temperatures for the entire parameter space.
in
of biorthogonality
t-J chain.
With this
properties
complicated systems
of the reduced
developed
have
we
more
application of this
by the inclusion of
rithm
For
right-eigenvectors
and
anisotropic Heisenberg model and for the half filled Kondo
for the
one
a
the
object-oriented
wants to utilize these programs for more than
major
effort to
Part of this
of Monte Carlo simulations has been
ensure
code-reusability
code, namely
developed by
a
and all the
portable library for
M.
Troyer,
E.
Heeb,
the
and
6
the present author
[33]
and
can
be obtained
freely [34].
7
Loop Algorithm
Quantum Monte Carlo
2.
Introduction
2.1
Quantum
Monte Carlo
(QMC)
interacting systems. They
addition, they
In
model.
exact within
given
can
statistical
be used for
This
update Monte Carlo (MC) simulations
point. This problem
which construct
generalization
investigate phase
The
[10,35]
model
QMC
can
of
an
in many
give unbiased results that
large
slowing
down also for
a
spin systems
not useful if the
cases.
errors
Classical local
diverge at
spin systems by
performing
review
QMC
are
interesting
of these cluster methods to quantum
For
are
phase transitions suffer from "critical
clusters instead of
transitions in quantum
see
cluster
local
the critical
algorithms,
spin flips.
spin systems, the loop
Ref.
[40].
This method
simulations. It has made
[12-16,18,23,41-44],
far
can
possible
to
beyond the
of previous MC techniques.
loop algorithm
can
and
is that the results
second-order
developed [10,11,17,35-39].
problem of critical
possibilities
are
therefore be applied to almost any
ideal tool for numerical simulations
has been solved for many classical
global updates
has been
algorithm,
solve the
a
thus
happens
near
can
the autocorrelation time and with it the statistical
slowing down:"
Recently
and
large systems
They
errors.
powerful tool for the investigation of strongly
generalize
easy to
large.
become too
errors
are a
major problem, however,
A
of complex systems.
statistical
are
methods
be
can
be
applied directly
be simulated
generalized
to hard
by coupling
to
particle models.
The
original loop
core
bosons and to
two
spinless fermion systems [17].
method
spinless fermions [36]. A Hubbard
One problem in
simulations of the Hubbard model is that its dominant energy scale is the Coulomb
repulsion
scale J
=
U > t, while the
4t2/U
-C U.
To
interesting low-lying
investigate
simulate the effective low energy
the
excitations
are
low-energy properties
Hamiltonian,
the t-J model.
simulations for the t-J model have been carried out both in
a
at
a
much smaller energy
it is thus of
Previous
advantage
to
finite-temperature
determinantal formulation
[45]
8
in two
dimensions, which suffered from
worldline formulation in
one
sign problems and metastability,
serious
dimension,
updates [46].
with standard MC
such standard MC simulations suffer from strong
explicitly later,
They
limit the accessible system sizes and temperatures.
seriously
sion),
which
chapter
overcomes
present
we
a
loop algorithm
these autocorrelation
problems
will show
we
autocorrelations, which
also
are
like the determinantal simulations have to be extrapolated to continuous
In the present
As
and in the
nonergodic,
imaginary
(for
for the t-J model
and has additional
time.
any dimen¬
advantages
such
as
error
of
possibility
of
complete ergodicity, the existence of improved estimators, which further reduce the
measured
quantities by implicitly averaging
directly taking
in Refs.
[47]
the continuous time limit.
[48].
and
The
the two-dimensional
(2D)
However, this
uses a
do not
paper
explain
Quantum
over
loop updating
t-J model in the low hole
severely
the
density
the technical details of their
cause
can
and the
already
a
classical
exponentially
restricts simulations of
improved estimators
one.
presented
and small J /t limit in Ref.
[49].
unfortunately they
of frustrated spin systems
perform QMC
This
mapping
error
for
a
can
given
fermionic models.
simulations
introduce
amount of
with system size and inverse
higher-dimensional
we can
or
In order to
cancellation effects. The statistical
then increase
been
algorithm.
Monte Carlo simulations of fermionic models
which
dates with
results have
different representation of the t-J model and
have to map the quantum system to
tional effort
our
configurations,
method has also been used for the simulation of
always suffer from the "negative sign problem."
weights,
many
Some of
and
nearly
we
first
negative
computa¬
temperature. This
By combining loop
up¬
reduce the variance of the observables and thus lessen
sign problem.
This chapter is
organized
and review the worldline
chain.
In Sec.
2.3
we
as
Finally
in Sec. 2.6
first results obtained for
First
QMC algorithm,
describe the
estimators is discussed in Sec.
method.
follows.
a
we
t-J
2.4,
discuss the Trotter-Suzuki
we
and the standard
loop algorithm
in Sec.
discuss the
chain,
a
2.5
we
The
use
a
Heisenberg
of
improved
derive the continuous time version of
performance
frustrated
loop algorithm
for the t-J model.
decomposition
for
of the
new
algorithm
Heisenberg chain,
and show
our
some
and t-J ladder models.
9
Background materials
2.2
To establish notation and formal
[50]
the worldline representation
take the ID
background
loop algorithm.
and the standard
we
spin-1/2 operator
boundary condition Sl+i
Worldline
We
the Trotter-Suzuki
use
diagonalize.
Z
tre-"*
=
••
by
site i, J > 0 for the
antiferromagnet,
and the
periodic
decomposed
and
a
path-integral
into two terms H
=
formulation in
Heven
#odd>
+
eacn
imaginary
of which is
Then
Jimjr
f3
=
x
((c-M*«n+«odd))M)
<i3|e-ATfl'CTen|i2>(J2|e-ATHodd|i1)
\jT
is the inverse
We may consider Eq.
one
application
function Z in Eq.
sical system.
can
we
L
decomposition [9]
tr
=
((e-Artfevene-Arffodd)^
|ij.)
be
is also
systematic
extrapolated
also be formulated
(2.2.2)
as
is taken
=
over
(Ar2)
^2)
+ Q
/3/M,
(2.2.2)
,
and M is called the Trotter number.
complete orthonormal
the evolution of the initial state
of the time evolution operator within
(2.2.2)
The
+0
temperature, At
The summation with respect to
with
example,
E (ii\e-ATHe\i2M)(i2M\e-ATH°^\i2M-i)x---
=
where
=
an
of
representation
time. The Hamiltonian H is
easy to
descriptions
Si is adopted.
=
2.2.1
on
detailed
The Hamiltonian is defined
L
a
the worldline representation
more
loop algorithm [10,11,38,40]. As
and the
Heisenberg antiferromagnet.
where Si denotes
briefly describe
We refer to the literature for
to
At2
directly
formally
error
->
0
the
of order
by fitting
to
a
1^)
of
a
imaginary
polynomial in At2.
-»
0
The
The
[18].
time
partition
(D+l)-dimensional
to the finite time
in the continuous time limit At
in
time step At.
a
partition function
At2 due
sets of states.
clas¬
step approximation
loop algorithm
can
This will be discussed
in section 2.5.
The
our
decomposition
ID system with
leading
to
a
of the Hamiltonian has to be chosen
only
nearest
neighbor
checkerboard structure
as
interaction
shown in
Fig.
we
according
take
2.2.1. As
a
to the
i?even/odd
second
=
problem. For
Yli
example
even/odd
we
H^i
show the
10
—
up spins
-
down spins
-
application of
_-AtH«
space direction
FIG
Example of
2 2 1
horizontal
The solid lines represent up spins
axis
application of
decomposition for
exchange
of the
hmM-+ootr((e
In
even/odd
Fig
2 2 2
In these
we
of
a
with
show
plaquettes
Eq (2
le,
2)
Heven
Si
Sl+i
conserves
and the real space direction
[26]
h^M)
a
=
J(S,
J
Sla
+ J'
to
f(jl£
ladder system
ff«
ladder system
Sl+i
The shaded
,
is
a
evenness
,
e
along the
plaquettes
magnetization
St
=
more
+
-r70dd
+
and
The natural choice for
-f^rungi
Hrung
=
where
J'
£,'
HLven/oM
j
S,
1
=
St
2
decomposition for ladder systems
2
2)
the Hamiltomaii acts
only
factor wp to the matux elements
Y.W(C)
on
m
(223)
{C)
V
expand the
generalization
=
Fig 2
a
the
is
=
of this
2 2 1 and
E HWP
we use a
coupled
(J')
wheic J
2,
than two terms, H
§l+l 2),
Sl+l l+Sl2
two
of the approximation still holds Z
Heven
=
p, earh of which contributes
we
1
At"*) "^ +0 (At2)
graphical representation
model
Si
consisting of
a sum
thus H
12M
Y^=\
consisting ol
For the ladder systems
that the
tr
=
Heisenberg
-ff0dd
+
H
Heisenberg
model with the checkerboard
Heisenberg
axis
the dashed lines down spins
(lungs)
the chains
Z=
For the
of the
the vertical
a
graphical representations (Fig
the shaded
2
=
£o-i Sj=i
be shown
At£*->
decomposition
#W
=
H
can
along
r
along
decomposition
Ylk-i Hk,
r runs
the Hdmiltoman of
interaction
and it
time
e
Heisenberg chains, HH
Ei
configuration
world line
The imaginary
show the
the
a
decomposition
states
Therefore there
are
\ik)
in
only
an
six
Sz
eigenbase
nonvamshmg
Each
/fW
=
matrix elements
11
V\,u<i9
FIG. 2.2.2:
board
Graphical representation
decomposition
of
The imaginary time
axis
The shaded
for each shaded
and down
over
runs
site in the
The
along
can
C
=
{\ik}}
the
as
as
plaquette
states.
be
ladder system similar to the checker¬
rYeven
=
Fig.
2.2 3
Thus the
of contmuoub worldlmes.
corners.
Accordingly,
+-£f0dd + -^rung
along
the horizontal
represented by solid and dashed lines connecting
a
set of
One
The
wp in
Thermal averages of observables O
configuration
Eq. (2 2.3)
can
(O)
=
Eq. (2 2.3)
is shown in
example
±1/2.
Fig.
2.2.1.
each defined
namely,
p,
a
up
is taken
magnetization conservation.
given shaded plaquette
a
in
sum
binary variables 5f
restriction due to the
the local state of
variables at its four
a
consists of three terms H
application of e~ArH
can
checkerboard, with the
Cp
decomposition for
shown in
be identified with
convenient to define
of the
decomposition
the vertical axis, and the real space direction
plaquette, which
spins, respectively,
Note that C
the
chain
plaquettes show
configurations
binary
r
a
set of
on
a
It
is
four
C is then identified with the union of
can
be written
be written in
a
as
w(Cp).
similar way
as
^X>(C)0(C),
(2.2.4)
{c}
where
0(C)
If the
value
is the value of the observable in the
weight
|W(C)|
of
a
configuration W(C)
to construct
the
can
probabilities
configuration
take
C.
negative values,
one
for the Markov chain of
has to
a
MC
use
its absolute
procedure (see
12
n
cr
g-ArJ/4
w(Cp)
FIG. 2.2.3: The six allowed
(Eq. 2.2.1).
below),
plaquette
states of the
the last two
since these
Heisenberg
shows the
row
The solid lines connect two sites
spins. (We have assumed
weight of
eArJ/'1ch(ArJ/2)
The second
conservation condition.
occupied by
respect
probabilities
need to be
may
with
now
and
)\w\
denotes
increasing system
exponential blow
up of the
be taken
configurations CM
p(C«)
=
This
sign,
(sign)^,
/?. For fixed
errors.
algorithms
(Markov chain)
configurations
expectation values with
size and inverse temperature
constructs
many
(2.2.5)
stems from the fact that the average
an
given by
W.
can
finitely
then
(signal
Eqs. (2.2.4, 2.2.5),
sequence
are
(sign •£>),„,i
The thermal averages,
a
connect down
—Sx'y to make the
e>(C)
W(C),
weight
effort this then leads to
Local worldline
2.2.2
of
sign
"sign problem"
decay exponentially
computational
ones
£{c)|W(C)|sign(C)0(C)
£{c} |W(C)| sign(C)
_
Z
absolute value of the
a
—>
positive. Expectation values
Z{c]W(C)0(C)
stands for the
In many cases,
magnetization-
plaquettes for the Hamiltonian
spins, and the dashed
up
E|c)|w(c)|sign(C)
T.{c}\w(C)\
to the
model that fulfill the
plaquettes positive.)
E(c> |iv(c)|sign(c)
Efoiwtcji
sign(C)
e&Tj^sh{AT.J/2)
of the
weights
M
bipartite lattice and rotated the spm-operators Sr"y
a
(O)
where
M
n
n
of
by MC importance sampling. One
such that
in the limit of in¬
their distribution agrees with the correct Boltzmann distribution
W(CW)/Z.
can
be achieved
by satisfying
two
conditions:
ergodicity
of the Markov
chain, and
detailed balance
p(C
->
C)
W{C)
_
~
p{C
-»
C)
W(C)
(2.2.6)
13
where
p(C
—>
C)
in the Markov
is the
chain,
a
sign problem
=
configuration C
the
configuration
the value of the observable in the
with
cases
choosing
expectation value Eq.
(O)
In
of
when the current
Then the thermal
averaging
probability
lim O
(2.2.4)
the next
configuration
is C.
of
an
observable O
can
by
be estimated
configurations CW:
O
,
the averages in
as
=
N
i
J2 °(CW)N
Eq. (2.2.5)
are
done
(2-2.7)
separately
for the numerator
and for the denominator:
g__*s£i(Big"-o)[£i
(2.2.8)
>\w\
algorithms
In standard local
by proposing
a new
an
configuration C
lines. The candidate C is
update from
configuration
one
that differs from C
accepted with
a
by
C to the next
small local
a
change
is done
one
of the world-
probability that satisfies detailed balance, e.g.
the
Metropolis probability [51]
p(C->C)=
or
the heatbath
mm
(l,^|jl),
probability
M^-wwww
otherwise the
There
strongly
are
configuration
two
quantity O, typically
on
average
a
number
r
of updates to arrive at
This autocorrelation time t, which
increases
quadratically
with
depends
correlation
spatial
on
factor t, which
Secondly,
a
desired statistical accuracy, the MC simulation has to be
can
easily
reach orders of
in contrast to classical MC
line
algorithms
nor
their
in
general,
winding numbers
the total
since local
106 and larger
simulations,
magnetization
and the
practical
local
updates
cannot
change
example,
when
updates only
in the simulation. For
in
a
are
statistically
the measured
length £
A-1 (respectively system size L and inverse temperature /? when £
To achieve
model,
C is kept.
It takes
independent configuration.
/3).
(2-2-10)
;
major problems with local updates: First, consecutive configurations
correlated.
energy gap
(2.2.9)
and inverse
> L or
A-1
lengthened by
>
a
cases.
are
not
ergodic
for world-
the number of worldlines
applied
spatial winding number remain
to the
Heisenberg
constant.
Many
14
quantities
In
addition, it
conventional local
therefore,
we
pointed
was
that
small,
is often very
a
are
complicated quantity
have to include
some
Also,
which is another
Both kinds of difficulties
configuration changes
global updates,
ad hoc
of
which tends to make the
global updates
long
autocorrelation times.
in
mulated
stochastic
one
loop algorithm, which achieves large nonlocal
in the
update. Autocorrelation times for the loop algorithm
smaller than those for the conventional
magnitude
ergodicity problems,
it does not suffer from the above-mentioned
addition,
directly
In the
loop algorithm, each update
configuration
G. G
{Gp}
=
which combine to form
with
mapped
Let
us
(1)
a
explain
simple
Since the worldlines
of
construct.
an
the
a
the
example
the
for
all the
a
case
are
of
an
spins
to
—>
defined
the
In the second step, the
a new
worldline
Heisenberg
exclusive-or), i.e.,
on
on a
can
be for¬
In the first
a
model.
step for the
loops.
Heisenberg
Eq. (2.2.3)
separately
for each
is
a
to
a
graph
plaquettes
configuration
Two observations
the location
set of closed
on
(?)
p,
of loops is
configuration C.
two
a
product
are
of
(2)
impor¬
worldline
in any allowed
the
loop flip.
model.
are
arbitrary
of spin flips
These
closed loop will be called
function Z in
therefore fulfill detailed balance
probability p(C
continuous, the difference between
loop-update
partition
a
graph segments Gp
loops.
of the
is located
configuration
Flipping
locally,
sense
C)
->
and
consists of two steps, both stochastic.
consists of local
set of closed
probability p(G
configurations (in
update
a
In
algorithm.
[18].
in continuous time
step, the current worldline configuration C is mapped with
acts
algorithm ergodic,
the acceptance rate of such ad hoc
cause
are overcome
found to be orders of
show
exists that does not vary in
Loop algorithm for the Heisenberg model
2.2.3
tant:
then very difficult to estimate.
the XYZ model. To make the conventional
code rather cumbersome.
resulting
are
[38]
out
updates for
usually
superfluid density,
like the
physical interest,
of
In
loops
we
Fig.
2.2.4
will
we
Since the Hamiltonian
plaquette
plaquette, provided
the
We
terms.
global
can
constraint
of closed loops is satisfied.
By inspecting the
each
six allowed local states
plaquette, spin flips
must
occur on
Cp
pairs of sites,
at another allowed local state. We connect the
together by
solid lines: these
are
on a
pairs
loop segments.
plaquette (Fig. 2.2.3),
not
on
of sites
single spins,
on
Since there
several
see
that for
in order to arrive
which spins
are
we
are
to be
flipped
possible pairings
of
15
FIG
finite
probability
up spins
for
these
loop update step for the Heisenberg model (anisotropic
Some of these
loops change
unchanged loops by solid
loops
lines
The
are
which
selected
We denote the
direction
figure
the
on
(a)
On the left
(solid lines)
and down spins
figure (b)
along
a
diagonal graph segments)
(dashed lines)
the middle
spins
Example of
2 2 4
(c)
(b)
(a)
show
we
mapped
is
to
an
(with probability 1/2)
right (c)
flipped
to be
configuration
shows the spin
C of
loops
in
the
i e
lines
flipped by dashed
will be
loops that
G of
configuration
a
which has
case
configuration
initial
after the
the
loop
flips
sites, the lines can,
be
can
the two
The
in
principle,
in
flipped simultaneously without violating the
loop segments
union
G
=
UpGp
breaking field)
an
are
p
a
given
We define
Cp
"freezing
It
graph,
in
we
function
belongs
will thus
to
speak
A(GP Cp)
to
interacting plaquette individually
Gp
as
the
one
It
so
is
the
If
we
c
shown
m
Fig
on
an
probabilities p(Cp
additionally
have
—»
For
Gp
is
e
allowed
belongs
to two
plaqucttes, except
on
a
those for which
onfiguration CL (l
Specifying Gp
overall
giaph Gp
invariant
Eat h spin
each of these
connected
to
2 2 5
p
(without symmetry
since
plaquette
plaquette
on
possible, namely,
are
stands for
We call the
that it takes the value 1 when
loop segment
are
giaph
G
plaquette weight wp
graphs Gp
all four sites
symbol Gp
The
"frcc/mg' graph,
automatically lead
specify
of
another allowed
to
which all foul sites
only need
the
certain
and the value 0 otherwise
plaquette individually
Therefore,
a
diagonally Also,
restriction
4 spins will leave the
plaquette configuration Cp only
^ 0)
We will
or
complete giaph configuration
a
grouped together
flip of all
interacting plaquettes
the
plaquette
update along connected points leads
w(C'p)
for
the
on
constitutes
which all four spins
given
horizontally, vertically,
run
for
each interacting
configuration of closed loops
Gp)
and
p(Gp
—>
Cp)
"freezing" graphs Gp
on
for each
some
of
16
plaquettes,
the
They
"frozen" into
are
loops passing through
then the
one
since then the whole lattice
Thus
tems without
Boltzmann
magnetic
field.
is the
as
To include
is used for
defining
into account in terms of the
be invariant
[40]
probability p(Cp
flipped together.
too often
occurs
is done
by
the
update.
flipped
are
with
case
general
XYZ quantum
symmetry breaking field,
a
probability 1/2
with
we
when
spin sys¬
factorize the local
in the form
weight
Wo(Cp)
problematic if it
is
change of weight
no
loop algorithms such that loops
\w(CP)\
where
freezing
have to be
freezing.
symmetry breaking field,
no
This
"freeze" and
might just
want to avoid unnecessary
we
We may construct
there is
loop-cluster.
plaquettes
these
under
Gp)
->
MCP) w^Cp),
=
probability
the
of
flip of all four spins
is constructed
as
choosing Gp,
loop.
of the
flipping probability
at the
plaquette
follows. First
^2v(G„) A(Gp,Cp)
(2.2.11)
we
whereas
weight wo(Cp)
The
p.
choose
Was(Cp)
Using
this
needs to
factorization,
weights v(Gp)
for all
w0(Cp).
=
is taken
the
graphs
(2.2.12)
GP
One solution to this set of equations is shown in
unique; depending
C
->
H, it
on
may also not
as can
be checked
easily.
In
general
not
detailed balance for the overall update
The construction of the
graph Gp
is chosen
loops l%.
Then
on
we
all
partitioning
loop
means
with
is then
that
we
a
can
plaquettes
We
to
flip
can
C)
->
we
have chosen
be
performed
p, and
we
obtain
think of this variant
that
a
in
a
loop
a
unique partitioning
w
=
as
w
=
1 for all
(2.2.10).
multicluster scheme. In this case,
to
picking a single
plaquettes.
and
In
can
also
use a
cluster l% of the above
loop \lt\
Ilpgi, Was(Cp)
a
of the lattice into
Eq. (2.2.13). We
to the size of the
weights
if
detailed balance with respect to
heatbath probability like Eq.
loops lt according
all
with respect to
always flip
satisfy
needs to
probability p(\k\) according
flipped
will
loops
attempt
single-cluster variant [52].
Here
(2.2,3)
nP«>as(Cp)+rip «"»(<?)
general, p(G
Hp w3S(Cp)A(Gp, C'v).
weight
This
solution is in
«G^=WQ«G»V
;
wo(cp)
m
exist.) Then,
(The
C is fulfilled by
PiC^Gp^^^
the
2.2.5.
Fig.
an
=
£slte
(rj)el,
*•
Eq. (2.2.9), which
implementation
of
17
Gp
_
v(Gp)
HiriAr-tdT
e-ArJ/4
,;JAr/2
_
eAr''/4sh(ArJ/2)
X
X
(JAt/2
eJAr
(J<l/2)At
Jedr
-eJAr/2
v(Gp)
(l-£)(7/2)dr
1
A(GP,CP)
cP
w0(Cv)
i"as(Cp)
lil 1
e-ArJ/4
1
1
0
1
0
1. ii ]
e4rJ/4ch(ArJ/2)
1
1
1
0
1
eAT'//4sh(ArJ/2)
1
0
1
1
0
FIG. 2.2.5:
Plaquette configurations Cp and graphs Gv for
The upper part of the
There is
weights.
a
figure specifies
free parameter
diagonal graph segments
section
2.5).
function
will
occur.
The lower part of the
A(Gp,Cp),
which
e
the
graphs Gv
and
in this solution.
The third
figure
row
shows the
specifies whether
a
If
the
one
e
antiferromagnetic Hcisenberg model.
solution
v(Gp)
of
Eq. (2.2.12) for their
is chosen to be zero,
no
freezing
shows the continuous time limit of
and
no
v(Gp) (see
spin configurations Cp and their weights, and the
configuration Cp and
a
graph Gp
are
compatible.
18
algorithm
this
building
need
we
construct this
we
an
only proportional
effort
The t-J model is denned
h
=
-tJ2J2
where cia creates
YLa ni,°
=
a
be
represented
and
several
are
configuration.
-
a
a
fermion with
One
example
component of spin
z
(i, j)
worldline formulation
—1, representing
For the t-J
sign
of
hole,
a
[46]
up
spin, and
bc
—
in the overall
across
Eq. (2.2.11).
It will also
The
play
In the last
binary
=
Fig.
sign
constructing loops
role for the
section,
we
improved
have
are
a
MC step into three
left unaffected
values +1 and -1
a
weight
The t-J model
given
are
=
2.3.1. The overall
sign
possible
in
Fig.
2.3.1.
\W(C)\ sign(C)
can
of
a
will
worldlines, bc
use
decomposed
as
(2.3.2)
=
+1 for periodic and
and ribound is the number of
we
be
;
particles hopping
the absolute value of the
according
a
loop algorithm is constructed for
loop algorithm for the t-J model
substeps.
(inactive)
and
weight,
to
Bq. (2.2.8).
a
model with
estimators treated in sect. 2.4.
how
seen
c\aCi^
=
nj,_CT) prohibit
—
spin, respectively. The
down
weight W(C)
(-1)"p«»> (6c)n>>°»nd
with trivariate variables into three
divide
holes,
a
i, ni,a
neighbor pairs.
will be taken into account in the MC simulation
variables. In order to construct
problem
we
For
the boundary.
at site
all stem from anticommutation of fermion operators.
they
in the third line of
antiperiodic boundary conditions,
1 for
a
nonzero
where nperm is the number of permutations of fermion
=
(2-3-1)
>
in terms of variables that take three
plaquettes with
sign(C)
a
projection operators (1
denote nearest
an
negative signs
model
The
conjugate.
site. The brackets
sources
Hence
-
-
spin-1/2
in
is the
path.
the
the Hamiltonian
matrix elements for the 15 different
There
plaquettes along
the t-J model
aQd H.c. the Hermitian
values, 0, +1
any site of the lattice and
[(* nh-M^^1 "*.-»)+H-c] +JY,(§iSi i>w
double occupancy of
can
by
the
on
length \l{\.
to the
Loop algorithm for
2.3
ni
single loop by picking randomly
single loop by choosing graphs Gp only
a
while
subproblems
In
substep I,
attempts
(active variables). Similarly,
are
with
binary
we now
reduce the
variables. To this
variables with the value 0,
made to
in the second
(II)
flip
end,
namely
all the variables with
and the third
substeps (III),
19
w(Cp)
Cp
HDD
1
Lj
pAtJ/2c1i(At 7/2)
s m
-eATj/2sh(ArJ/2)
ch(Art)
00S0
sh(Ari)
FIG
spins
by
according
to
a
line, down
solid
Eq (2
plaquettes Cp
spins
with
by
a
of the
binary problem
The sign of
broken line
a
loop algorithm
w(Cp)
Therefore,
binary problems,
To each of these
substep
We denote
'active
as
plaquettes"
those
plaquettes, the resulting algorithm
On the active
active
Up
will be taken into
8)
2
each
m
t J model
nonvamshing weight w(Cp) for the
keep variables with the values +1 and —1, respectively, unaffected
with
are
denoted
are
account
we
The 15 different
2 3 1
deal
apply the idea
we
which all four variables
on
substep I
for
we
is
identical to the
loop algorithm for the 5=1/2 antiferromagnetic Heisenberg model, while the algorithm for
substeps
the
same
II and III turns out to be the
as
are
of
Since
we
loops
that for free
fermions),
affected
course
by the
problems,
in
choosing
a
computational simplicity
resulting loops
such that the
Ref
[48])
set of
those where two
loop updates
only
XY model
loops
with the active
(which
flipping probabilities
is
of the
is
one
another
loop algorithms
simply by
inter¬
The detailed balance condition holds for each
achieved by the combination of them
flipped independently
Therefore
plaquettes,
,
1/2
need to construct three
we
graphs and graph weights
may be
see
deal
The
=
It is,
we
must have
may be
flipped
plaquettes
The
in
a
can
weights
a
scheme
multicluster variant
weights iDas(Cp)
This
We have
however, advantageous
and the reduction of autocorrelation times to (hoose
different choice
i e
below
being transformable into
substeps whereas ergodicity
ample freedom
see
the 5
plaquettes
the roles of the values +1 and -1
of the three
for
ones
will
we
inactive
have three different binary
with the second and the third
changing
as
loop algorithm for
=
1
on
(for
a
the active
be achieved
by letting
the
of the other
plaquettes
are
20
put into the global weight function
if all four variables
iw(Cp)
if all four variables
{1w(Cp)
case we can
where C' denotes the
consider
us
the
one
plaquettes
now
variables,
only
state after the
(holes)
inactive and
are
active variables
are
weights v(Gp)
Fig.
2.3.2
shown in
remain
satisfy
unchanged
symmetry of the Hamiltonian. Thus
spins
are
to the
is
consider the
for the S
[40])
connected. The
to calculate the
=
1/2
to each
plaquette
to the
magnetization
netization results from
particle
a
loop lz
kept
substep I in which
The
algorithm
the
plaquettes
verify
It is easy to
of
as
equiv¬
far
as
with inactive
variables, if
weight equation (2.2.12).
flipping
model
is
any,
that the
are
graph
algorithm,
In this
a
loop because of the spin-inversion
loop flipping probability of 1/2.
a
(since
number
or
same
plaquettes
as
antiferromagnetic Heisenberg
are
shown in
of the loops
local-update
the
Fig.
2.3.3.
according
in the first
worldline
the
is
equivalent
for free
algorithm
model.
Again,
a
unique
change
magnetization
to
substep I,
we
Eq. (2.3.4), since there
are
have
is
no
algorithm.
algorithm,
in the
wrap around the lattice in
the
Contrary
to
grand canonical ensemble,
in the present method. A
loops that
the active
that is the
spin-inversion symmetry
or
on
with inactive variables such that any active variables
flipping probabilities
formed in either the canonical
times. If the
(2.3.3)
for
algorithm
unaffected.
concerned. As for the
obtain
XY model
In contrast to the conventional
variable
active
in the
time, the algorithm
This
graph weights v(Gp)
symmetry similar
are
Fig. 2.3.2).
upon
we
rather than that for the
assigned
plaquette
algorithm for substep II (or equivalently substep III), where all down
kept unchanged.
one
fermions
graph
we
the
to each of them such that active
lower part of
Next,
on
1/2 antiferromagnetic isotropic Heisenberg
=
(see
weights tt>as(Cp)
active
flipping.
probabilities
connected to each other
the
are
ripgloopi, Was (Cp)
Flpeloop J, was(Cp) + Ilpgloop I, was{C'p)
unique graph is assigned
a
plaquette
otherwise.
in detail the
for the S
with
_
plaquette
variables with the value 0
alent to the
the
flip all loops independently with the flipping probability for
,q
Let
on
otherwise,
1
In this
w^:
simulations
particle
fixed,
be per¬
with either constant
temporal
should be
can
number
direction
we can
or
or
the mag¬
one or more
simply
disallow
21
xi: X
GP
1
(J/2)dr
1
1
1
0
1
1
1
A(GP,CP)
\ ] n
1
1
1
0
1
0
0
0
n u
eAr-'/2ch(ArJ/2)
1
1
1
0
0
0
0
eArJ/2sh(ArJ/2)
1
0
1
1
0
0
0
1
ch(Arf)
0
0
0
1
0
0
1
sh(Art)
0
0
0
0
1
0
1
1
0
0
0
0
0
1
in)
the t-J model
The upper part of the
equation (2.2 12)
and the function
The first six
case
The lower part of the
A(GP,CP),
which
configurations Cp
of the
figure specifies
specifies
figure
graphs Gp
shows the
whether
and the solution
the
a
v(Gp)
configuration Cp
and
restricted to these
antiferromagnetic Heisenberg model. The
brackets, the corresponding graph Gv given
(flip
and the
in the
spin up
«->
freezingless
spin down) of
solution
v(Gp)
spin configurations Cp and their weights,
a
graph Gp
figure
ones.
(For
has to be
are
compatible.
configurations correspond
open circles in the
represent active variables whereas solid circles stand for inactive
in
I
Plaquette configurations Cp and graphs Gp for substep
FIG. 2.3.2:
Cp
0
W'as(Cp)
nr:<
the
HmAr^dTv(Gp)
(eATj
l)/2
-
u>o(Cp)
r
of
v(Gp)
1
the
diagrams
in the top
to
row
plaquette configurations
flipped spatially.)
22
_
v(Gp) (l
+
e-Art)/2 (eA"
hmAr-tdTw(Gp)
-
l)/2 (l-e-Ar')/2
C/2)dr
(</2)dr
1
; V.
X
Gv
•
•
1
1
1
1
1
1
A(G„ C„)
u>o(Cp)
">as(Cp)
1
1
1
0
1
0
0
0
ch(Art)
1
1
1
0
0
0
0
sh(Art)
1
0
1
1
0
0
0
li <[ :i)
eATj/2ch(AT7/2)
0
0
0
1
0
0
LiO
th(Art)
0
0
0
1
0
0
m^
rAT-'/ish(AT.//2)
0
0
0
0
1
0
0(0)
sh(Art)
0
0
0
0
1
0
1
0
0
0
0
0
1
Cp
D
00
FIG. 2.3.3:
t-J model
Gp
and the
v(Gp)
Fig
Plaquette configurations Cp
The solution for
freezingless
restricted
232
to
substep II
solution
these
v(Gp)
and
graphs Gp for substep III (flip
is
equivalent
of
Eq (2
2
The upper part of the
12)
configurations correspond
The first
six
spin up
configurations Cp
to the XY model
(free
*->
hole)
of the
the
graphs
hgure specihes
hardcore
and the solution
bosons)
See also
23
flipping these loops
rithm is
no
difficulty
subspace of
one
subspace
chooses the
us now
discuss
"improved
by implicitly averaging
loop algorithm,
define
a
over
series of i
a
graph G?W
any member of
a
However,
as
Since the
winding
loop algo¬
spatial winding number,
here the
sign problem
negative
a
is not
system size becomes larger. It
the
of constant
probability p(C)
1,...,
of 2"
*
really
can
a
also be
number.
according
probabilities.
An improved estimator
averaging
over
C'1',
set
a
nW loops.
0f
configurations by flipping
6
rW
2'
measuring only
the value in
£
C
states
one
G
rW
state
probability p(C)
of the
flip probabilities pa\p. (Actually,
from the
balance
flip probability
requirement
as
subset of the
a
by
the
that
can
reach
loops.
loop flip
randomly
be constructed
by
be reached from the
can
CW:
JV
i
_
0(C')p(C), Oimpr
=
Jj520impr,
(2.4.1)
«=i
configuration C
choose
we can
used in the MC
pmp.
loop update
we can
will then be chosen
cerw
where the
graph
;s determined
(2.2.7)
quantities
construct, with the
we
Prom this
configuration £(t+1)
one
of measured
In the first step of each
for the expectation value Eq.
<Oimpr>, Oimpr=
=
£W
error
MC simulation
a
configurations C
the value in each of the
instead of
(0}
Ojmpr
step
In
reduce the
configurations C^.
worldline
for each of the
In the second
to these
N
that consists of
rW
set
a
configurations.
many
=
[53]. They
estimators"
probabilities p&p.
state
constant
Improved estimators
2.4
The
significant
because it becomes less
avoided if
we
the detailed balance condition.
may appear also for the ID t-J model.
sign
Let
longer
violating
without
restricted to the
some
updates;
Thus there is
can
it
actually
be calculated
probability
just
a
needs to
as a
p'aip
product of the loop
here that is different
satisfy
large variety
of
the
same
improved
detailed
estimators
available.)
To
time
really gain
comparable
an
improvement
we
to the time needed for
Particularly simple improved
estimators
need to calculate the average
a
single
can
measurement.
over
Fortunately
often be found in the
case
2n'
states in
that is
that pmp
=
a
possible.
\
for all
24
loops.
In that
case
the above estimate
Oimpr
simplifies
to
f>-"W
=
E
as
all of the states in
p'm
loops have
possibilities
now
have the
loop flip probabilities
Even if the
some
rW
=
If Pflip <
the
51
FW
of
<
on
with
loops
the fixed
nM loops
can
us
examples
show two
algorithm (or
for the
Oimpr
The
takes
only
fixed in
flipped
a
a
still choose
just
pjy
a
certain state. There
loop with
a
such that
many
are
probability
of
1|.
(2.4.3)
in the old state and
with
the inactive
as
ifpmp
|
in the
flipped
spins. The remaining
probabilities pimp
new
>
=
=
the locations of the
We
case
provide
4S*iTS*,T,
=
at momentum
|.
a
of substep I of the t-J
more
detailed discussion
-k
0
if the
spins
are on
different loops
I
1
if the
spins
are on
the
loops
thus
corresponds
±1. When
(O)
same
to the
same
is small
see
expectation value
(e.g., {O)
~
(2.4.4)
loop.
spin-spin
estimators is easy to
Yet it has the
4 for
is
I
{
using improved
from
set
sign problem
spin-correlation function
the values 0 and 1.
=
treated
-
can
Eqs. (2.4.1) and (2.4.21), the improved estimator (multiplied by
potential gain
estimator O
|2pflip
.
improved estimators for the simple
4(Sr*,TSrV)impr
=
Remarkably,
tion.
are
Heisenberg antiferromagnet).
for the
in section 2.4.3. Prom
convenience)
are
then be
of
=
probability pgx
Simulations without
2.4.1
Let
loop is fixed
spins
n/W
loops
we
to do that. We have chosen to fix the state of
Pflx
state. The
equal,
not all
while the other
5,
(2-4.2)
probability 2~n'
same
are
°(C'),
cert')
»=i
correlation func¬
in this
as
exp(-r/£)
the
at
case.
0\mpi
unimproved
large r),
then
the variance of O is
(O2)
whereas the variance of
(OLpr)
-
0jmpr
-
(O)2
=
1
-
{Of
«
1
(2.4.5)
,
is
<eW)2
=
(0.mpr)
"
(Oimpr)2
«
<Oimpr>
=
(O)
« 1
.
(2.4.6)
25
For
a
given
correlation
is
largest
gain
from
using
whereas the
gain
from
distance r, the
length f,
thus the computer time
estimator may,
An
<X>
=
required for
however, have
can
a
cancel part of this
especially simple estimator
gives (x)
numbers
autocorrelations with the
wt(l)
large
a
factor.
variance, and
The
nonimproved
summing over
all lattice
magnetic susceptibility
by using
E
E
(loops 1) ((r,r)
(Ximpr) simply
of the
by
also be derived for the uniform
can
r
=
accuracy,
loop algorithm
therefore reduce the
can
at small
gain.
E^?E*,=
t
given
a
largest
estimator appears
sizeable amount of self-averaging from
^ ((5bEr,T5rV)2)
which
reducing
large £. Using the improved estimator
at
sites, which
improved
the
the the
as
in
j^*, !
E
=
I)
sum
loops
««<'>.
(2-4-7)
I
of the square of the temporal
winding
loops I:
X,mpr
=
^£^)2-
(2-4.8)
loops /
Here V is the number of
position (D
=
2 for
a
spins
in the
lattice,
nearest-neighbor chain)
Thus VDM is the total number of
spins
single
the
cluster
Since there
we
variant,
we
pick
a
D is the number of terms in the Trotter decom¬
the
sum over
single loop
in the classical D + 1-dimensional lattice.
loops
I with
a
and M is the number of Trotter time slices.
in
Eq. (2.4.8)
is also calculated
probability \l\/(VDM) proportional
In the
stochastically.
to its size
\l\,
have to compensate for this extra factor and obtain
{x)
The
improved estimators for
=
ljM{nMWt{in
general spin
more
and
charge
(249)
correlations
are
derived in section
2.4.3.
Simulations with
2.4.2
In the
case
of simulations with
puted according
variance,
Let
to
a
negative sign problem, expectation
Eq. (2.2.8). Improved
and thus the
us
sign problem
a
error
of the
restrict ourselves to the
sign(C)
=
estimators
can
values have to be
again help here,
as
they
com¬
reduce the
sign.
case
of the t-J model
(bc)n°
on a
single chain,
((-l)^'-1)"' (-1)""*
,
for which
(2.4.10)
26
to
or
(possibly frustrated) spin
models
sign(C)
+1 for
periodic
Here
6C
total
particle number,
Nneg
are
=
nx
the number of
decompose the sign
and
bc
=
—
lattice, for which
any
on
(-1)^"-*
=
antiperiodic boundary conditions, iVtot
1 for
particles hopping
denotes the number of
plaquettes with negative weight.
J[
a
product extends
plaquette
In the canonical
the
boundary
ensemble,
s(Cp),
plaquettes
sign of
across
is the
and
we can
as
sign(C)=
where the
(2.4.11)
.
all
over
is defined
plaquettes
(2.4.12)
p
6C(—l)"'0'-1
of the lattice. When
—1, the
=
as
(
8(C,)
ticle
=
—
hops
if for
1
1
to
a
across
the
signs of the
estimators
<
border, but
w(Cp)
spin model
0
not
or a
par¬
both,
(2.4.13)
0
<
otherwise
satisfy Eq. (2.4.12). (When 6C(-1)W«<«-1
Improved
w(Cp)
if for the t-J model either
—1
=
1 it
can
be formulated for the
can
similarly).
be defined
sign
if
express it
we can
as a
product of
loops:
sign(CW)
11
so
=
siSnW'
{2A.U)
!£<•>
where so is the
the
case
sign
where the
estimators
are
of the
plaquettes
that contain
flipping probabilities
are
all puip
only
=
inactive spins.
1/2,
since
We consider
only
especially simple improved
available there:
(sign(CW))impr
=
s0
2-"W
J]
(si6nW
+
siSnW)-
(2-4.15)
lecM
This estimator is
very
simple
The
inactive
the
zero
if at least
one
loop changes
its
sign
when it is
flipped. Again
this is
a
estimator.
signs
of the
loops
are
constructed in the
spins, its sign contributes
plaquette
is
assigned
to that
to so-
loop.
If
only
The
following
one
loop
way: If
a
plaquette
threads the
only nontrivial
case
contains
plaquette,
is where two
only
the
sign of
loops
thread
27
of signs scp
Assignment
TABLE 2.4.1.
model, when the bipartite transformation
the table
in
are
all
equal
to
not
is
done)
(or
t J model
substep I of the
for
Gp{p)
The values of scp gp
[p)
for the
for the
Heisenberg
plaquettes
not
1
SCpGp(2)
SCj,G,,(2)
sCpG))(l)
5CPGP(1)
Gp
Cv
o-—-o
m e
1-1
-11
2
°
°
ia
1
1-11-1
1111
I-1-
n n
a
We denote these
plaquette
are
chosen for the
the
flipped
equal
scp
f°r
t°
m
Overall
change
loop algorithm
we
and
is
sCr
gp(2)
They
like
be
can
Eq (2
4
we
assigned
15)
ora
is
to
2 4 1
the
graph
6p(l)
loops
sCp Gj (2)
the
assigned similar
as
show
a
(The
solution
loop whose flip
causes
for all three steps of the
problems,
such
we
are
get additional sign changes
possible
For fiustrated spin
model,
or
and sCp
on
both of the
corresponding loops flipped
the
canonical ensemble
In this
In
case
since
we can
the
substeps II and III
still
signs appear
only
in
to Table 2 4 1
ladder systems
or
higher
di¬
sufficient to have
most of the
similar way
we can
an
improved
possible reduction
also
measure
the improved estimators constructed
use
corresponding spin flips
more
equal
do not
complicated improved
at least foi summing measurements over
already obtain
a
way that if
plaquette ) In Table
therefore always be
updates of substep I,
number
probably
In
can
Gp(2) depending
plaquette
lattices, then sign(C) gets additional contributions from winding of fermion world-
above for the
structed,
10)
4
assignment
an
lines inside the boundaries
winding
(2
Eq
grand
on
the sign of the
case
scp
a
Gp(l)
with the spins
In the
In that
such
in
sCp
that part of the
on
I
G„(2),
"
Gp(l)
<>cp
=
consider different geometries for the t-J
mensional
it
s(Cp)
for the ID t-J model
smgle-plaquette weights
If
sCp
plaquette
spins
substep
changes of JVtot
from
JVtot
Gp(p)
g„(T)
products sCp
flipped
and "2
This must be done
plaquette
to the sign of the
bars denote
for
loops by "1"
has to be divided into two parts
s(Cp)
Gp
1111
some
of the
substeps
particle-particle and
spin spin
one
the fermion
can
possible loop flips
of the
estimator for
in
change
estimators
be
con¬
However,
in
order to
variances
time
correlations,
28
magnetic susceptibility
the
susceptibility
estimator for the uniform
If
no
loop changes
the
and other observables.
exactly
requires
in
if
zero
one
one
or
^sign(C«)
=
X)«pr
we
algorithm (or
then
flipping,
£
present the improved
for pure
spin models).
have
we
wt(lf.
(2.4.16)
loops (££<*>
^Msign(e« W*H(0,
=
sign change
of
improved estimator),
the
estimators for
more
a
a
set of
loops (all
improved
flip
those whose
algorithm
multi cluster
correlation functions
general
(2-4.17)
As this
loops change the sign.
than two
more
to know the
constructing
improved
I of the
upon
example
an
loops I and V change the sign, it is
two
(sign
and it is
substep
sign of configuration C^'
(sign.X)«pr
If
in
As
is
estimator
is considered
advantageous.
The
derived in the next section
are
2.4.3.
Improved estimators
2.4.3
In this section
consider the
we
at
we
imaginary
show
improved
for correlation functions
estimators for
spin- correlation function
times
t
and
t', respectively.
charge-
(S}TSp T,)
The
and
spin-correlation functions. First
between two
improved
spins
at sites
r
and r' and
estimator is
£ S*r,T(C)Szt,,AC)p(C).
(2.4.18)
csrw
As each spin
spins
are on
can
be
on one
different loops
[(1
where
a
is the value of the
loop only,
this
sum can
simplified substantially.
be
If the two
it is
-
pflip)a
S*T
+
in the
pflipCT] [(1
original
-
p'flip)<7' + p'mpc'}
state
CW,
and
o
(2.4.19)
,
the value in
a
state where the
loop containing the spin is flipped. The flip probability of this loop is given by pmp. Similarly
the
primed symbols
refer to the other spin. If both
[(1
The
equation for the
cases
inactive in the t-J model
where
-
Pflip W +
one or
algorithm
or
both
spins
pflipoff']
spins
are on
the
same
loop
it is
(2.4.20)
.
have been
fixed, either because they
because the loop has been
fixed,
are
straight
are
forward.
29
Let
and in
make the above estimators
us now
I of the t-J
substep
estimators
very
are
algorithm
I
for the
located
the
on
estimators
are
change
as we
=
as
inactive
\
=
the
and
a
of a pure
case
In this
—a.
—
0
if the spins
are on
different
aa'
if the
are on
the
spin
case
Hamiltonian
the
improved
(down) spins
look
more
loops
(2-4-21)
'
spins
For
but
all pfnp
are
a
substeps
=
versa.
be
can
loop.
same
+(—)cr'
=
In this
equal,
not
case
The
spins
are
algorithm
the
and
loops
some
the fixed
on
spins
a
=
±5—<r,
the improved estimators
simplified by fixing
1/2.
when the
II and III of the
flipping probabilities are
There the
complex
have
we
into holes and vice
remaining flipping probabilities
just
i
I
(different) sublattices).
slightly different.
Eqs. (2.4.19, 2.4.20)
have pmp
Heisenberg antiferromagnet,
same
up
specific. In
simple, namely
(SrVSrV)impr
(Moreover,
we
more
loops
so
are
that the
treated
spins.
Similar improved estimators
can
be used for
Y,
charge-charge
correlations
(2.4.22)
nr,T(C)nT,y(C)p{C)
csr(')
with
a
then
only spin degrees
suitable
changes
to
n
=
of "a."
reassignment
1
—
of freedom
They
are
are
trivial for
or
pure
For steps II and III the
changed.
n, because these steps
I
substep
exchange
a
hole with
an
occupation number
or
up
that the calculation of improved estimators of correlation functions
spin models, since
can
down spin. We
n
see
performed
with
be derived.
For
be
effort similar to that for the nonimproved estimators.
For simulations with
the two-site
Both spins
same
loop,
spin
or
are on
the
a
sign problem
similar
charge-correlation functions
the
same
improved
loop I,
or
they
improved
two
are on
estimators
different
two
cases
can
have to be
different loops
1,1'.
If
distinguished:
they
are on
the
estimator is
[(1
-
Pflip(0)sign(Z)CT<r'
x «o x
„
Y[
loops 1^.1
+
Pflip(/)sign(I)CTo']
((1-Pflip(«))sign(t)+Pflip(*)sign(i)).
(2.4.23)
30
estimators for
Improved
TABLE 2.4.2:
for simulations with
a
sign problem,
spin correlations
from
I
Only loop
in the t-J model and in pure
Eqs. (2.4.23, 2.4.24)
changes sign
in the
Both
case
pflip
=
spin models,
1/2.
loops change sign
Any
other
loop
changes sign
Step
I
or
pure
spin models
both spins active
different loops
Spins
on
Spins
on same
both
One
or
Step
II and III
0
0
0
loop
spins
sign(C)CT<T'
0
0
inactive
0
—
both spins active
Spins
Spins
on
on same
Spin a'
±isign(C)(a±i)
different loops
Sign(C)[±l(<T
loop
Improved
TABLE 2.4.3:
estimators for
for simulations with
(The improved
sign(C)(<r±i)(a'±i)
estimators
are
a
0
0
—
charge correlations in substeps II and
sign problem, from Eqs. (2.4.23, 2.4.24), for
trivial for
Only loop
I
0
<T')-|]
±sign(C)(a±i)a'
inactive
algorithm,
+
III of the t-J model
the
case
p^p
=
1/2.
substep I.)
changes sign
Both
loops change sign
Any other loop
changes sign
Both spins active
different loops
Spins
on
Spins
on same
loop
Spin n' inactive
sign(C)|(n-|)
sign(C)i(n + ra'-l)
sign(C)(n
-
|)n'
sign(C)(n
i)(n'
-
—
—
-
1)
0
0
0
31
In the second
it is:
case
[(1
-
PmP(l))siga.(l)a
[U
x
-
p&ip(l)sign{l)a]
MipCOJufentfV
Yl
x
xs0
+
+
PmP(l')sign(V)o'}
(2.4.24)
(Q--PBip(i))8ig*(i)+Pmp(i)sig*(t))-
loops i^l,V
In simulations with
1/2.
sign problem it
a
Then the last term in the
Pflip(i)sign('))i
simple. If
vanishes if
sign,
(Eqns. (2.4.19, 2.4.20)).
The
only
estimators
In this section
we
briefly
loop algorithm [18]
depends
on
for
and how it
a
to
finite limit,
At-1,
the
a
now
nonzero
improved
imaginary
0 the
is well-defined
to the above
estimators
ones
occur
if
the
frequency
of
a
worldline
a
are
is therefore best
well
as
configuration
to
we
loop
time
even
hopping from
as
of the order
a
hop
O(Ar).
specified through
the initial
0. All the
—>
implementation
one
site to another tends
can occur
is
proportional
In the continuous time
the time values r, at which
configuration
reduces the memory
discrete time
over
in the limit At
have well defined values in this limit.
describe the continuous time limit of the
implementation
on
—>
configuration
specifying
equivalent
be used for the t-J model. The continuous time version
can
hopping probabilities
magnitude compared
We
in continuous
loop algorithm
spin configuration changes,
way of
rii((1_Pfiip(i))siSI1(i) +
which makes the estimators
are
because the number of time slices where such
and the
formulation
=
in Tables 2.4.2 and 2.4.3.
choosing graphs Gp
Note that for At
to
with
product
sign,
its
estimators
cases
flipping probabilities pmp(i)
review the main idea behind the continuous time formulation of the
the fact that the
probabilities
improved
other two
presented
are
Loop algorithm
2.5
changes
the
loops going through the spins under consideration change their sign. These
both of the
improved
the
to have all
(2.4.24),
and
of the loops
loops flip
one or
advantage
Eqs. (2.4.23)
one
their
no
is of
at the time
r
requirements by about
at
loop
all time slices and decide the
a
=
an
0.
This
order of
typical value of At.
construction. In the discrete time
graph segments
for each
plaquette
this time slice.
In the continuous time limit
having
a
graph
we
need
segment that forces the
a new
loop
to
procedure. We
"jump"
note that the
to another site is
probability
proportional
for
to the
32
infinitesimal time step At: p
the other hand. The situation is therefore
on
"decay constant" A. This decay
only
at the time
decay
Instead of
(i.e. jumps)
site
A special
channels."
to the
I
they
These
are
neighboring
The
a
independent
are
neighbor.
to
we
case
can
The forced
are
site. This is called
"forced
weights
straight
configuration.
calculate
decay
can
loop "decays"
time" after which
processes to the various
=
points where
algorithm.
decay"
in Ref.
a
neighbors
worldline
There the
loop
of
"decay
jumps
has to
to
a
jump
[18].
in the continuous time
are
change
for finite At.
not the
"decay
a
O(dr),
We have listed the
probabilities
or
—
process with
spin configuration and
step dr whether
we
decay
is 1
well-defined continuous time limit.
a
even
contribute ratios such
the inverse of it. For
all infinitesimal
a
also have
«a<y/«.<5)
or
radioactive
with the
is the finite number of time
in steps II and III. There
decays
a
on
calculate independent decay times for each of these
always 1/2, which holds
probabilities
As the
site.
in the
treated like in the discrete time
loop flip probabilities
are
probability dr
to
the
together
at each infinitesimal time
neighboring
on
change
2.3.2 and 2.3.3
to another site with
continuing straight
equivalent
depends
constant
r, where there is a
Figs. 2.2.5,
deciding
loop "decays"
the
a
steps
constants in
for
probability
Adr. The
=
case.
The
two contributions from the
In
only
substep
nontrivial
weights w^Cp).
as
2j«
eATh(A^J/2) i
^
=
worldlines between two
decays
at T\ and T2 the
product
over
has to be considered. The continuous time limit is
Krn^ n "-(Cpin+jjj)).
(2.5.2)
j=o
In
particular
lim
TT
ch(*^)
=
1
(2.5.3)
eJ^-T^2.
(2.5.4)
i=0
lim
TT
eJ^ch(J-|7)
=
j=0
Thus the forced
decays
tribute the classical
contribute terms like Jjit
Ising weights
While this continuous time
time
version,
it has two
or
2t/J
and the
straight pieces just
con¬
of the worldline segments.
algorithm
is
more
significant advantages.
complex
One
to
advantage,
implement than the discrete
mentioned
already above,
is
33
that the memory
requirements
This is crucial if
implementation.
the
constraints become the
The main
tematic
error
systematic
then
restricting
associated with
a
to At
algorithm
=
0. In
about
a
up to
an
order of
wants to simulate
magnitude, depending
huge systems,
algorithm
that in the continuous time
finite time step At.
could be controlled
extrapolating
one
by
on
where memory
factor.
advantage is, however,
error
the discrete
reduced
are
In the discrete time
experience this need
to
slower, depending
factor 4-8
no
sys¬
algorithm
this
for several values of the time step At and
by simulating
our
there is
several simulations makes
run
on
the hardware
platform
and
implementation.
2.6
Results
We will
discuss the
now
performance
autocorrelation times of the local
improved estimators.
chains,
J
and three
update
frustrated
2.6.1
Autocorrelation times
We have determined the
t-J chain. Let
us
first
Heisenberg
integrated
give
model
or
Eq. (2.2.7)
the
as an
are
on a
on
how
we
average
denominator of Eq.
over
(2.2.8).)
The
error
T®t
is
iW"
loop update with and without
single
t-J
T?t
of
chain,
two
coupled
coupled
t-
t-J
algorithm applied
our new
MC procedure. It
we
and
chain.
can
estimate the value of
(We
of the estimate is
=
errors
do the
o/\/N
same
to
OM
be the
be either the
simple
have calculated these times. Let
these JV measurements.
a2
The autocorrelation time
single
our
improved estimator. As usual,
a
the
the first MC simulations for
autocorrelation times
the details
estimate of the observable O in the ith step of
estimator
algorithm by comparing
examples:
coupled t-J chains (these
and
a
new
method and the
We will consider four
chains),
a
of the
an
observable O by
for the nominator
or
1, where
—
2rgt(^-e>2).
(2.6.1)
given by the autocorrelation function T(t)
(0(«o)0(<o)>
-
(0(*o)) (O(*o))
•
V*-V
as
*&
=
*
+
££! r(t).
(2.6.3)
34
In the MC
of
I
length
Then
simulation,
N/n
=
and
t°,
have calculated
we
computing
by grouping the
the bin averages
Ot(l)
can
of bin
lengths
n
6=1,
bins
...,n.
/:
^6=1
"
and the autocorrelation time
\ Ej=(H)i+i 0^\
=
0f,(/)
have calculated the variance of these averages
we
N measurements into
be estimated
as
[54]
la(lf
t(0
(2-6.5)
£ftj,
=
2a(l)2
expectation value of which becomes equal
the
/
As
—)• oo.
When statistical
of
independence
T®t(l) approaches
r®t
the estimate for
value
was our
For
a
estimate for
subdivided into four
sublattice. In
one
loop.
performed
(only
bin size I.
where
a
that with
times
we
In the conventional
nondecreasing
and the
ceases
increased.)
lengths
T?t
a
a
All
M
=
called
one
we
I
=
The
2.5
"sweep"
simulations
x
statistically
were
106 MC steps
for the conventional
In these
expectation value
asymptotic
constant
1,2,4,8,....
also need to
give
our
definitions of
plaquette flip algorithm,
sequentially attempted
over
complete decomposition
our
function.
finite number N of measurements,
a
when I is
We have taken bin
MC step
MC steps for the plaquette flip
cases
generally
the increase
increasingly
algorithms.
is
the lattice. For
update
to
all sites of
sweep with the
one
of the three steps I, II, and III at random and chose graphs
This results in
each
(Note
r®t(l)
in the limit of
the lattice is
sublattices, which allow the simultaneous modification of all sites of the
single
a
generally
chose
r®t.
length I,
approached,
constant value.
will fluctuate
"one MC step" for both
we
is
bin
comparison of the autocorrelation
a
which is
increasing
function of
a
given by Eq. (2.6.3)
to Tjnt
cases we
of the lattice into
algorithm),
r®t(l)
where
as a
reliable estimate is still
Since the value of
T?t
depends strongly
the internal energy, the static
Sc(k)
algorithm
these very
and M
long
=
x
general
106
as a
to
we
to 70
simulation times
rPjJ) keeps increasing
lower bound for
attempted
In
30
plaquettes.
we
flip
have
x
106
found
function of the
Eq. (2.6.3) with the largest I
possible.
on
the observable
O,
we
have calculated Tjnt for
charge-charge correlations
1
=
We then
sublattice,
loop algorithm,
for all
Gp
done in the canonical ensemble.
for the loop
algorithm. Despite
took
loops.
a
L
l*E,elk<J~mH(n3,t + n3,i)(nm,t + nm,;)),
(2.6.6)
35
and the spin-spin correlations
L
4
Ss(k)
In
Fig.
We have
a
larger
at
2.6.1
performed
quarter-band filling
-iY. e'k^-m){S]S^).
on a
smaller
64 sites and Art
=
and for ratios of
loop algorithm
we
lattice,
=
J/t
with L
=
This is in contrast to the conventional
point
most
are
even
larger
than
out here that the values of r,„t for the
likely poor
able to reach
effective in
algorithm
a
for
a
for
r,nt(/)
in these
no
significant
algorithm,
for the t-J model works
t-J chain.
0.25 and
on
performed
increase of 7}nt with
where Tjnt is between 100 and
104 for the spin-spin correlations. We have
larger
lattice with the conventional
Obviously
cases.
the autocorrelation times for the
reducing
=
and 15 sweeps for all
one
lower bounds of the real autocorrelation times T;nt, since
plateau
single
1 and 2.
obtained values of rmt between
1000 for the internal energy, but
to
/?
16 sites and Art
—
All measurements have been
0.125.
observables and all parameter values. Especially there is
increasing /?.
(2.6.7)
function of the inverse temperature
as a
simulations
lattice with L
For the
r®t
show
we
=
successfully, saving
loop algorithm
the
correlations.
spin-spin
magnitude
orders of
algorithm
have not been
we
is
especially
Thus
our new
in computational
effort.
Next
estimators
Improved
2.6.2
we
show
some
results and the effect of
with the multicluster
algorithm.
The results of the measurements
We have considered different correlation
quantities,
correlations. For all observed
proved
estimators. The variance of the
than without the
use
of
improved
improved estimators for the t-J chain, obtained
functions, such
as
the
can
large self-averaging
when
cancels part of the
gain
In order to
have simulated
and
from
investigate
a
simple
improved
measurements is up to
the correlation measurements
a
in Table 2.6.1.
charge-charge
application
of im¬
factor of 1.7 smaller
unimproved measurements,
Thus here
over
the
we
can
large lattice,
have
which
improved estimators.
the
improved
estimators for simulations with
frustrated spin system,
next-nearest-neighbor
and
the variance is reduced with the
measurements. Note that in the
summing
seen
spin-spin
have measured the correlation functions from each lattice site.
we
be
interactions.
namely
the
Heisenberg
With the notations of
a
sign problem,
we
chain with nearest-
Eq. (2.2.1) the Hamiltonian
36
(c)
O
0000
L=16, J/t=1
L=16, Jrt=2
iL=64,J/t=1
O L=64, J/t=2
t
1000
•
100
t
10
===—.
FIG. 2.6.1:
filling
on a
Integrated
lattice of L
of the plaquette
filled
symbols.
algorithm
(b)
for
arrows
gains
we
autocorrelation times r-mt for the
=
16 sites with Art
flip algorithm
For the
used the
is shown with open
loop algorithm
we
charge-charge correlations Sc
at k
=
increasing gains
magnitude
at low
in
Uf
computational
symbols,
condition.
denote measurements where *rmi is only
orders of
=
dimensional t-J model at quarter band
64 sites with Art
the results of the
=
0.125.
The results
loop algorithm
with
antiperiodic boundary conditions, for the plaquette flip
took
zero-winding boundary
one
0.25, and L
=
—
a
-?r/4
lower
effort
Figure (a)
and
(c) spin-spin
bound,
over
shows r-mt for the internal energy,
see text
the traditional
temperatures and for large systems.
correlations Ss at k
for details. The
=
fcjr. The
loop algorithm
plaquette flip algorithm,
with
37
chain t-J model
single
surements for the
charge-charge
correlations
spin-spin correlations at
r
conditions and 64 sites, J
comparison,
pJ
at
—
np =32
U
np =32
I
=
48
=
tt/4),
=
L/&
=
and
t, (3J
—
the last two
r
L/4
=
16, Art
rows
We have considered
The number of
0 125
=
a
the results for the
and
unimproved (U)
mea¬
the internal energy e, the
are
Ss{k
the spin-spin correlations
=
tt/4),
and the
real-space
system with periodic boundary
particles np
Heisenberg
32 and 48
are
chain of the
same
For
length (hb),
Error
(s>s,4.0
Error
(S,S.l)
Error
0 78092
0 00202
0 00533
0 00015
0 00088
0 00011
0 78012
0 00161
0 00535
0 00012
0 00086
0 00008
0 00066
0 71221
0 00096
0 01046
0 00029
0 00220
0 00021
0 00056
0 71135
0 00072
0 01078
0 00022
0 00230
0 00016
0 67635
0 00151
0 04659
0 00064
0 00761
0 00054
0 67480
0 00105
0 04739
0 00053
0 00797
0 00044
S„(k
5C(A:=J)
Error
-0 7S295 0 000292
0 28284
0 00053
0 75254 0 000277
0 28238
0 00045
-0 81528 0 000322
0 23708
0 23636
Error
e
U
rip =48
in
Sc(k
The measured quantities
performed 105 updates for each simulation
We
16
Alg
Model
np
show
we
Comparison of improved (I)
Results for the t-J chain
TABLE 2.6.1:
I
-0 81563
0 000288
hb
u
-1 38017
0 00046
hb
I
-1 37965
0 00037
_
f)
=
reads
Hjr
=
X>S, Si+i + J'S,
•
Sl+2).
(2.6.8)
i
We have
have to
for
implemented this model by the continuous time loop algorithm
use a
freezing (see Fig. 2.2.5);
appear).
of J
Note that
we are
improved
factor
e
we
able to
and
value is
the ratio of the
e ss
a
factor three
in
sign
which
implies
ergodic (and
a
a
relatively
0 0054 ± 0.0007 for
0J
this small. In Table 2 6.2
T/J
=
01 and
depends
on
T/J
=
0.2.
The
the value of the
errors
freezing
probability
weak
=
we
0.1
frustrating
on
20 sites.
show the results
freezing
and the improvement
factor e,
in
this
case
improved
estimators
of the
improved
and the conventional measurements increases
Note that the factor of the
CPU time.
perform better
improvement
as
we
negative sign will
no
The
0 2
errors
measure a
=
not
Even with
here.
(sign)
is
finite
measurements of the sign, for different values of the
estimators
temperature is lowered
of
reliably
algorithm
severe
have obtained
unimproved
improved
diagonal graph segments,
is very
and temperatures of
due to the
optimal
10J'
=
for
otherwise the
problem
The sign
coupling
for
probability e
finite
For this model
the sign
decreases,
of 1.75 leads to
a
as
the
and
the
reduction
38
TABLE 2.6.2: Results for the frustrated
of the
ratio
sign
e
and
improved
for the J-J'
0J
—
and the
Heisenberg model
01 and
(3J
unimproved
=
0.2.
Heisenberg
on
chain.
20 sites for J
In the last
column,
Improved and unimproved
=
we
measurements
10J' for different values of the freezing
show the ratio of the
errors
between the
errors.
e
Improved sign
Error
Unimproved sign
Error
Ratio
0.1
0.1
0.00587
0.00109
0.00651
0.00162
1.48622
0.1
0.2
0.00535
0.00075
0.00570
0.00133
1.76066
0.00536
0.00146
1.74388
T/J
0.1
0.3
0.00581
0.00084
0.1
0.4
0.00479
0.00084
0.00379
0.00134
1.60041
0.1
0.5
0.00444
0.00105
0.00390
0.00148
1.41069
0.1
0.7
0.00614
0.00223
0.00638
0.00263
1.17623
0.2
0.1
0.08953
0.00201
0.08985
0.00243
1.20779
0.2
0.2
0.08681
0.00151
0.08632
0.00195
1.29734
0.2
0.3
0.08933
0.00152
0.08886
0.00202
1.32511
0.2
0.4
0.08387
0.00166
0.08250
0.00222
1.33664
0.2
0.5
0.08564
0.00219
0.08465
0.00256
1.16866
0.2
0.7
0.08290
0.00372
0.08259
0.00400
1.07410
39
1
a=2
1
a=1
t',J'
y\
-o-
Ch—oi=1
2
o—
3
4
*-
x
FIG. 2.6.2: Schematic
rung
a
a
breakup,
calculation for the
the first
are
we
have
performed
the interactions
QMC
on
picture, following
on
bonding
or
breaking
plaquette lattice,
Refs.
coupling
[56,57].
orbital
on a
bond J' and
on
of the order of
a
the
2t
in first
the
where
those
unpaired
=
along
that
t'
In
show the
Fig. 2.6.2,
we
the
A
single
same
hole
For the Trotter-Suzuki
again
we
could be
AJ
=
J, t. In this limit,
=
legs.
we
doped in such
At,
obtained
a
lying bonding
gain
orbital is
of t'
rung also break
simple
are
regime
is
and intuitive
weakly coupled singlet pairs
a
ladder will stay in either
given
to first order
along
the rung and t
bond
J',
one
model
applied unchanged.
This parameter
have
a
where J' and t'
rung, while the rest of the system will remain
unpaired holes
a
unchanged
by
along
the cost
the ladder
but their kinetic energy
while two holes bound
E(0)
in the ladder is of the order of
on a
is the energy of the
single
given by
x
=
2
3^
In
rung have
an
Fig.
region
a
2.6.4
E(0)
energy of
corresponding Heisenberg
and thus two of the rungs will stay in
Curie law is then
so
loop algorithm
therefore expect that the two holes in the parameter
remain
[55].
we
gain
At2/J' (Fig. 2.6.3(c)).
(Fig. 2.6.3(d)),
order,
are
kinetic energy
Hence the total energy of two
It'
our
The undoped ladder consists of
The energy of the lower
direction. Two holes
—
and
limit J' 3>
(Fig. 2.6.3(a)).
antibonding
a
calculations for t-J ladder systems.
each rung, and J and t
the rungs
(Fig. 2.6.3(b)).
only
single chain,
a
of the t-J ladder
simulations with two holes and J'
dominated by the strong
formed
magnetic susceptibility
split the Hamiltonian into bond terms,
checkerboardlike
We have
is
couplings along
J and t.
are
graphical representation.
These
of
and L rungs. The
legs
example of the loop algorithm for the t-J model beyond
results of
on a
two
Two-leg t-J ladder
an
show
J' and *', those along the ladder direction
are
2.6.3
As
picture of the t-J ladder with
ladder.
+ 2 J'
E(0)
We
+
—
J'
can
considered in this simulation
doublet state. The low-temperature
we
show
our
results,
which
are
in
40
—
(a)
T
t T
\\ t
-
t
(b)
—
—
—
(c)
FIG. 2.6.3:
into the
E(0)
holes
+
bonding
J'
on
t'
-
or
The
different rungs, with
an
1
t
case
(c)
o1
t
—
O1
^
t
—
t
t
tj tj
_
-
t
I
t
—
t
t
tj i
of the
low-lying
with
ground-state energy £(0). (b) A single hole goes either
anti-bondmg orbital,
t in first order,
t
t
-
tj
_
—
t
t
undoped
t
t
—
t
\
t
1
t
Graphical representation
J,t. (a)
-
t
t
_
—
tj
(d)
limit J' >
—
t
II
?
-
a
the energy of the ladder with
Two holes
energy
states of the t-J ladder in the
E(0)
on a
+
2J'
single
—
2t'
rung, with
—
2t.
a
£(())
hole
+
in
a
strong coupling
bonding
orbital
is
J' in first order, (d) Two
41
pf
T/f
(a) Magnetic susceptibility
FIG. 2.6.4: t-J ladder:
holes
16 rungs, for temperatures down to T
on
average
sign
for the
doped
=
of the t-J ladder with J'
t'/16 compared
agreement with the expected limit of 4Tx/site
16 rungs
as
-»
A
0.
Then two holes form
is much
sign problem
Three-leg
2.6.4
a
more
physically
bound state in their
worse
in this
ently
(*', J')
At low hole
tinger liquid
with
small
an
are
4J
=
=
it and two
(np
case
32). (b)
=
1/16
for two unpaired holes and
region
is J'
J
=
=
t/3
=
t'/i.
ground-state (Fig. 2.6.3(d)). Unfortunately
the
region.
odd number of
doping,
the
an even
equal:
three-leg
t
=
the
In this
legs (t, J)
t' and J
=
legs
paragraph
and the
in the two
and
we
differ¬
will concentrate
couplings perpendic¬
J'.
parity under reflection about the
Luttinger liquid
completely
behave
ladder consists of two components:
insulating (i.e. undoped) spin liquid phase
all holes enter the
number of
legs [56,58,59].
couplings along
assumed to be
in the channel with odd
doping,
remains
an
t-J ladder. The
three-leg
ular to it
t'
t-J ladder
than those with
the
=
realistic parameter
Several studies show that the ladders with
on
=
undoped
case.
excellent
T
to the
repel
a
center
even-parity
each
other,
conducting
Lut-
leg, coexisting
channels
while the
[59].
At
spin liquid
undoped.
The energy gap A between the
have been calculated
by
exact
odd-parity
channel and the even-parity spin
diagonalization
liquid
states
of very small ladders of only 3x6 sites in
42
[59]. Using
Ref.
the
QMC loop algorithm
odd and even-parity channels for much
doped with
ladders of 3x40 sites
along
the ladder and set
fit
down to
J/t
one
periodic boundary
hole. We have assumed
0.5. With this choice of parameters,
=
Figure
2.6.5 shows the
of the
three-leg
ladder.
ladder. At
are
The
also have
density
ncenter ieg of the
at T
odd-parity
ncenter leg
=
0 is
only
nonvanishing weight
higher temperatures,
t/7,
T >
higher-lying even-parity
and
ncenter leg
ncenter ieg is
estimated from the MC data
The two
[59].
parity.
lowest-lying
The states
two different transverse
is determined
gets
that the
an
but
only
3
approximation
change qualitatively
can
as
of the
on
a
length
by
one
of the outer
[60].
odd-parity
channel
states
as
parity,
2.6.5 shows that at
This is caused
of ncenter leg below
T is lowered.
doped
with
one
$£jjj,s
and
the center
vector
k)
^trSJs
leg
~
0.45
*n
a
can
first
by
admixture of
T/t
=
0.5 shows
The gap A
can
[60].
are
seen as
of odd
Bloch
approximation.
a
value
diagonalization
nj£?nter ("center)
of the ladder gets
be
's
Fig.
shown in
are
band
for all states in the odd
and equal to
From the exact
n^ter
hole
(second-lowest) lying
have cosine forms and
independent
k
0.2 and
^
the
be described
of
on
3x8 ladder
a
1/3. Figure
1/3.
than
the
over
As these states all have odd
sharp drop
8 ladder
wave
<t>°?£is (0SS)-
n°^ter
x
functions
wave
distributed
dominantly
0.2 for
~
the center
temperature, however, the
zero
smaller than
larger
to the lowest-
approximately
(independent
only by
estimate:
"center leg
a
uniformly
on
be
simple two-band low-energy model:
a
of the hole to be located
band is constant
sees
bands of
belonging
These two bands
probability
using
hole to be located
states with the hole in the
clearly
channel states. The
At
channel and is
decreasing weight of the even-parity channel
Ref.
1/3.
in the thermal average.
suppressed
ncenter leg is
single
the hole is
high temperatures
low, but finite-temperatures other
density
the
ladder,
negligible.
Therefore the density nCenter leg is equal to
At very
the
40
>
probability
hole is in the lowest state of the
legs.
of the
sign of
width, the number of doped holes, and the fraction J/t. In the temperature range
considered, finite-size effects for L
leg
reach temperatures
we
length
t-J
conditions
is smaller than 0.01. Note that the
sign
the MC simulations of these t-J ladders is not sensitive to the
to their
three-leg
We have considered
ladders.
larger
7. Below this temperature the
=
able to estimate the energy gap A between
we are
a
3
x
valid within
(even)
waves
of
Then the
(even) parity
nj^er ("center)
of
5 of
which
8 ladder
10%,
and
one
one
Since this situation is not supposed to
longer,
this two-band model also in the
the
low-temperature
case
of
a
long
behavior of
ladder. Therefore,
43
0.4
a?
J8
J
0.3
l
J/t = 0.5
0.2'
l
1
'
T/t
Temperature dependence of
FIG. 2.6.5:
^center leg of
a
three-leg
a
t-J ladder
the
probability
doped with
one
(diamond)
3x40 cluster and the zero-temperature value
diagonalization (Ref. [60]).
considering
the
density
of the hole to be located
hole. The filled circles
are
is calculated for
The dashed line shows the fit calculated
by
a
the
on
3x8 cluster
a
low-energy
of states, the expectation value of racenter leg
can
A
This fit is shown in
Fig.
«
0.25(5)*
=
even
parity band in
0.5(1)J
(J/t
=
to those of
physically reasonable
is
bigger
a
(w°*fter, reenter
3
x
be calculated
range, the gap A
than that of the 3
x
6
even
hardly changes.
ladder, obtained
one
by
only.
At
low-energy
are
The value obtained for A
where it
was
low-energy
used for
fitting
model described above is not
diagonalization (A
the MC results.
so
an
higher temper¬
=
bands)
precise
are
varied in
a
(Eq. 2.6.9)
0.15i) [59].
This difference may result either from strong finite-size effects in the small clusters
the fact that the
get
a
model results to
if these paramaters
exact
can
as
ladder:
and tne bandwidths of the two
8 ladder. But
exact
(2.6.9)
2.6.5 and is reasonable at low temperatures
the MC data all other parameters
equal
long
using
0.5).
atures other bands have to be considered too. For this fit of the
assumed to be
a
leg
data for
two-band model.
function of the temperature T and compared to the MC results. Prom this
estimate of the gap A between the odd and
the center
QMC loop
or
from
in the temperature range
44
2.7
Discussion
In this
chapter
discussed the
with
a
we
use
have introduced
the
especially
of improved estimators,
for simulations of t-J type models and
the
of
use
improved
estimators for models
sign problem.
We found many
local
loop algorithm
a
MC
updating
loop algorithm
The
algorithms.
lattice, without the introduction of
algorithm
it is
possible
to
perform
of the autocorrelation time
studied, the reduction
or
study
much
magnetization
in
loop algorithm
is
free
This
bigger systems
huge
at much lower
previous
grand
loop
With the
canonical
ensemble,
natural way.
the
great reduction
2.6, where for the parameters
This
magnitude.
is up to four orders of
or
a
to
for any geometry of
certainly
in Sec.
examples
We have shown
and lower temperatures.
larger systems
allows to
t.
fully ergodic
updating procedure.
any additional
important improvement of the
The most
is
simulations in the canonical
(constant winding number)
with fixed
algorithm compared
for the loop
significant improvements
gain
will increase further for
reduction of the autocorrelation times
temperatures than before with the
same
amount of computer time.
The
loop algorithm
for the t-J model
ferent additional terms
loops.
For
rithm is
some new
can
be
terms it
easily adapted
can
also be extended to various other models. Dif¬
incorporated easily
might
into
be favorable to
to other lattice
geometries.
change
This
underlying geometry
of the lattice in the simulation and
terms in the Trotter
decomposition.
With the
limit At
->
loop algorithm
0
(see
section
it is also
possible
2.5). Therefore,
to
an
the
can
flipping probability
weights v(Gp).
be done
The
of the
loop algo¬
simply by changing
simulations in the continuous time
eliminate the
errors
due to the finite time
steps At without making simulations for different values of At and extrapolating to At
afterwards.
Again
we can save
a
large
amount of
the
introducing corresponding additional
perform
we can
overall
computer time compared
=
0
to discrete time
simulations.
The
use
of
improved
introduction of
tigation
variance
improved
of many
presented
estimators further reduces the variance of measured
new
systems with this
t-J ladders and
by
the
estimators also for models with
improved
a
frustrated Heisenberg model
estimators
sign problem
a
method, e.g., frustrated spin
depends
very much
as
examples.
on
quantities.
The
allows the inves¬
problems.
We have
The reduction of the
the model and the observable
45
under consideration. For the systems
we
to reduce the variance of the observables
Although
we can
simulate much
have studied
by
here,
the
improved
estimators
helped
about one-third.
bigger systems
much faster than before with these
new
techniques, the sign problem still remains and limits the application of the loop algorithm
systems where the negative sign problem is
ladder systems in Sec. 2.6.
many
new
problems
tackled due to the
that
not too
severe.
to
We have shown examples of t-J
Despite this drawback for higher-dimensional fermion systems,
are
advantages
far
beyond
of these
the scope of previous local MC techniques
new
simulation
techniques.
can
be
46
47
Random ferromagnetic-antiferromagnetic
3.
Heisenberg chains
Introduction
3.1
One-dimensional quantum spin chains
variety
rich
of
typical examples of many-body systems with
are
they
Over many decades
physical properties.
a
very
have attracted much interest in
theory and have motivated the development of various calculation schemes, both analytical
and numerical. The number of real
is
A few
growing.
such
compounds
possible
as
are
Sr3MPt06 (M
these systems
were
are
investigated
supposed
like NENP and NINO
organic systems
=
Ni,
as
to be
an
Zn) [63].
Cu and
Recently
realizations of Haldane gap systems.
Sr2Cu03
and
examples
compounds containing quasi-one-dimensional spin systems
example of
regular
chains
Already
or
a
the
low-energy
properties. A
discovered
by Nguyen
and
chain
pure
was
recently
compound SrsCuPtOg
provided by
Pt-ions
is
a
each Cu-ion.
along
chains.
forms
In this
compound
peculiar
(Jf
<
composition
example of
are
a
0)
and AF
correlation among the bonds in the
a
(J^
>
0),
which
in sequences of
numbers, since each Ir-ion makes such bonds with its
generic
and coworkers
[22]
model to
who
study
neglected
the
properties
of such
a
a
spin 1/2 is
are
that FM bonds
system
The
spinless
spin 1/2, then the system
a
Cu-ions. A
[21].
alloy SrsCuPti-^Ir^Oe
sense
have
can
with the
distributed. There is
even
All
disordered spin
chain where
alternating
Ir which carries
chain. Therefore the random
system with two types of bonds, FM
(Ca)2V03
Loye: the alloy SrsCuPti-^Ir^Oe
the Cu-ions
as
However, in practice disorder
antiferromagnetic (AF) spin
replaced by
If all Pt is
ferromagnetic (FM) spin
an
zur
as
valence bond system.
little disorder in the
systems.
considerable influence
on
resonating
inorganic
or
investigated
are
also ladder systems such
ladders.
must occur in most of these
[61,62]
Some of them
was
a
considered
=
spin
randomly
always
two
the correlation among the bonds and used J
is
occur
neighboring
by
Furusaki
| Jf\
—
\ Ja\-
48
spin system has the following
This quantum
H
=
neighbor Heisenberg
nearest
Hamiltonian
^2jt8i-8i+i,
(3.1.1)
i
probability
with the bond
distribution
P(Ji)
where 0 < p <
AF
or
FM
The
pS(Ji
=
+
JF)
(1
+
-
p)S(Jt
-
1, and <5 is the Kronecker delta function. For
JA),
p
0 and 1
=
we
have
purely
a
spin chain, respectively.
analysis
by high-temperature expansion [22]
of this model
(RSRG)
malization group
[20] suggests
scheme
the
First
they align
among the segments is weak.
within the segments of
purely
FM
Thus, the spins in each segment
of freedom which is rather
in the
large
by
a
real space
case
kBT
to
or
as
an
of FM bonds and 5
effective
=
0
coupling
single spin
1/2
or
independent
as
are
inde¬
J the spins start
AF bonds. The
create
bonds. At intermediate temperatures these effective spins behave
~
renor-
regimes
behave
spins essentially
pendent degrees of freedom. If the temperature is lowered down
to correlate.
and
that three different temperature
high-temperature regime
present in this system. In the
degree
(3.1.2)
for AF
due to the
thermal fluctuations. However, they begin to correlate at lower temperatures. The interme¬
diate and
low-temperature regimes
continuous random bond distribution
The limit of T
0
—>
lowered,
spins
on
a
[20,64].
growing
the average, which form
spins
average number of
exponent
a ss
0.22
was
of the clusters behave
in
as
perature regimes,
we
a
leading
C
=
S(S
+
=
l)/3
=
random
as
will
we
number of
large spin
<*
spin
with
C/T
a
T
model with
basically
a
spin sizes (effective spin sizes).
and coworkers
briefly
spins
by
means
[20,23].
of
a
RSRG
review their main results. As the
is correlated in clusters
S. The spin size S scales with S
essentially independent
as n oc
Because the
under thermal
dependent Curie
consisting
<x
n1/2
of
n
and the
T~2a. The scaling
large
effective spins S
fluctuations,
one
expects
a
constant C.
conclude that, in principle, this system exhibits three different tem¬
each with its
j??,
a
Here
random
a
cluster scales with the temperature
high-temperature regime (kBT
to x
well
determined numerically
Curie-like susceptibility x
From this result
as
by
analyzed by Westerberg
was
scheme for this type of model
temperature T is
described
are
>
where hb
1/4.
own
J)
=
Curie-like susceptibility and Curie constant.
the Curie-behavior
2mc
's
tne
In the intermediate
originates
Bohr-magneton
from the 5
=
In the
1/2 spins
and the Curie constant is
temperature regime (kBT
~
J)
the effective
49
spins of the segments give
regime
with
Curie behavior
a new
constants should be
decreasing
The
or
p-dependent
a
a
at very
occurs
decreasing
sequence
as
low-temperatures. Obviously
the number of available
separation into three regimes is expected
degrees
of freedom.
lower temperature. In the
independent large spins S
The
ined
by
crossover
the
to be visible in the
-»
0 has been
spin scaling regime
more
scaling regime
high-
in
by demonstrating
specific
not
for
specific heat
as
too. Peak-
they
are
sign
of
J and the second at
~
oc
T2a_1| lnT| [20].
regime
has been
however, only
effective Hamiltonian with
an
yet been analyzed
the
exam¬
limiting
broad
a
ran¬
are
so
know that the effective
we
hard to observe
regime
far. In this
that the intermediate temperature
heat and the
susceptibility by
can
by experiment.
be
chapter
regime
investigated
we
would like
is well-defined
exact treatment of the
an
Hamiltonian.
The correlations among the FM bonds in
realistic model where the FM bonds
uncorrelated model in the
SrsCuPti-^Ir^Og
investigate
always
the distribution of the Pt-ions in
investigations
an
are
occur
magnetic susceptibility
accessible temperature range. These data will
which is
of freedom is
which is in contrast to the discrete distribution of
temperatures, which
which has not yet been examined. We will
Our
cv/T
from the intermediate to the low-temperature
and observable in both the
original
scaling
temperatures, the assumption of
SrsCuPti-^Ir^Oe- From these results
easily by experiment, but has
to close this gap
at very low
kgT
For low temperatures,
investigated
starts at very low
crossover
occurs near
and intermediate temperature
couplings J% [20,23],
spin couplings
The actual
boundary
of correlated clusters leads to
between the
dom distribution of
the initial
One
high-temperature expansion.
behavior for T
a
the three Curie
degrees
shoulder-like structures indicate the boundaries between the regimes
some
to
a crossover
lowering temperature.
with
correlation of
a
Finally
Curie constant Ceg.
based
the
are
implications
pairwise and
and the
provide
a
another
important aspect
of these correlations in
study
specific heat
its differences to the
in
an
experimentally
sensitive test for the randomness of
SrsCuPti-^Ir^Og.
on
Monte Carlo
excellent method to simulate
(MC)
accurately
the
simulations
by
the
thermodynamics
loop algorithm [10],
of large
spin systems.
50
—
IU
i"
i
'
\
104
*2
%
.
"a
'S.
103
V
u
o>
(0
b,
10*
"o
n
E
3
.
N
101
;
i*
c
af segments
10°
—
Ofm segments
ideal distribution
—
X
X
n"1
.
5
10
15
segment length
FIG. 3.2.1: Statistic of the total distribution of the
in the
3.2
The
configurations
we
curve
shows the ideal distribution.
Numerical methods
QMC loop algorithm [10]
decomposition [9]
putation
is
of various
finite temperature method based
of unfrustrated
on
the Trotter-Suzuki
spin systems. It allows the direct and
thermodynamic observables,
the internal energy u, without
Metropolis
a
of the partition function Z and is ideally suited for the calculation of
thermodynamic properties
or
segments with la fm (squares) and af (circles) bonds
have used for the simulations. The dashed
world-line
introducing
algorithm [51],
the
e.g.
any
the uniform
approximations.
loop-algorithm
exact
magnetic susceptibility
further reduction of the variance. This enables
us
to
does not suffer from
investigate
x
In contrast to the classical
prohibitively
long auto-correlation times. Additionally, the introduction of improved estimators
a
com¬
much
[53] gives
bigger problems than
with previous MC methods.
For the calculation of the random bond
of different random bond
periodic boundary
length ls
in
range from
our
configurations,
conditions. In
Fig.
=
1/60
to
T/J
=
numbers between 20 and 120 and
each
3.2.1
we
10.
we
have considered up to 400
consisting
performed
Unless otherwise
extrapolated
on
the
a
chain of L
=
time-steps
x
samples
400 sites with
(AF) segments
of
calculations in the temperature
mentioned,
to Trotter
104 multi-cluster updates for thermalization, followed by 2
updates for measurements, depending
of
show the distribution of FM
We have
configuration samples.
T/J
models,
10s
we
have used Trotter
AtJ
up to 2
x
=
0.
We made
106 multi-cluster
temperature T and the Trotter-number.
51
The value of
from the MC simulation:
Oj coming
by
magnetization M,
efficiency
Due to the
accurately enough
internal energy
a
of
configuration,
the average
have measured
susceptibility x
numerically
heat
configurations
directly
for each
measure
as
is determined
error
the internal
configuration.
the internal energy
the first derivative of the
has been taken. In order to
configurations
have taken into account the variance of the observable
we
and the
all
over
of the observable for
error
the error-bars for the observables of
general,
and tne
have been able to
we
specific
Y^jLi Oji
we
and the uniform
to calculate the
correct estimate of the errors,
for the set of
M~l
=
In this way
MC algorithm,
our
O
~
M successive measurements
over
with respect to the temperature T. After the calculation of all observables for
u
each individual
get
{))
the variance of the measurements.
energy u, the total
u
observable O is estimated by averaging
an
variance of the observables for the set of
a
single configuration
configurations,
a
single configuration.
are
In
much smaller than the
and therefore
neglect
we can
them
completely.
Results
3.3
We have studied two different models.
model with H
=
Yli^i^i' &i+i
P(Ji)
where 0 < p < 1 and
a
FM
as
an
|Jf|=|J,4|
AF bond.
[20, 22,64] .In
methods
First,
and trie bond
=
pS(Jt
JF)
+
for the
case
we
have
investigated
probability
+
of p
(1
=
the rest of this
chapter
we
-
random bond
Ja),
0.5, i.e., the
This model has been studied in
generic
distribution
p)S(Ji
-
a
a
same
(3.3.1)
probability
number of papers
for
using
finding
different
will call this model the "unconstrained"
model.
In the real
previously
FM bonds
SrsCuPti-^IrxOe alloys,
mentioned in the
are
there
are some
additional restrictions.
couplings,
Ir-ions
we
have
corresponds
systems
as
an
estimate
generated
to the
a
we
have
introduction, by replacing the spinless Pt-ions with Ir-ions,
created between the
spin 1/2 carrying
gives |J>|
=
probability
of
couplings
4|Jyi|. By generating
configurations equivalent
finding
a
the "constrained" model in the
a
are
stronger than the
random sequence of Pt-
to and Ir substitution
FM bond with p
following
two
Ir-ions and its neighboring Cu-ions. In
addition to this pairwise correlation of FM bonds, the FM
AF
As
=
text of this
2/3.
of
x
=
an
0.5, which
We will refer to these
chapter.
52
Susceptibility
3.3.1
Let
independent single spins
have
we
susceptibility Xci(T)
the
//2/(4A;bT)
=
total spin
possible
(5
0 for
=
8E?s
ex
J/ls
for AF and
intermediate temperature
intermediate
on an
spectra in the
SEps
Jjl2s
oc
Ceff
=
regime
the segments
segment, i.e.,
seen
energy-scale (< SEps)
I22].
can
the total
always
spin
of
a
=
1/2
for
Curie-law for
a
lowered,
collectively
an even
an
a
the
lowest
number
parallel).
all spins
J,
finite size gap
ls),
Due
SE^s
the excitations remain localized
In this
uncoupled
effective spins
second Curie-law of the
susceptibility
behave
as
effective Curie-constant Ceff
depending
jU2Ceff
length
'
kBT
of
a
denotes the effective
FM/AF segment.
have to be counted to the AF
FM segment with
=
0.5 and S
a
=
length
1/2
spin of the FM/AF
The
spins
segment and
of ls bonds is S^ot
one
on
obtains
Ceff
=
the
seg¬
boundary
not to the FM
=
1/8
&(h
for
~
a
1) [22].
chain of
length.
Fig.
3.3.1
we
show
xJT
as
function of the temperature, such that the Curie-law
a
be
can
seen as a
plateau.
From
our
QMC
simulations
0.138 ± 0.003, whereas the value of the effective constant Ceg obtained
of the bond distribution is
analysis
graph.
in
=
(Sfm/af)
where
*ne average
1S
behavior of the effective spins
Ceff
S
=
S>
spins
For the unconstrained model with p
infinite
essentially
with
FM and AF segment
In
or
segments),
be
g|f)+|f^)
ment, and (tIpmiaf)
a
form the
they
AF/FM spin segments (the
for FM
X
of
and
chain,
collectively largest spin (by aligning
the average size of the effective
with
FM
As the temperature T is
oo.
In the AF segments,
Seff. These uncoupled effective spins
on
-»
or
segments and the interactions among different segments remain very weak.
in the
X
for T
odd number of bonds ls
an
while the FM segments form the
to the misfit of the discrete
high-temperature regime ksT
in the uniform AF
as
individual spins start to correlate.
is
In the
consider the unconstrained model first.
us
Ceg
=
0.1252, and it is shown by
Previous results by the transfer matrix method and
[22]. However,
obtained
Ceg
from the
extrapolation of
=
by considering
0.13
to which
to be counted in the
the Pade
segment
statistical
a
in the
spin
analysis:
on
In
There is also
the
spins
statistical
dashed line in the
some
a
possible
error
stems
range for modification
the border between FM and AF
general,
get
high-temperature expansions
high-temperature expansion,
approximants.
a
by
we
segments has
at the border between these
53
kBT/JA
FIG.
are
3.3.1:
Figure of
the uniform
magnetic susceptibility
the results of the MC simulations for the
SrsCuPti-iIr^Oe (diamonds).
Heisenberg
chain.
For
comparison
The Curie-law behavior
results of the statistical
analysis,
so are
can
generic
we
be
temperature T.
(squares)
The
symbols
and the realistic model of
also show the results for the uniform fm and af
seen
the limits of
x times
model
by horizontal lines, the dashed lines show the
%T for
T —> 0.
54
segments tend to be included in the AF segments, but if
FM
segment,
increase of
some
Ceg.
shows that if
boundary spins
of these
spins
count the
bonds next to FM segments
get CejF
=
on
the border of AF segments
for AF segments
segments consisting of three and less bonds
consisting
QMC
our
consisting of
find Ceg
we
of five and
=
short
an
calculations
six and
we
more
rather
bonds next to FM
more
instead,
0.138
a
segment, resulting in
of three and less bonds to the FM segment,
consisting
similarly
0.131 and
AF segment is next to
to the FM
analysis of the bond distribution used in
A detailed
we
long
a
couple
rather
in agreement with
QMC results.
our
The temperature range of the
region
of this
of the
spin segments SEpg.
regime
compares
of
regime
plateau,
extension of this Curie-law
uncoupled
it starts at
favorably
ksT
effective spins is determined
0.16J and ends at ksT
a
model with He$
The
start to correlate. These interactions
spins
couplings J*ff
Seflf is random
]Tt ^eff^eff ^eff1'
=
as
their interactions
'
are
well.
J|ff
where the
sign
random in both
[20],
Following Ref.
sum
<
more
0)
or
spins
minimal
is very broad. Hence two
S\s
as
depends
the
~
Fig.
0) spin
now
described
taken
temperature
3.3.2
we
have
a
with the
a new
a
power law
0.21 ± 0.01 of Ref.
plotted JyT
as
a
where so is the
=
s0 +
function of
groundstate expectation
—
a
Heisenberg
spins 5|ff
largest
with
a
energy gap
As
more
and
original spins
original spins
T~2a,
and
of their maximal
effective spin 5efj.
~
spins
=
in
a
cluster
0.22±0.01
in
n
[22],
[23].
low temperature behavior has been calculated in Ref.
XT
n
gap
spin segments.
the
over
groundstate, consisting
The average number of
T and scales with
=
by
the
relevant and
and the size of the effective
neighboring spins
state and form
n1/2 (random walk).
consistent with the result
In
>
again
freeze out, the effective spin size Seg scales with the number of
the cluster
on
(Jefj
is
spins become
the distribution of the effective
to their first excited state will be locked into their
(Jeff
i
are
magnitude,
and
by
0.06J. The
simple estimate given by the finite size
to the
At very low temperatures, interactions among the effective
the effective
~
+
Ta/J
for different values of
a.
The
[23]:
0(T3a),
value of the total spin per site
(3.3.2)
55
(a) generic model, p=0.5
0.14
r-
^
0.10
~3
©
©cc=0.18
Q
Qa=0.22
<S
0a=O.26
A—Ao=0.3
s„
0.06
0.0
0.2
0.4
0.6
T"/J4
0.35
-
0.30
-
(b) Pt-lr,
x=0.5
&ar
x
q—ea=o.i8
<
0.25
B
Ha=0.22
$
0a=O.26
-
.
A—Aa=0.3
0.20
0.0
0.8
0.6
0.4
0.2
1.0
r/jA
FIG. 3.3.2: Low temperature
(b).
The
scaling
behavior of
low-temperature scaling regime
denote different values of the
yT for the generic model (a) and the realistic model
is characterized
scaling exponent
a.
by xT
=
so
+T"/j
+
0(T3a).
The symbols
56
52+i
where
=
—
4sgn( Jt)
Si
and
=
beginning
1. The
of the low-temperature
[23],
estimated to start at T < 0 05 Jo for the effective Hamiltonian
of a broad random distribution of couplings. This distribution of
effective
couplings Jeg
order of
magnitude
original couplings [22].
the correct zero-temperature value so and
reached for T
temperatures prevents
Next
In this
we
us
concentrate
case we
Starting
Unfortunately,
0.02J.
=
from the
same
due to the stronger
increase of
as
in the
If
an
we
on
critical
reaching temperatures
the
susceptibility
high temperature
by aligning parallel
coupling \Jp\
JxT first
=
4\Ja\
Then the AF
spins
calculate the average effective
at
as
low
=
higher
obtain
a
Ceff
=
as
neglect
can
be obtained
segments
we
would obtain
remain
get Ceff
«
by
assuming that
uncoupled
unconstrained
model,
large
the spins
due to their weakei
FM and small AI-
causing
si
an increase
gives Ceff
=
0.34 if
couple
we
to the FM
count the
spins
t
peak
near
generic
at low
case, the FM
large spins
ksT
previous
*ne
m
completely
are
bonds,
leads to
plateau
a
J.
=
we
case,
a
obtain
lower bound
correlated.
The
is very important, if
An upper bound of
C^*
=
we
0.40 ± 0.02
the FM segments correlate and the AF
spin
6",rr
QMC
our
data in
Fig.
3.3.2
segment statistic is larger than for the
coupling slienglli
of ol
are
s
I
lie
twolold
lie border of
segment
on
in
yield
SraCuPti-^Ir^Og.
too, but instead of
SrsCuPti-^Ir^Oe
girients and make
on
in the
as
couplings From
Ct,n of the
for the unconstrained model, spins
FM segments tend to
more
only
QMC simulation
0.33 ± 0.017. This value is
0 2731 instead
since the difference in the
subtle. The effects
same as
~
0.36. The deviation from
the boundaries of
more
Ceff
a
has not been
temperatures T than the AF
broad
an
that
of the FM and AF interactions.
\J/T
{Spm/af)
spins
we can see
scaling regime
the
about
are
0 22 does not yet
The formation of these
effect of the constraint among the FM bonds in
we
=
to the
T < 0.005 J.
as
strength
of the Curie constant, where the spins within the segments
that
in
start to correlate
we
estimate of the effective Gunc tonstant
x/L
4J
=
down
a
of the constrained model of
limit
of the unconstrained model
case
slowing
have to take into account the different
bonds start to correlate
an
from
scaling exponent
conclude that the
we
couplings
Fig. 3.3.2,
In
is
Jo is the maximum
couplings corresponds
of the intermediate temperature regime. These
smaller than the
extrapolation of the data points with the
linear
where
scaling regime
large
A del.nled
the bender ol A I''
favors quantum effects
simple
m
I
Ins
ca.se:
the first is the
AK segments next to
analysis
on
statistical estimate
a
short
of I lie spin distubution
segments consisting of five and
bonds next to FM segments consisting of four and less bonds to the FM segment, and
57
Ceff
=
0.345 for AF segments
consisting
The second
of three and less bonds.
of four and
reason
for
more
a
completely
spin-1/2 degrees
for the
Additionally
as
previous
uncoupled
form
a
case
it
effective
investigate
strained
model,
Cu- ions.
In this case,
are
is taken
sum
into four terms
i, while
£sPi„s<^>
culate the
over
we
=
2
use
using
probability
the
fa^iSiSj)
value
and that each
(SiSi+n)
following
=
+
limit
Ceg
original
some
yTJ
(1
distribution
same
we can see
1/4;
=
again
but in contrast
0.33 ± 0.017 of the
=
how
regime
start to correlate and
effective spins is established.
Eq. (3.3.2) for the
behavior of
Eq. (3.3.3).
notation:
X^iSiS;)
missing
Let
us
consider
+
x)L(Sf)
we
+
ground
state expec¬
^2spias{SiSj),
Sspins(^^j)
con¬
chain of L
a
we
where the
divide the
enumerate all Cu-ions with the in¬
Ir-ion
=
=
=
+
Ej^S,)
+
E;<j'<^>)-
into account that
changes
the
sign
an
Now
Ir-ion exists
of the next
we can
only
^
spin:
(3.3.4)
'
\ £ *ma *r-m(-ir-mQ
=
-
^
cal-
with the
\j^xm(l-xY-m{-lY-m(^\=\(2x-lT
'
f (2* -1)"
"
(3.3.5)
jE^m(i-xr1-m(-ir1-m("~1) J(2a;-ir-1
(3.3.6)
fEa;m(1-K)n"1"m(-1)n"1~mQ
(3-3-7)
=
m=0
<5;%r>
23. Here
For the evaluation of
{SiSj) by taking
m=0
<$$+»>
=
Ir-ions in the chain and the
=
yet
have demonstrated
simulation for the
2\J^\
independent
((Sspins^)2)
m=0
<5^>
=
as we
not
the index i for the Ir-ion between the Cu-ions i and i + 1 and write
expectation
x
spin
is:
all Cu- and Ir-ions.
sum
QMC
low-temperature scaling
the
spin 1/2 carrying
Lx
tation value of the total
a
Ceg
are
before.
spin segments already
some
have to calculate so from
we
There
by
is that due to the
of the AF segments
further increase of
reaches the lower bound of
never
spins.
some
high-temperature
collective spin before the regime of
In order to
dex
a
segments consisting
Ceff
AF segments still consist of
coupling strength \Jp\
increases from the
susceptibility \T3
to the
rise to
long
have also determined Ceff
we
above but with different
the
of
freedom, giving
the
completely uncorrelated AF segments
of
case
of
especially
Thus
correlated.
value of
larger
relatively weaker AF couplings than the FM couplings,
a
bonds next to
^
'
=
i(2^-1)""1-
58
Prom this
we
obtain
2.
L(l+p){S?}
£(2z
1)»
-
+
.n=\
gives
Fig.
In
in the
3.3.2
extrapolate
our
£*(2x
groundstate
we can see
=
((Espins Sif) / (ZkB{l
with the correct
peak
below
fcjjT/J
expected
(as
for the uniform
specific
chain for
on
~
e.g. the
a
spin
The
In the
second
or
2/9
=
get for
a rj
0.222222....
»
so is too small if
0.22. Thus
we
we
conclude
1/100.
<
specific
In the
one
AF
heat
Heisenberg systems,
spins
start to correlate.
S
original
data, the
=
1/2 spins
one
broad
Two
peaks
start to correlate
where the segments of the effective
Curie-plateau ksT/J
large
inset,
of the different
and broad peak is the
one
spins
regimes
clearly
can
1/20).
~
see
signature
In
Fig.
3.3.3
we
spins
show
the second
of the correlation of the
peak
for the unconstrained model.
and could not be
scaling regime
at very low
investigated
at low
temperatures of
This confirms the clear
with the
previous methods,
temperatures T, the assumption of independent large
spins S of correlated clusters [22] leads
T-2a\ lnT|.
By
/
high-temperature expansion.
In the
we can
l)-^
cy of the random bond systems and of the uniform AF and FM
the correlation of the effective
separation
and
the end of the
(at
heat per
spins.
-
regimes should be visible by peak-like structures,
1 appears, where the individual
systems),
comparison.
individual
we
the unconstrained model first.
in the random system. One where the
are
the
x)L)
+
T/Ja
start to correlate. In the uniform FM
start to correlate
J>(2*
heat
concentrate
spins
+
ra=l
scaling exponent
starts below
between different temperature
where the
I)""1
-
that also for this model the value
linearly
data
s0
Specific
crossover
(]>>*
x
(3.3.8)
3.3.2
us
+
L,
low-temperature scaling regime
let
1)"
\7i=l
that the
Again,
-
n=0
=
which
+
From this
one
not reach such low
the relation
finds
cv/T
/0 dTcy/T,
entropy per site
T"2"-1! lnT|.
oc
temperatures
to the
as
we can see
But
to determine the
from the
a
oc
ln(2S(T)
for the magnetic
as
scaling exponent
divergence
of
cy/T
as
T
a
-¥
+
l)/n(T)
oc
susceptibility,
from
our
0 that
a
data.
large
fraction of the entropy is at very low temperatures, due to the broad spectrum of energy-scales
59
FIG. 3.3.3: The
model.
The
For
specific
comparison,
symbols
are
heat per
we
spin
cv
of the
generic
random bond model and the
have also shown the results of the uniform fm and af
the results of the MC simulation for the
generic
SrsCuPti-zIrsOe model (diamonds), dashed and dotted lines
expansion (HTE). In the inset,
one can see
the
peak
in the
are
model
(squares)
the results of the
specific heat
of the
Sr3CuPti_iIrxC>6
Heisenberg
chain.
and the realistic
high-temperature
generic model and
cusp for the realistic model which indicate the onset of correlations among the effective
spins.
a
60
KBT/JA
FIG. 3.3.4: The
results,
specific
The
bond systems.
area
heat per
of the effective
entropy S. If
Only
for the
More
model of
seen
«
to the
T
«
model is
we
always
temperatures,
crossover
by
the
a
a
to the
entropy S.
The
are
in
perfect agreement
are
the MC
can see
that the
symbols
high-temperature expansion (HTE). One
down to ksT
~
0.2J, then the
divergent behavior.
show
cv/T,
Fig.
the
3.3.4 in
smaller than
we can see a
sharp
area
more
cv/T
below the
detail,
corresponds
we can see
of the uniform AF
crossover
is due to the very different
curve
to the
to the
that the value of
Heisenberg
chain.
divergent low-temperature
energy-scales of the effective spins and
high-temperature expansion.
important for comparisons with experiments is the specific heat of the constrained
SrgCuPti-^IrxOe- Here,
due to the different
T/J
Fig. 3.3.4
to
consider the data in
generic
This
not be
can
we
at very low
behavior.
In
spins.
crossover
by the temperature T for the uniform and random
corresponds
results
high-temperature expansion
MC results show the correct
divided
cv
curves
the dot-dashed line is the result of
MC and
cv/T
spin
below the
the
spins
coupling strengths.
start to correlate at different temperatures
Hence there is
no
clear transition and the
T/J
peak
at
1 is much weaker than for the unconstrained model. Also the peak at the transition
scaling regime
0.05J
as
the
is much
weaker,
corresponding peak
but
in the
we
can
specific
interpreted
heat. In
the
cusp-like
Fig. 3.3.4,
where
structure
we
show
near
cv/T,
61
finds that there is
one
behavior
no
peak
We have
before the
crossover
model, but cy/T
for the unconstrained
as
investigated numerically
chains with random FM
to the
is rather
divergent low-temperature
continuously increasing for
(constrained model).
mined
or
AF
The
by three different
the AF and FM
spins
3>
energy scales: at
The
a
susceptibility obeys
a
realistic model of
Curie law of free
can
be
seen
by
as
a
free
peak
spins. The
in the
deter¬
given by
the
high
at very
start to correlate within
spins.
heat and
a
spins
Curie-law of the
by
is visible
a
peak
At still lower
temperatures,
spins become relevant and the spin segments gradually
freeze out into clusters of correlated spin segments. Also this
previous
of the individual
crossover
specific
the interactions of these effective
the interactions among the effective
than for the
spins
spins
are
of the effective spins. The energy scale of this intermediate tempera¬
given by
scaling regime
SrsCuPti-^Ir^Oe
the energy scale is
the temperature, the
spin
unconstrained model with
of the random bond systems
high temperatures,
weakly, essentially
magnetic susceptibility
is
and
an
dimensional
one
segments and form effective spins. At intermediate temperatures, these effec¬
spin segments
regime
couplings
ksT). By lowering
interact very
to effective
We have considered
couplings.
thermodynamic properties
original spin-spin couplings.
temperatures (T
thermodynamic properties of
the
of the AF and FM
equal magnitude
ture
cyjT
Discussion
3.4
tive
in
in the
crossover
specific heat, although
to the
this
low-temperature
peak is much smaller
transition.
Our results represent the first exact treatment of the original Hamiltonian
showing
a
clear separation into three different temperature regimes for the unconstrained model. This
separation
uniform
can
be
by
seen
two marked
magnetic susceptibility
comparison
with
SrsCuPti-^Ir^Oe
at
high
experimental results,
with
a
peaks
the
crossover
magnitude
heat and two Curie laws in the
investigate
for the first time
a
realistic model of
constrained distribution of the FM bonds exactly.
are
of the FM and AF
between the three
the correlation of the
specific
and intermediate temperatures. In order to allow the
we
We find that the three energy levels
the different
in the
regimes
original spins
not
so
clearly separated
couplings,
is continuous.
in this
the energy scales
Therefore the
are
case.
Because of
overlapping
peak originating
and
from
is very broad and the onset of correlations among the
62
effective
in
spins
only be
can
seen
xT, the effective spins form
Overlapping energy-scales
mation of
a
by
a
a
strength
for systems with
overlapping
Curie-constant of the
heat. Instead of
be found in many random bond systems,
can
of the
specific
However,
bonds.
energy scales and
we
have
the
given
general picture
can
be used
as a
regime
Ir-ions is not
enhance the Curie-constant
constants
can
We have
T
=
also be
given
0.027 for the
efficient MC
with the
regime
algorithm,
original
starts at
at low
model and T
we
=
be needed to reach this
on
beginning
with the
spins, providing
a
of the constrained
in
experiments if
of bond-distributions
0.017,4 for the realistic model. But
original
a
can
regime
[23,24].
the very
low-temperature scaling regime
completely
Hamiltonian.
different
However,
is well described
firm basis for the
despite
original Hamiltonian,
For the
low temperatures, such that
regime
also
low-temperature scaling regime,
of the
have not been able to reach the
confirm that the intermediate temperature
for the effective
cluster-analysis
Hamiltonian for both models.
extremely
the
temperature. The calculation of these Curie-
in the statistical
upper bounds for the
generic
applies
really random because of large FM clusters which
drastically
applied
still
on
for¬
universal tool for the anal¬
ysis of the bond-distribution, e.g. much larger Curie-constants will result
or
indeed, the
upper and lower bounds for the
in the intermediate temperature
model. The calculation of these Curie-constants
the distribution of Pt-
clear plateau
regime depends subtly
intermediate temperature
FM/AF
susceptibility
a
peak.
broad
clearly distinguishable
distribution and
small cusp in the
investigations
by
an
of the
our
scaling
the
approach
will
numerical results
effective Hamiltonian
scaling regime
based
the effective Hamiltonian.
We would like to finish by mentioning that
exists for very
an
analogue
dilutely randomly depleted Heisenberg
the parent material, the Curie law of the effective spins
easily
of the random bond spin chains
ladders
can
[65].
Due to the
be observed
spin
gap in
experimentally
and the value of the Curie constant is in agreement with the theoretical value
more
[66].
63
Finite Temperature Density Matrix
4.
Renormalization Group Method
Introduction
4.1
discovery
Since the
of
high-Tc superconductivity
related fermion systems is the most
thermodynamic quantities
tween
are
actively
of great
stimulated the
development
of
a
systems. Quantum Monte Carlo methods
large systems
ulation of
at finite
problem,
exceptions
the Hubbard-model at
trated
spin-systems,
new
are
they
when
half-filling).
cor¬
comparison
tool for these
be¬
systems has
numerical methods for the treatment of these
among the most efficient methods for the sim¬
some
more
The
than
special
same
one
dimension due to the
symmetry cancels the
problem
also
occurs
anti-ferromagnetic Heisenberg systems
e.g.
strongly
physics. Especially
allow the direct
analytical
of
temperatures. Unfortunately these methods usually fail
lower temperatures for fermion systems in
with few
since
The lack of
number of
physics
studied field in solid state
interest,
theory and experimental results.
in the cuprates, the
at
negative sign
negative sign (e.g.
for the simulation of frus¬
with next nearest
neighbor
interactions.
Another method which is also based
matrix method
with
[26,69].
on
This method does not evaluate the
Monte Carlo sequence, instead the
a
diagonalization
method
the Trotter-Suzuki
[27].
partition
partition
The
function
problem
rather
or
by
partition function by sampling
function is calculated
exactly by
some
Therefore the transfer-matrix method does not suffer from
negative sign problem. Thermodynamic properties
the
is the transfer-
decomposition
are
direct calculation of the thermal
obtained from derivatives of the
expectation value of
an
observable.
with this method is its limitation to small Trotter-numbers M and therefore
high temperatures.
Other methods include the
method
[67]
and different
high-temperature expansion,
density
the finite
matrix renormalization group
temperature Lanczos
(DMRG)
methods.
The
64
high-temperature expansion only gives
reliable results if there is
Lanczos method
gives
phase
a
transition at
accurate results for
also at low temperatures, but
=
on
[68].
chains in Ref.
3/2 Heisenberg
number of excited states t>f
method,
the DMRG
formally
can
White showed how
[8].
be extended to finite temperatures
improved
DMRG method has been
thermodynamic
and S
give
lower temperature. The finite temperature
thermodynamic quantities and correlation functions
publications
the zero-temperature DMRG method
This
some
and fails to
high temperatures
for other Lanczos methods, it is limited to small system
as
in the very first
Already
sizes.
accurate results at
and
applied
Since this method
for the
essentially
spin S
1/2
—
calculates the
finite sized system with the DMRG method, it
groundstate
and
cannot treat
accurately the low temperature region of a physical system where the correlation
a
a
length diverges.
Most of these
problems
are overcome
in
a new
approach by combining the
accuracy and
infinite system size of the transfer matrix method with the DMRG method in order to
tend the number of Trotter
algorithm
transfer-matrix
time-steps
The idea of
M.
developed by
has been
applying
the DMRG method to the
independently [28,29,70,71].
several groups
The first attempt to combine these two methods has been made in Ref.
the dimerized XY chain.
[70]
and
not take into account
However, these authors did
ex¬
applied
correctly
to
the
nonhermitian properties of the transfer-matrix in the truncation procedure, such that af¬
number of DMRG steps, the
ter
a
are
unreliable.
tian
The first
density matrix
developed
at the
are
time
same
culations show that this
DMRG method and
limit of infinite
been
by
correctly by using
in Refs.
show that this method
The crucial
between the left- and
much
[28]
and
right-eigenvectors
[29].
has been
The results of these cal¬
thermodynamic
methods down to very low temperatures in the
thermodynamic
point
more
calculations for the Kondo-insulator at half
and various spin systems in Refs.
also be
algorithm
chain. However, this
unstable and the results
accurate results than the
[30, 72],
can
numerically
the left- and
different groups in Refs.
algorithm gives
QMC
became
where the nonhermitian transfer-matrix and nonhermi¬
system-size. Recently
performed
problems.
algorithm
treated
algorithm
applied
to
more
stability
right-eigenvectors
for the zero-temperature DMRG
of the
of the
algorithm,
as
can
not be
algorithm
density
the
have
These results
complicated systems than the Heisenberg
is still unstable and
for the
[73, 74].
filling
is the
matrix.
density
applied for
No
many
interesting
biorthogonality
relation
stability problems
occur
matrix is symmetric in this
case
65
and the left- and right-eigenvectors
is reduced to
a
normal
finite-temperature
biorthogonality
sion of the
of
a
are
orthogonality
DMRG
DMRG
re-biorthogonalization algorithm
we
method,
explain
is based
briefly
cause
loss of
the first stable
on
the
ver¬
incorporation
the zero-temperature DMRG method and the
in section 4.3 how these ideas
we
method for fermion systems away from half
will show
filling
a
can
test of the finite
be applied to the
temperature DMRG
in section 4.5 and compare these
findings
QMC results and the conformal field theory.
Background materials
4.2
The basic idea of the renormalization group methods is to
physical system
and
neglect (or integrate out)
lowest in energy, and
Wilson's
method
original
soon
adding iteratively
[75]
method
was
keep
the irrelevant
impurity Kondo problem by calculating the spectrum of
more
a
one.
on
investigated
keeping
proved
ladder systems
[78,79], strongly
by
Baxter's
a
one
the states
While
impurity Kondo problem, the
one
to be stable and accurate also for
to
a
number of
correlated electron systems
corner
the
[75].
sites to the renormalized system
very successful for the
the other hand
and 2D classical systems
transfer-matrix DMRG
[80,81],
White's
interacting
spin systems in
2D systems
[82]
[83].
Zero temperature DMRG
4.2.1
In this section
we
Wilson
small system,
quantum systems. This method has been applied successfully
[76,77],
the relevant information of
turned out to be unreliable for most other quantum lattice systems.
DMRG method
ity
often
errors
developed
algorithm. Our approach
finite temperature DMRG method. Then
ID
matrix of the
nonsymmetric density
numerical roundoff
relation
that controls the numerical instabilities.
will review
and
biorthogonality
the
case
and numerical instabilities therefore. We have
In the next sections
to
In this
For the
relation.
algorithm, however,
finite-temperature
transfer-matrix
identical.
we
want to
explain
the basic zero-temperature DMRG-method.
restrict ourselves to quantum
although
the method
can
tions in momentum space
also be
[84].
systems
applied
to
on a
more
We start with
an
For
simplic¬
lattice with two-particle interactions only,
complicated
interaction terms
or
formula¬
occupation number representation of the
66
two-particle
terms
where Cj]a and
type
c\a
the site
a on
site i and the
are
the
destruction, respectively construction, operator
i, the operator n^a
sums
a,
j3
taken
are
this notation the Hamiltonian of
The
basis is
single-site
(w.a,)-1^2 (cIai)
dim<5>j
the
=
'
claCi,a
+
1)
=
a
particle of
denotes the occupation number of
the set of all different
particles Ti
on
a on
the
the site i.
In
system with 2L sites reads
{<S>aie^ I'iVv*)
=
:
Vw
=0,1,...
*• The dimension of the single-site basis
IlieJ7, W.<»i
single-site
over
a
given by Si
=
of
<fc- A basis of the Fock
space is
,di,Qi}, where \i;<Pi,ai)
must be finite and is
given by
the tensor
given by
product of
states
'
2h
B
Let
J® ®
=
illustrate the notation with
us
Hamiltonian is
Hij+i
=
—icjc]+1
attractive interaction term, and
respectively
Now
we
a
think of the system
part and the right
the system
BSL
S£
These
=
are
=
YliLi <k,
M)
as
-i
=
and
BSL
C
Me
=
1......M.}
the states that
the block states of the
we
j®*|-
0 for
=
being
where t is
\i
—
j\
case.
=
The
is
BeL
]li=z,+i
—
j. The
single-site
<&
given by
states
j (g)<% |
on
set of
basis is
chain. The
a
a
repulsive
or
particles Ti is
given by
a
hole,
{|i;0), \i\ 1)}.
divided into two parts of
B, respectively
=
(4-2.3)
hopping term, U
a
> 1 or i
half the environment part,
and the environment
restricted to the system
Ms
Hij
Urnrn+i,
fermion, occupying the site Si
the system
these
+
=
simple example for spinless fermions
fermion in this
single spinless
limited to the
a
l*;W«>?
as
Fig.
the left half being
4.2.1.
the states of the above basis
the environment
by
length L,
shown in
a
BL
BeL
c
A basis of
Eq. (4.2.3)
B, and enumerating
suitable enumeration convention
=
will renormalize later and
M)
we
:i
=
l,...,M.}
will refer to these states
\ipf)
as
system (environment) in the following. The Hamiltonian of the system
=
67
environment
system
(a)
2
1
L
...
L+1
2L
...
superblock
/
\
enlarged
enlarged system
(b)
1
2
•
•
...
L
environment
•
•
L+1 L+2 L+3
...
i
•
2L+2 ,
enlarged superblock
FIG. 4.2.1:
Graphical representation
figure (a) represents
of the
the division of the
system and environment
and the environment is
are
density
matrix renormalization group method. The upper
physical system
being enlarged by
one
in
a
site each
system and environment part. Then the
as
shown in the lower
figure (b).
given by
2L
(4.2.5)
i,j=L+l
Then
we
need to know all the relevant operators in the basis of the block states, in this
the Hamiltonian restricted to the system
or
nsL(i,j)
environment
<tf|«£ii#>,
=
the construction and destruction operators of
Clji,])
and the
=
case
a
particle
C&(t, j)
Wlc«|$>
(4.2.6)
a on
=
site I
mc{ JVrj),
(4.2.7)
occupation number operators
KUihJ)
=
(4>!\ni,a\i>sj).
(4.2.8)
Similar matrix-elements need to be calculated for the environment too. For the above example
of the
spinless
fermions the operators needed
tonian restricted to the system,
respectively
are
given by
to the
the matrix elements of the Hamil¬
environment, and the construction and
68
destruction operators for the sites L and L + l:
Cl(iJ)
=
The
WfIcl+iIi^),
called
so
and
superblock
C${i,j)
=
is
as
given
CsL(i,j)
{i/>'\clM), Cs£{i,j)
=
=
(ipf\c'L\il)j),
mA+l\^)
the combination of the environment and the system
part and consists of 2L sites. With the above definitions, the Hamiltonian of the superblock
reads
i<L,j>L
(4.2.9)
where
of
Is,e
7?J
denotes the
.
of the system
identity
A pure state of the
superblock
V>
For the measurement of
an
where
ps
=
13» i r«jl^?) Wl
's
=
7?;
and
the
complex conjugate
represented by
observable As defined
Ie|V>
®
(environment)
$>«#?) «> h/>|>-
=
on
(4.2.10)
the system part
only,
we
introduce the
ps
concept of the reduced density matrix
WAS
is
Yallaj,,{ri,Asi>sj)W,fi)
density matrix,
the reduced
tr(pM»),
=
its coefficients
are
(4.2.11)
given by rij
=
Ei ai,iaj,lIn the next step
system-
Fig.
and
we
environment-part
4.2.1. Therefore
must
we
the size of the
enlarge
want to
one
assume
translational invariance. A
system is given by the tensor-product of the block
the site added
first,
new
then
states
\ipf)
new
in between the
each,
shown in
as
basis for the
with the
single-site
enlarged
states of
\<PL+\,aL+1)
B£+i
For the
superblock by adding
site to the system and environment
basis of the
we can
l¥'i+2,ai+2>'and
=
{ W>
enlarged
also form the
the block states
S£+i
=
:
i
=
1,.
basis
•
we
,dL+iMs)
=
{BSL
®
SL+1}
must relabel the old site numbers
tensor-product
of the
single-site
(4.2.12)
.
by
i
->
i + 2
states of the additional site
\1pf1
{m
: t
=
1,...
,dL+2Me]
=
{SL+2
®
BeL}
.
(4.2.13)
69
The
enlarged system, respectively environment,
Hamiltonian for the
new
reads
KL+\
(4.2.14)
L+l<!<2t+3
(4.2.15)
where the site numbers i in the Hamiltonian
The Hamiltonian for the
%£
have been substituted
enlarged superblock given by
by
i
the combination of the
->
i + 2
well.
as
enlarged system
and environment part reads
H2L+2 =nsL+1
®
ii+2,e
+
neL+1
iS)£+i
®
ns
+
®
hl+i,l+2
®
ie
KL+l
(^iJ^+i,aC;t, 7«f+1C;,,cl+lja 2^fljnL+1,Q^)
E
+
+
(4-2.16)
+
]>L+l
E
+
KL+\,]>L+l
where
ls,L+i (Ii+2,e)
(L + 2).
site L +1
sparse matrix
the
we
denotes the
The extreme
diagonalization
groundstate
Then
(7f/C,V,
can
®
identity
eigenvalues
method such
be written
calculate the reduced
=
Sjc*,icj,i-
defined in Eqns.
by
each
^n tnis
(4.2.12-4.2.16),
+
t^C*
as
^
the system
on
of
®
H2L+2
are
+
2</M!a ® *&)
(environment)
then calculated
the Davidson
[85]
or
and the additional
numerically by
Lanczos methods
we
can
some
[5],
and
density matrix of the groundstate of the enlarged system,
wav we
=
E^IW*S|.
could continue
(4.2.18)
iteratively
the
enlargement procedure
but the number of states increases with
a
factor of
enlargement step and the dimension of the Hilbert states would
number of states that
,
as
PS
where rtJ
C*
treat
exactly.
Therefore
we
will limit
our
soon
dL+i^L+2
exceed the
knowledge
to the
70
relevant information about the system. But which
measurement of observables
Eq. (4.2.11) that the
psAs,
of
those states with
eigenvectors
m
density matrix
oi, 02,
ps
as
renormalized basis,
6
[01,02,.
=
•.,
om].
om
•..,
the
we
in the
large weight
corresponding
[8].
basis
new
define the
and similarly for all other operators. In the
ff^
The matrix
The
so
product Pm
tr(&Ops)
=
try
an
The reduced
state
i\>q,
lowest
we can
lying
by
matrix O formed
new
basis
are
performed by keeping
of the
these
m
new
column-vectors
then obtained by
&HsL+1d
(4.2.19)
6t(C&®IL+i)d
(4.2.20)
dHh®$+1:a)6,
(4.2.21)
same
we
way
have to truncate the basis of the envi¬
pe
of the environment this time:
e
is
a
=
projection operator which is optimal
is
a measure
J2z,j ^fjl^JKVf |,
otherwise the number of states
scheme
kept
equally
a
mixed state
tpk of the system,
optimal description
temperature, then
we can
equal weights ipK
density
=
ipK-
a
Pm, and tr(Pm)
that
=
m.
procedure
has to be increased
J2k=i wk'll>ki
matrix
and the
corresponding
case.
pure state like the
For example if
calculate the reduced
of these K states in this
we
m
sense
or one
[8].
consider
of the reduced
=
in the
for the accuracy of the truncation
matrix p needs not to be calculated for
states
P^
is maximal under the constraints
density
of these states with
eigenvectors
—
&O
error e
small,
extrapolation
=
1
=
called truncation
and is usually very
can
density matrix
reduced
give the largest
T,ic*,ici,i-
=
tr(&p~sd)
matrix will thus
in
to the trace
largest eigenvalues Ai, A2,..., Am
to the
rectangular
S,ct
where
density
The effective operators for the
HL+1
using the
equivalent
the system is
seen
For the transformation of the operators to the
n/S
ronment
on
The selection of the relevant states is therefore
general.
contributions in
a
the relevant states? We have
are
If
we
we are
density
weights
to the
matrix for
are
Wk
=
J
a
%
mixture
[8].
The
largest eigenvalues give
want to consider
have to choose the normalized Boltzmann
ground-
interested in the K
weights
as
a
system
the
an
at finite
weight
factors
Wk-
The dimension of the
of
dim('H2L+2)
=
superblock
iTi^di+idL^-
renormalized basis
depends
on
for the
enlarged system
is thus limited to tractable size
The amount of work needed for the transformation to the
the number of
operators needed, the number of states kept
71
m, and the dimension of the
single-site
basis
for systems
on
chain with nearest
a
also allows the treatment of ladderinteractions.
The method
dz. If the interactions
independent
then the number of operators needed is
neighbor
or
where both the system and environment part
are
more
complicated
infinite system method,
called
so
It is minimal
but the above formalism
the inclusion of
have described here is the
we
only,
interactions
2D-systems,
or
only short-ranged,
are
of the system size 1L.
being enlarged by
site at each DMRG
one
step. The finite system method iteratively improves the effective operators and ground-state
approximations
for
environment part
2L of the
are
superblock
environment the
length
a
being enlarged by
is reached.
order to increase the
algorithm
we
In this section
of the system.
only
two
the
the latter
method,
the system and
the infinite system method until the desired
only
the system
part
is
being enlarged,
being used,
are
sites, then the
length
while for the
such that the total
enlarged
until the
process is reverted and the environment part
previous system descriptions.
This process
precision of the algorithm vaxiationally. For
a
can
be iterated in
detailed description of
[8].
refer to Ref.
we
interactions
=
want to
terms H
QMC algorithm,
=
given by Eq. (4.2.1)
1 for
a
chain, further
7?/
some
explain the transfer-matrix method for
As for the
easily diagonizable
for
By
Transfer matrix method
4.2.2
JV + 1
Then
2L.
length
of the superblock is kept constant to 2L. In this way the system is
being enlarged, using
this
of fixed
previously calculated operators "H\_l
environment contains
is
superblock
=
constant B with
split
For
simplicity
\i
£
-
j|
>
that there
B,
rfif
are no
=
cluster-decomposition
the Hamiltonian into
consider
we
and periodic boundary conditions
we assume
0 for
L/B
Heven + %°dd.
we
a
on a
a
sum
of two
only two-particle
lattice of N sites, e.g.
long-range interactions,
0 for
\i
-
j
\
>
B,
(4.2.22)
N. We set
N/{2B)
^even
=
\p ^(2.)
(42 23)
2=0
N/{2B)
«odd
=
Y^ H&+1),
i=0
(4.2.24)
72
%W
where each
denotes the Hamiltonian for two clusters
B
W
2J (^«B+P,(»+l)B+9
=
+
+
of B sites each
B
1
^(i+l)B+p,jB+g)
consisting
2_/ (MiB+pjB+q
g
+
~H(i+i)B+p,{i+l)B+q)
P,9=l
p,q=l
(4.2.25)
The left
incorporates the inter-block terms, and the right
sum
1/2
block terms. The factor
interactions
are
only,
teractions
decomposition reduces
and each cluster consists of
decomposition
neighbor
in front of the intra-block terms accounts for the fact that these
single
a
ID system with nearest
Fig.
4.2.2
only
ftM.
Next
one
define
we
a
basis
each cluster
on
1
space is
denotes the basis of
given by
notation
we can
Eq. (2.2.2)
the
and next-nearest
%W
de¬
consisting
of the sites
single-site
write the
\
((i+l)B
JJ
l,...,
=
=
(g) Sj\,
i
states defined in
(4.2.26)
J
(j=iB+l
Eq. (4.2.3). A basis of the Hilbert-
of these cluster-states
tensor-product
{\ip)}
=
{®ji0
^}-
With this
partition function Z by applying the Trotter-Suzuki decomposition
as
ZN=
or more
neighbor
half of the interactions among the rungs has
dk\
!{i+\)B*=«B+1 J
Sj
examples for the cluster-
\)B by
\a):oi
where
in¬
decomposition (see Fig. 2.2.1)
show
we
neighbor
A cluster consists of two sites then, and each Hamiltonian
to be included in each
+
a
to the checkerboard
site then. In
scribes four sites. For ladder systems,
(i
For
of ladder systems and for systems with nearest
interactions.
iB + 1 to
'Hodd.
counted in both %even and
this
term consists of the intra-
lim
M->oo
tr
[e-AT"eveV^°ddr
L
lim
=
J
M-too
(4.2.27)
ZMN,
explicitly
M
V^TT/
n
2?-l
2.7-11
2.7-1
-AiJ?eve"i
2?
2i
2?\
(4.2.28)
{a} 3=1
x
where
with
j of
we
{a},
of
"
"
CTiV le
have denoted the insertion of 2M
At
=
0/M
denotes the
Fig. 2.2.1,
\CT1 °2
this
\"1
complete
°2
••°N
sets of the basis of the
is the size Trotter-time step for the Trotter-number
Trotter-time, and the lower index
equation
can
be
seen as
h
Hilbert-space
M, the upper index
i the site number. As
the time evolution of the initial state
we
have shown in
\a\a\
...
aj^) along
73
a) Decomposition for
a
chain with nearest
b) Decomposition for
a
chain with nearest and next-nearest
neighbor
interactions
(checkerboard decomposition)
neighbor
interacions
8
>
<$
-
-
9"
denotes
-
for
-
9--
-
9
this
the
/^0<jd
case
cluster consisting of B sites
Graphical representation of
In
the
(a)
W(>
system.
on
&how the
the
a
right
only half
(see
text
for details)
the cluster
hand side
decomposition
cluster-decomposition
decomposition for
->
,9---9,
that have to be counted
sentation of the total system,
and
i
o
10=2
7
ladder-system
interactions
symbolizes the
FIG. 4.2.2:
a
\
.'
i
a'
o
10=2
c)Decomposition
9=1
for
a
decomposition. On the
we
show the
chain with nearest
reduces to the checkerboard
chain with additional next-nearest
left hand
decomposition
side,
is
a
repre¬
into two terms ifeven
neighbor interactions only,
decomposition.
neighbor interactions
In
(b)
and
we
(c)
in
demonstrate
shows
a
ladder
74
(b) Transfer-Matrix along x-direction
(a) Transfer-Matrix along beta-direction
S;MI
IS;M'si"'s,M'
T, T2 T1 T2
>
1^2 Tx T2
n-
along
the
modified
the
of the transfer-matrix
Graphical representation
FIG. 4.2.3:
real-space direction (b)
plaquette
imaginary
ta
time
order to
in
In
(b)
measure
the local observable A
the time and space direction, and consider
Fig.
Let
the
4.2.2
us
we
show
algorithm
The transfer-matrix
s/3.
a
the Trotter-time direction
this
uses
plaquettes
Tl((T.
2ii
"
2j-l
lCT2. + l
-J2i+
T2(<72t+l><T2.+ llCT2.+ l'<J2.+2)
inserting Eq. (4.2.29)
=
ffeven/oM
Eq. (4.2.27)
N/B
Zmn
of
sums
in
and
by the
we
can
exchange
in the real space direction
x.
In
exchange and of the transfer-matrix.
-At//"
2j
2j
/
|
\°2i'°2H-lle
11
2j+2x
2j
2j + l,
The block-terms in the
(a),
n
Eq. (4.2.26)
next
by defining
of the local virtual transfer-matrix
T2
2i-l
single plaquette
the fact that
transform the local matrix elements of the cluster-basis
T\ and
a
See text for details
propagation
graphical representation of
a
along
replacement of
do also the
we
we
,2,+.
(^2.+ l
.
CT2,+2
11
2j-l
r2j-U
le
^^^V^,,^)-
commute since
they
act
on
(4 2.29)
(4-2.30)
different sites, and
by
obtain
M
e n nn^r1.^^;.1.^!^^..^1^..-
2j+2x
2i4 2 I
i=l
tr
j
=
l
((7,72)"/*)
=tr(TM),
(4.2.31)
75
where
we
have introduced the virtual transfer-matrix T
=T\Ti'-
M
Tl{«lu---,°\°l+i,---A)
=
Un(o22r\4!\4£Al+i)
r2(4+1,...,4^1l4+2,---,4+r)
=
\{^2i+l,a2iXl\al+1,a2iXl).
(4-2.32)
M
(4.2.33)
3=1
By applying
system size N
energy
interchangeability
Suzuki's
=
-^InZ
—
per site in the
/
=
-
a
lim
--5
p M->oo
diagonalization
71 and 75
are
Hermitian, their product is
the standard Lanczos-
apply
in Sec. 4.3 in
more
limit:
-^-lntr(Tw/B)
j3N
(4.2.34)
lnAmax,
of the transfer-matrix 7" and the
Basically
ergy u, the
the two factors
commuting. Hence
heat cy, the
magnetic susceptibility
accurately,
we
one
virtual transfer-matrix
2j,
not
interesting thermodynamic quantities
directly by modifying
2J-1
algorithms.
are
impor¬
diagonalization
We will discuss this
obtain many
derivatives of the free energy
derivatives
71 and 75
two
point
we
later
general,
in
sparse matrices for the Hamiltonian H.
we can
specific
not since
Davidson
Although
are
detail. The second difference is that the matrix T is dense in
typically
contrast to
or
the limit of the
and obtain for the free
—¥ oo
Hamiltonian %: First, the transfer-matrix 7" is not Hermitian.
cannot
T.(
lim
lim
interchange
Amax is the maximal eigenvalue of the virtual transfer matrix 7". There
tant differences between the
of
we can
thermodynamic
N-*oo M-xx,
=
where
[26,69],
with the limit of the Trotter-number M
—> co
density /
theorem
2j-l
density /.
calculate
or
the
thermodynamic expectation
tj,
/
s
_
2j
for the local observable A
2,
particle density
Since it is difficult to evaluate
\l(Ae-f>H>/M
+
n
en¬
from
numerically higher
values for local observables
of the virtual transfer-matrix plaquettes.
plaquette
2]
x>
like the internal
We define the modified
by
e-0H^«/MA\
I
2,-1
2j-l\
(4.2.35)
In the
product for 71
we
replace
one
plaquette
t\
by
t^ and
get the operator
M
Ai
=
r^(ai,4|aiJ+1,4+1) JJr1(a22r1,a22f|<7^-11,<72^+1).
3=2
(4.2.36)
76
With these definitions
we can
expectation
calculate the thermal
{A)th
value
°ftne observable
A from the left- and right-eigenvectors corresponding to the largest eigenvalue Ai of the
transfer-matrix T
{A)th
^ocaI^oc
=
where
we
the last
have
applied again
the
{1)tf!^\
*»
=
(4-2.37)
theorem for the limits N
exchange
Prom these measurements
equation.
tr(T"/*)
we
calculate the
can
—>• oo
specific
and M
—> oo
heat cy
in
the
as
derivative of the internal energy
<*
magnetic susceptibility
and the
external
magnetic
x
=
^>
(4.2.38)
tne derivative of the
as
magnetization with respect
to
an
field h
a(Efgf)
(4.2.39)
dh
h->0
where S" denotes the z-component of the
Let
concentrate
us
dimension of the
by
a
large
numbers.
on
the role of
each
factor if
we
With
symmetries
the space- with the
quantities
Sfj
we
of
But
cases.
H^:
Let
denote the total
st,k+i
can
rewrite this
equation
+
implies
that S
=
want to
J2j(^Y+^i,j
+
si,k
=
a
+
=
~
be reduced
of conserved quantum
the usual conservation
spin
is
\a\)
locally
conserved
with the
by
space-time
reads
(4.2.40)
si+i,k+i-
St,k
The
similar conservation laws for
conserved quantity
conserved
and it has been shown that the maximal
that the
plaquette
on a
£f+i,*+i
's a
subspaces
of the cluster-state
spin
si,k+i
diagonalize can usually
we can recover
assume
us
in order to obtain
-^«+l,t
This
in the transfer-matrix method next.
imaginary-time direction,
coordinates i,j. The conservation of the spin
We
the site i.
limit the calculation to invariant
By exchanging
conserved
%(*>.
on
Hilbert-space of a Hamiltonian we
of quantum numbers is lost in most
locally
spin
along
the space direction
Si,k+1-
(4.2.41)
quantity along the real-space direction,
eigenvalue \max is
in the 5
=
0
subspace
for
77
rotationally symmetric systems [86,87]. Similar arguments
number with M
space-time
ensemble,
we can
winding
and
even
find
number
sites in the
conditions
are
similar argument
a
can
Finite
be calculated
largest
studied.
in the
largest eigenvalue
will
Throughout
thermodynamic
of
a
we
the
calculation, the
temperature T is thus given by T
a
subspace
justified by
to the
apply
to the
keep
=
transfer-matrix T with
as
wc
shown in
Eq. (4.2.26)
that
we
start
Fig.
treat
the transfer-matrix of the system for
M, and with T*(oi,ipe, a^cr", 4>'e, v'2)
especially
for the
tne
are
environment,
see
useful for systems with
matrix of the environment
they
it'"
is
boundary-
a
increasing
enlargement
kept
constant and the
density
by dividing
4.3.1.
exactly,
us
i/)sie
we
we
denote with
numbers of the Trotter-
matrix of the environment. For odd
T°(o", i/>'s,a"\<Ji, t/>s, 02)
Fig.
and with
notation, let
even
the transfer-
With crj and 02
4.3.1 for
a
for the system, and
graphical representation.
This
reflection symmetry in <7i, since the transfer-
are
kept
on
needed in order to evaluate the trace
imaginary
an
exactly
given by the transpose of the transfer-matrix of the system
Non-renormalized states
matrix in the
0
(JWAr)-1.
system and environment part,
T°(u", ipe, o"\o[,$e, a'2)
as
=
the dimension of the transfer-matrix T limited.
numbers M of the Trotter number the notation is
then.
subspace of N
The basic scheme is to calculate
zero-temperature DMRG algorithm,
T^(a[,il>'s, a'^a" ,ips, a2)
Ts
zero-winding
odd sites
number is
the ideas of the zero-temperature
denote the renormalized block states. In order to establish the
notation is
canonical
on
the fact that
size of the Trotter-time step At is
denote the states of the basis defined in
number
of
Since the number of states grows with each
the DMRG-method to
Equivalently
grand
limit anyway.
want to
good approximation
use
matrix T into
in the
particle
state with the
spin by considering the winding number.
This is also
method,
number of Trotter-time steps M.
we
performed
restrict the calculations to the
DMRG method to the transfer-matrix method.
step,
particle of the
to the
Temperature DMRG
In the finite temperature DMRG
the
applied
the difference of the particle number
as
one we can
are
important
for the
as
are
time direction. Since the
imaginary
properties
not
where iV?- is the total
i,j. Although the calculations
important and
unless transport
4.3
^,-(—l),+JJVt?j,
coordinates
The
the most
=
also be
can
both sides of the system and environment
over
periodic configurations of the transfer-
time direction. The transfer-matrix T and is obtained
by summing
78
(a)
P-direclion
V,'
o,'
a,'
a,
o.
v
jtj
a,
a,
vs
o2
V,
°[
P-direction
(b)
V,'
".'
1
T
o
J
r
.
111
h
'.^
1
V,
FIG. 4.3.1:
Graphical representation
temperature DMRG algorithm
(a)
the left hand side
a
M,
on
on
the
right
we
M,
on
n 2,
T^ (a i,ipe,
the left hand side
hand side the environment
we
a2
summing
\a[',ip'e, <T2)
show the system
T°(o", t/;e, (T2'|<ri, tpe, o'2)
Ts/e
over
of the finite-
numbers of the Trotter-number
(b)
shows the transfer-matrices for
T°{o", ip's, a2'|<7i, ips, U2),
the intermediate states
o'{
and
a2'
on
the
right
The shaded squares represent the transfer-matrix
the dashed frames represent the renormah/ed block-states of the system,
the environment, which consist of many renonnahzed
by
even
graphical representation of the system part Tg(a[, V'si(T2 |°"i\ ?Ps, 0*2),
hand side the environment
odd numbers of
plaquettcs
show
of the system and environment transfer-matrix
shows the transfer-matrices for
plaquettes
7~i,2
respectively
The transfer-matrix T
is
obtained
79
over
all intermediate states <r" and a" of the system and environment parts
'
TmW\,n's,a'o,n'e\ai,ns,02,ne)
<
=
£T°(ai^>>i,^.,a2)TeV^tk,^K>#.^)
M odd
"if*
I EI7W.^0i,l°i,.*«^)3?^>*«^'K.*'..o/a)
Meven.
(4.3.1)
In the initial step with M
given by
the
product
of the
=
2, the system and environment transfer-matrix T/ and T/ is
plaquettes
=
^tiW.^K.Ots^.^I^.oJ)
(4.3.2)
ree(<Ti,^,<x2'K',V>>2)
=
Y,T^a'^'e\c'i^e)T^e^2\^o2)
(4.3.3)
\ips/e)
are
given by the
same
cluster-states
as
for o\$,
Eq. (4.2.26).
iteratively
The Trotter-number M is increased
system- and environment-part,
define the
and T2:
T/(<7'„^,4'|<Ti',Vs,<T2)
In this initial step the block-states
defined in
t\
enlargement
as we
have shown
then
by inserting
graphically
in
t\ and t2 between the
Fig.
4.3.2.
Formally
we can
as
rs0K,&,4Vl>^2)
^r2K,a'Vl,a)Tse(<T',^,a2V",Vs^2)
=
(4.3.4)
a-"
W.&XKl.lM
ETl(ff''<7ilCT"'CT'/)Te(cr^e,4V",^,4)
=
(4-3.5)
a"
Z?(*i,#,ff5K,&,o4)
=
^nK.aXVO^V'.Mla,^,^)
(4.3.6)
2?(<ti,&,o2K,#,o/2)
=
^(aV^^i^V^e^V-C^),
(4-3.7)
where the basis of the
the environment and
the
enlarged
|^e)
=
\ij}e)
states is
®
Icrx)
for the
tensor
product \ips)
environment, equivalently
=
to
\o\)
®
\if>s)
for
Eq. (4.2.12) for
zero-temperature DMRG method.
When the number of states of
truncate the basis
by keeping
\tps)
the most
As for the zero-temperature DMRG
to the
largest eigenvalue
cannot
can
given by the
use
use
a
important
method,
targeted
power method
procedure [5].
by iterating i$>k
number of block-states m,
states selected
we
calculate the
Amax of the transfer-matrix T.
the standard Lanczos
simple
exceeds the
If the
=
by
the DMRG method
only.
eigenvectors corresponding
Since the T is nonhermitian,
eigenvalues
Tm4>k-i
we
for
are
an
well
separated,
we
we
initial random vector
80
(a)
o
w
l-jl
-•-*-
°i
P
a.
o.
a
a,
^
fb)
a
°
°
%
1 I
1
-
°2
*
-j
^[
a:
°
direction
a
a
1111
!:*2
*j:
'
FIG.
4.3.2.
defined
in
Graphical representation
Eq (4
3
4)
renormahzed block states
enlargement
plaquette
the left
is
n
for
2
is
even
of the of the
The shaded squares
t/;, respectively
TrotK r-immbers
r<
prr sent
the
renormahzed the
enlargtd
new
block-states
t/i
n 2,
i('normalized block states
(erivironnit nt)
then the mUiimdlate states a"
of the transfer-matrix T
plaquettes
M, (b) for odd numlxrs of M
added to the ixistmg system
(right) figure
enlargement step
transfer matrix
are int<
See text for details
(a)
ip
In all the
shows the
figures,
a
single
ti.msftr matrix with the block-states
gi.it{ d
out
as
the dashed frames
and the system
tp
in
(environment)
81
4>o
until
i>K
=
well
i'R
AmaxV>j<r
reach convergence
we
=
E<r,,^,<r2,^ c?1,*.,<72,^lai)
separated,
prefer
we
[73,89].
to the
problem.
For systems with
i>R by
<4 A,^.
be modified
The
reflection
a
slightly,
since
ri>s,i>',
The reduced
\ij>s)
=
\a{)
have left- and
diagonalization
and the
Q
=
method
we
[qi><l2)---,<lm]
and
Q,
eigenvalues
algorithm [88]
are
or
implicitly
are more
accurate
direction,
matrix
ps
obtain
we can
defined in
and
ipR
ipL
Eq. (4.2.18)
in this
from
has to
case
(4.3.8)
^ C°-l,*s,!T2,V>eC<Ti,V'i,0'2,1<>e'
=
not so
be obtained from the transpose of T.
right-eigenvectors \j>l
is nonhermitian in this
O
are
formed
right-eigenvectors
These
largest eigenvalues \.
can
then calculate all the
matrices
rectangular
ipL
right-eigenvector
o2,i><:
\ips). Obviously ps
<8>
If the
number of parameters has to be tuned
a
density
i>e
where
foM-
at g\ in the time
2^lC1pa,i>aCi^,'Pe
=
®
Jacobi-Davidson
method, but
symmetry
we
k2)
®
left-eigenvector
c*u1>e,<t2,1>,-
=
IV's)
and obtain the desired
We have found that the latter methods
and converge faster than the power
according
®
recently developed
the
restarted Arnoldi methods
TmiPk
=
eigenvectors
by
O
too.
case
eigenvalues Ai
X2
>
By
dense-matrix
>
>
the column-vectors of the
=
a
\d„m
left-eigenvectors
[oi,o2,... ,om] corresponding
fulfill the
biorthogonality
ps,
of
relation
to the
(u,,Oj)
=
m
S%].
The transformation to the truncated basis is obtained from
T/K^^'K,^,^)
=
rs°K,^,4Vi,^,^2)
=
The transformation of
Te
by *&,&
E„,^
is estimated from
cyclicity
0(^i.^)3?(*i.^.O2'|oi.^.«a)OW.,^).(4.3.10)
by Eq. (4.3.9)
finite-temperature
e
=
of the trace
in this case- where
DMRG method
A\ for the observables .4.
we can
Trotter-time step i, but it is
afterwards.
$«>
tr(ps/e)
<
=
performed equivalently
density
We)
®
matrix
ki>-
pe given
Since
ps'e
1, and the truncation
effort than
are
is
error
also calculated at this point. Due to
insert the modified local transfer-matrix
more
accurate to
step and transform the operators .Ai similarly
computational
with the reduced
{Y?K)lto(ff,e)-
The transfer-matrices
the
J2
^U.^^a'^^^
nonhermitian in the
e
Q(<,ti)T?(e[,ti,a%W';,i>s,<J2)d(i>sAs) (4.3.9)
to the truncated basis of the environment is
to the transformation of the system
=
J2
enlarging
replace
to
n
in
Eq. (4.3.9).
Eq. (4.3.4)
plaquettes
at each
T4 at any
enlargement
This method does not cost
the old operators A\ and
transforming
to the
new
more
basis
82
Systematic
the truncation
At
—>
0
errors
originate
The
errors e.
by fitting
to
a
from the finite size of the Trotter-time steps At and from
error
polynomial
in the Trotter-time step At
(At)2.
in
The truncation
best method to check the accuracy of the calculation is to
different numbers of states
errors are
depends
4.4
typically
not
only
rather
kept
m
small, i.e.,
Breakdown of the
e
finite-temperature
be eliminated to
can
is hard to estimate. The
error e
perform
the DMRG-method for
10-5,
<
kept
but the
m
magnitude
but also
on
of the truncation
error
the system considered.
algorithm
The nonhermitian transfer-matrix T and
and make the
0/M
and to test for convergence of the results. The truncation
the number of states
on
=
density-matrices
DMRG method
more
ps/e
cause
many technical
problems
complicated than the zero-temperature
version.
Three
major problems
eigenvalues,
loss of
due to the
occur
biorthogonality
results), especially
to calculate the reduced
accidental
degeneracies
The first
problem
density
eigenvalue A,
to each
is the
other,
greater
one
is
large (as
many
it is needed in
degenerate
symmetries of the system
as
subspace separately.
states.
possible
and
This avoids
and alleviates numerical instabilities.
occurrence
than one,
often obtains
of
complex eigenvalues and eigenvectors of the density
density
eigenvalues.
for the real-valued
is
as
matrix for each invariant
matrix. If all matrix-elements of the
eigenvectors
kept
if the system considered has many
Therefore it is very important to exploit
complex
algorithm.
These numerical problems aggravate if the number of states
order to get reliable
the appearance of
non-Hermiticity:
and breakdowns of the
a
dim(Af(I
pair
of
—
matrix
are
real-valued,
we can
But if the dimension of
Aj))
>
1,
or
if two
an
find real-valued
eigenspace
eigenvalues A,
are
of
an
very close
complex conjugate eigenvalues
Xt
=
A;+«>,
(4.4.1)
AI+1
=
Aj-i/i
(4.4.2)
83
complex conjugate eigenvectors
with (j, <C 1 and
This is
no
problem
severe
since
«i
=
"i+i
=
o'i
=
o'i+i
=
we
can
iPi+U
(4.4.3)
Pt-*Pi+i,
(4.4.4)
Pi +
qj +
iqj+i,
(4.4.5)
and
(4.4.6)
<ii-m+i-
choose linear combinations of these
eigenvectors
to
get real valued eigenvectors, e.g.
However, it is important
llu^Mlo'll
our
=
u;
um
=
i(u;-uj+1),
6*
=
o|
6m
=
ifa-oj+i).
+
+
oJ-+1,
(4.4.7)
(4.4.8)
(4.4.9)
and
(4.4.10)
to choose linear combinations with
the above choice
Usually
-^ "•
u5+1,
Ui
or
interchanging
sufficiently large overlap
Oj with Oj+i will be sufficient for
purpose.
Another
more severe
of the reduced
density
problem is
matrix
ps'e
the loss
of biorthogonality
and breakdowns of the
(uJ)oJ)
occurs
due to the finite
overlap,
m,^1',?^,,,
to become
(u^,o') ^
<S
numerically
A breakdown
This blows up roundoff
In this
near
(4.4.11)
case
Q^O
errors
is not
a
to be
multiplied by
and leads to
a
loss of
a
large
number
biorthogonality:
projector and the algorithm becomes
consequently.
occurs
if
(u5,o{)
Then the vectors
biorthogonality
<y
1, the normalized eigenvectors have
^ j.
unstable
algorithm.
right-eigenvectors
Loss of
precision arithmetic. If two left- and right-eigenvectors have very small
biorthogonal.
0 for i
=
of the left- and
can no
longer
=
be
and
0
IK||2,||o;||2#0.
biorthogonalized.
In finite
(4.4.12)
precision arithmetic already
a
breakdown
0
KH2IKH2
and
K||2,||o;||2#0.
(4.4.13)
84
stops the algorithm.
Similar
numerical
problems
algorithms dealing with
[5,90,91]
method
with the loss of
Inspired by
only
The
If it occurs,
much
removes
reason
algorithm
methods,
these
numerically
This method
nonhermitian
for the solution of AX
the nonhermitian Lanczos
becomes
biorthogonality
we
for the
a
for failure is then caused
Lanczos
of the
biorthogonality
algorithm [92]
special
case no
a
length
variational
W
where the
{i, j)
larger eigenvalues
breakdown will
L with
0.1 and //
we
=
and
apply
for
a
very
occurs
of the reduced
some
density
encountered such
special system and
lookahead-algorithm
algorithm
rarely.
symmetry-subsector.
never
the
which
for the nonsymmetric
finite-system
method for hardcore
pair
a
a
of nearest
particle
on
neighbors, c'jcr
the site
problem
of
-
rn,-a)
and Cji<r
j with the spin
set to fj,
=
DMRG
spin-1/2
fail,
algorithm
fermions
complex eigenvalues
+
are
on
a
(4.4.14)
,
the usual construction and
CjiCTct
is the
occupation
Because of manifold
degenerate
a, nj:<r
0.
H.c]
occurs
=
a
especially frequently.
show the results of three different implementations of the
0, and different numbers of states kept
temperature DMRG
with
problem
the inclusion of
_tEE[(1- ^w)cj><V(l
states in this system, the
=
have
stable.
more
occur.
number, and the chemical potential has been
At
problem
eigenvalues
we
much
periodic boundary conditions. The Hamiltonian reads
destruction operators of
In Tab. 4.4.1
algorithm
improvement of the renormalized basis and hope that for this
re-biorthogonalization
=
denotes
density matrix,
DMRG
[5,92].
should be considered. In the worst case, where all these methods
We have tested the
chain of
algorithm
breakdowns. This
serious
matrices
or
re-biorthogonalization algorithm [91].
a
for very small
a
matrix,
n x n
nonsymmetric
and makes the
relation similar to the lookahead
it is necessary to truncate at
in order to obtain
of
to truncate the basis in this
But if breakdowns should be
large eigenvalues
relaxes the
occur
biconjugate gradient
finite-temperature
and in the calculations for realistic systems
breakdown.
occur
by
however, the simplest solution is
only,
for
we
known from various
nonsymmetric
a
diagonalization
biorthogonality
We have found that such breakdowns
matrix
where A is
include
are
e.g. the block
matrices,
have found that the
stable if
more
the loss of
B,
=
and breakdowns
algorithm
large overlap (uj,Oj)
>
with
no
modification,
10-8 for i ^ j
are
m.
The first version
in the second
algorithm
(I)
for
is the finite
algorithm (II)
the states
removed, and the last version (III) is the
85
TABLE 4.4.1: Free energy
steps (M
=
100)with
have set /a
we
(II)
=
0. The first column
shows the results of the
column
(III)
kept
Results where the
m.
density /
for hard
(I)
algorithm
is the
algorithm
=
algorithm
(see Eq. (4.4.14))
(the temperature
0.1
without any
with removal of states that
re-biorthogonalization
is with the inclusion of the
has failed
are
after
modification,
are
not
a
hundred DMRG
is therefore T
=
O.lt),
and
the second column
biorthogonal
and the third
method for various numbers of states
denoted with
\n,
where
n
is the number of DMRG
performed successfully.
that could be
algorithm
from the table, the
I
II
III
-0.6549984
-0.6549984
-0.6549984
m
=
10
m
=
20
t
16
t
16
-0.6722589
m
=
30
t
39
f
25
-0.6741314
m
=
40
t9
t
15
-0.6744112
m
=
50
t7
m
=
60
m
=
80
above discussed DMRG
f
17
f
50
t8
algorithm
algorithms (I)
hundred DMRG steps, but
few states kept
even
(II)
obviously
for this very
28
with inclusion of
and
-0.6745481
-0.6745787
t«
are
-0.6746244
re-biorthogonalization.
very unstable and
the desired number of DMRG steps has been
a
fermions
core
the Trotter time steps size At
performed, only
the results
simple model.
one
can
see
they almost always fail before
for
m
=
far from being
are
As
In contrast these
10
we
can
converged
implementations,
perform
with that
the finite
temperature DMRG algorithm becomes numerically stable with the re-biorthogonalization
and there is
larger
no
failure for all numbers of states
number of states
kept.
In the
the results among the different
stability
of the
algorithms
are
case
kept
m.
where all of the
algorithms
are
in
The results
algorithms
seem
to converge for
have succeeded
perfect agreement. Similar
(m
=
a
10),
results for the
obtained also for other values of At and the chemical
potential
86
The one-dimensional t-J model
4.5
In this section
we
fermion-systems
/i
^
away from
0 for this aim.
computational
which
were
effort than it
we
half-filling
spin-systems
or
to
a
half-filling.
have defined its Hamiltonian in
potential
algorithm
DMRG
finite chemical
a
test-case
have to include
we
needed for the previous
was
With the inclusion of the chemical
defined
and
The introduction of
restricted to
model,
ID t-J
finite-temperature
want to show how the
to
as
a
finite chemical
potential requires
applications
As
an
be
can
potential
much
of this method
example
we
applied
more
[28-30],
study
the
plaquettes
are
want to
Eq. (2.3.1).
yu, the local transfer-matrix
as
Tll°2i
'a2i\a2i+l'a2i+l>
\a2i
\a2i>a2i+l\e
~
'
c2i+l
I
(4.5.1)
T2\a2i+l>a2i+l\<J2i+l'a2i+2>
\a2i+l>a2i+2/>
\°2t+l °2i+2 le
~
>
(4.5.2)
where
Ne (N0) is the total particle number included in He (H0) only, and Neo is the total
particle number included
/
=
f(T, /j.)
T. During
=
a
He and H0. In the thermodynamic limit of L
in both
—4 limM^oo ln AmaX)
where
Amax
is the
largest eigenvalue
—> oo we
obtain
of the transfer-matrix
DMRG calculation, the value of the chemical potential fj, needs to be constant,
otherwise the renormalized basis is
Thermodynamical quantities
to include the chemical
can
no
longer
an
accurate
description of the original basis.
be obtained from derivatives of the free energy, but
we
have
potential this time
and
WL
As
we
are
potential
tial fj, for
interested in
fj,,
we
with fixed
will have to evaluate the
sufficiently
temperature. The
polation
quantities
methods.
[W„
interpolation
interpolate
accomplished by
The evaluation of
lT.
densities
n
rather than fixed chemical
thermodynamical quantities
many values of n and
is
particle
(4.5.4)
(-)
to the desired
standard cubic
at fixed chemical
particle density
or
n
poten¬
at fixed
quadratic spline inter¬
thermodynamic quantities by Eqns. (4.5.3
and
4.5.4)
87
involves many numerical derivatives. In order to avoid numerical derivatives if possible,
have evaluated the internal energy u, the
particle density
by Eq. (4.2.37), and interpolated these results
heat
was
Let
concentrate
gapless phase
the
field-theory
of
the ID t-J model next.
on
charge
by
cs
=
=
and spin
1
the central
are
velocity
can
vc
and
E\(q)
charges,
\vc
and vc
=
[E0(q
[Bk(q
2-k/L)
=
=
the lowest
are
Kp.
ity
classes
single
Alternatively
dimension,
one
2*/L)
the
the
-
charge velocity
a
or
K
with
Eq(L; N) being
to obtain the
ag of the
ac
limit T
-»
0 of
velocities. The
of small lattices at T
=
0
E0(q
of
2ir
=
=
a
0)] I—
(4.5.6)
0)] /—,
(4.5.7)
spin singlet, respectively triplet, sub-
Numerically
can
by
it is rather difficult to
Kp [7,94-98].
We
can
determine
product of the charge velocity
is determined from
a
=
on
the
boundary
obtained from the correlation exponent
belonging
Luther-Emery liquids [7]
K„
re
charge (spin)
the
long range fluctuations of systems
correlation exponent
compressibility
are
diagonalization
E0(q
-
eigenvalues
Tomonaga-Luttinger liquids
expression is given by
The
specific
given by
Eq. (4.5.6) exactly, since the results depend markedly
obtain vc and vs from
In
low-temperature
The free energy is
(vs)
space of total momentum q, and L is the lattice size.
conditions.
directly
vsJ
be obtained from exact
=
vs
Eo(q)
particle density
n.
[5]
the Lanczos method
where
m
The
magnetization
before phase-separation, the ID t-J model is described by the conformal
6
where cc
In the
Tomonaga-Luttinger liquid [46,93,94].
a
n
to constant
we
by Eq. (4.2.38).
then obtained
us
and the
Kp
to either of universal¬
can
be described
from two
vc and the
relations,
compressibility
re
the first
[97]
-^—.
(4.5.8)
finite-size approximation to the second derivative
(4'5'9)
~~
the
iV2
ground
charge velocity
conductivity
E0(L; N
is
vc
+
2)+Eq(L;N-2)- 2E0(L; N)'
state energy of
we use a
given by
by the
a
system with JV electrons in L sites. In order
relation for the Drude
the energy shift of the
weight
ground
erg.
The Drude
weight
state in the presence of
a
TABLE 4.5.1: The
and J
=
charge
0.5( and different
and
spin velocities
particle
densities
n
vc
and
for the t-J model
vs
on a
length
chain of
boundary conditions
with closed shell
(CSBC).
L
16
=
See text
for details of the calculation.
vc
Vs
0.5
1.06212628
0.47782319
0.625
1.05754460
0.59857716
0.75
0.90300219
0.68891167
0.875
0.73115912
0.74889384
n
[99]
field
by
and for ID systems also
-d2E0(<fi)
irL-
<T0
where
with
a
sites and J
Eq. (4.5.6).
=
we
charge
show the
The
boundary
energies
obtained
were
corresponds
one
to
this
boundary
condition the
ground
for N
conditions. We have
to the DMRG results in
Fig.
up to T
a
ss
are
state is
=
a
=
case are
For ID
Am, with
by
with the Lanczos method.
boundary
either
fully occupied
single-band models,
m
=
being
this
Am + 2 electrons
an
integer.
With
spin singlet. Alternatively choosing the opposite
4.5.1. The fitted
perfect agreement
for N
fit of the
velocities
chain of 16
conditions calculated
for systems with N
Am + 2 and PBC for N
a
on
One reasonable choice is the
noninteracting
(PBC)
(ABPC)
performed
parameter and the charge and spin
in
boundary
diagonalization
carefully.
conditions
conditions
boundary conditions, APBC
and
n
conditions) [93,100,101].
periodic boundary
antiperiodic boundary
asymptotics
exact
electron orbitals in the
and
boundary
by
conditions have to be chosen
empty (closed shell boundary
As
system with twisted boundary conditions
spin velocities for the t-J model
and
0.5t, and different particle densities
condition where all
or
(4.5.10)
2vcKp,
4>=o
phase factor cj>.
In Tab. 4.5.1
The
=
d<j>2
state energy of the
Eq (<f>) denotes the ground
[95]
the correlation exponent
Am will be called open shell
low-temperature asymptotics Eq. (4.5.5)
curves were
were
=
obtained with
/o
taken from Tab. 4.5.1. The
with the free energy
/
obtained
by
as
the
only
free
low-temperature
the DMRG
algorithm
0.25«.
second
example,
let
us
demonstrate how the uniform
magnetic susceptibility
x
can
89
-0.4
-0.5
-0.6
-0.7
-0.8
n=0.875
conformal field
-0.9
0.0
theory
0.2
0.1
0.3
T/t
FIG. 4.5.1: Free energy
density
n.
The
kept, the dashed lines
liquid
density / of
symbols show
are
the the
the t-J chain for J
=
0.5* and different values of the
finite-temperature DMRG results for At
=
0.2t, and
m
the low-temperature asymptotics of the conformal field theory for
model. The coefficients of the conformal field theory
small lattices. See text for details.
were
obtained
by
exact
=
a
particle
60 states
Luttinger
diagonaliztion of
90
FIG. 4.5.2:
At
L
=
Magnetic susceptibility
0.2t, and
~
m
=
60 states
64 sites and fixed
show the
=
O.Oli.
are
in
can see
n
< 1 at
For
and the
a
Fig. 4.5.2,
the
finite-temperature DMRG
3/4.
a
A weak
we
chain of L
magnetic
weak external
have calculated the
=
method for J
field of ft
=
t/2,
=
chain of
on a
O.Olt has been
applied
by the DMRG method.
x
x f°r the t-J chain with J
comparison,
for
in excellent
64 sites and fixed
=
magnetic
t/2
and
an
field ft.
In
external
magnetic field
magnetic susceptibility
x
particle
3/4.
density of
n
=
Fig. 4.5.2,
by
the
QMC
The
QMC
agreement with the finite-T DMRG results at all temperatures. As
the
magnetic susceptibility
high temperatures
for reduced
=
magnetic susceptibility
susceptibility
loop-algorithm
results
n
by
comparison with the QMC loop-algorithm
by Eq. (4.2.39) by applying
be obtained
h
and for
particle density
for the calculation of the
we
kept,
x calculated
T > t, the reduction
particle densities
n.
magnetic susceptibility
start to correlate and form
a
x is
corresponds
At low temperatures
shows
a
peak
2fci?-spin density
only weakly
(T
<
wave.
doping
to the smaller number of
J),
similar to the
The
reduced for finite
a
qualitative change
Heisenberg
peak intensity
chain
as
we
spins
appears
the spins
increases with
higher
doping and the maximum is shifted towards lower temperatures T.
4.6
Discussion
In this
chapter
matrix
we
have reviewed the zero-temperature DMRG
method, and
we
have
explained
the
finite-temperature
form of the Hamiltonian with short-range interactions.
gorithm
can
be
applied
to
a
algorithm
and the transfer-
DMRG method for
a
general
We have demonstrated how the al¬
fermion system away from half
filling by
the inclusion of the
91
chemical potential and tested
ment with the
These
QMC
findings
The
demonstrate that the
finite-temperature
thermodynamic properties
fast, accurate, and works
biggest advantage
Because of the
the t-J chain. The results
results and with the conformal field
method for the calculation of
The method is
algorithm for
DMRG method is
excellent
an
new
of low dimensional fermion systems.
limit of infinite system size.
is that this method does not suffer from the
sign problem, the application of
in excellent agree¬
theory in the low-temperature limit.
thermodynamic
in the
are
QMC algorithms
is
negative sign problem.
essentially limited
to
one
as
the
dimension for fermion systems.
However, the nonhermitian transfer-matrix
occurrence
of
merical instabilities
can
be
numerically
overcome
matrix and loss of
of
only
limitation of the
biorthogonality,
are
larger eigenvalues
nu¬
when
a
break¬
essential ingredients in order to make the
finite-temperature DMRG algorithm numerically stable and
Currently
such
re-biorthogonalization algorithm
a
matrix for
density
The inclusion of these methods
occurs.
problems,
unstable. We have demonstrated how these
by the application
and eventual truncation of the reduced
numerical
cause
complex eigenvalues of the reduced density
which often makes the method
down
can
finite-temperature
more
DMRG
accurate.
algorithm
is the dimension
of the transfer-matrix T. Due to the fact that the transfer-matrix has to be
enlarged by pla-
quettes
t
for
the
a
cluster
decomposition, the dimension of the transfer-matrix
and the number of block-states
kept
m
needs to be reduced
sites of the cluster and the number of states of
implement
the
believe that
of the
algorithm
currently
in
parallel
in order to
this method is the most
thermodynamic properties
a
single
then, depending
on
very
large
the number of
site. Therefore it will be necessary to
investigate
powerful
2D
one
problems. However,
for the numerical
of low dimensional fermion systems and
finite-temperature DMRG algorithm is the method of choice for the
systems for fermions and frustrated spin problems.
get
can
we
we
firmly
investigation
think that the
investigation
of ladder
92
93
Thermodynamic
5.
properties of the
t-J
ladder
Introduction
5.1
Ladder systems of
strongly
correlated fermions
are
simplest systems showing high-
among the
Tc superconductivity and they have been the subject of intensive and ongoing investigations
during
the last few years. Ladder systems
interesting phenomena
spin liquid background
of the
undoped
correlations in the
doped system.
of the 2D cuprates
can
ladder systems
also be
easier to
are
are
system, and
applied
to ladder
doped
of the
of A
comparison
ladder system
280-K-
R3
originally
systems. Due
investigate numerically
theory
between
and
Special
experiment
Sri4_ICaaCu24C>4i
(for x=9) [104]
and
single chains,
as
they
such
show many
the
as
gapped
singlet dx2_y2-like superconducting pairing
Thus many ideas
less finite-size effects than 2D simulations.
that the
attractive for theoreticians
of the 2D cuprates not present in
description
formulated for the
to the reduced
dimensionality,
than the 2D systems and suffer from
interest has been created by the fact
possible with the realization
has become
[102,103].
This material shows
superconductivity under high
a
spin
gap
pressure has been found
(P>3GPa) [104,105].
The
groundstate properties
extensively
in the past few years
Lanczos method
[81,108]
and
the
by numerical methods such
[56,106,107], density
analytically [109-111],
emerges from these
material,
of the t-J and Hubbard ladder systems have been studied
investigations
see
Ref.
and
[55]
for
a
large
be described
correlations with
amount of research done
(DMRG)
by
a
calculations
picture
The
spin liquid
short range
dI2_y2-like
on
the
that
groundstate of the undoped parent
the spin gap remains and power law
singlet superconducting pairing
In contrast to the
can
diagonalization by
complete review.
is rather consistent. The
antiferromagnetic Heisenberg ladder, consists of
Upon doping of holes,
exact
matrix renormalization group
gap and magnon excitations. This system
bonds.
as
the
with
a
finite spin
resonating
charge density
symmetry
wave
valence
(CDW)
appear.
groundstate properties,
much
94
properties
of these systems. In
t-J ladders
by
less is known about the finite temperature
doped
of
properties
of the transport
study
systems, the results presented in this chapter
properties
of
doped
ladder system of
investigate exactly
able to
are
a
the
magnetic susceptibility
n
=
thermodynamic
+ J
2s
3,0.
single
j
sum
The
index.
is standard.
( Sj,a.Sj+ita
-
^
runs
the rungs,
over
implements
trate
on
J'
V
=
a
a
roughly
otherwise,
the
at
studies.
we can
investigate
have studied is the
will
couplings
always
are
simply given by
a
This
are
Doping
than the
on
to be
coherent
the rungs,
single
a
hole
of
an
a
effective
rung
Fig.
terms
singlets
a
a
2.6.2.
are
the
t'.
groundstate
across
Unless not
we
will
concen¬
of this limit and
of the
undoped
the rungs in the limit of
of the rung
one
singlets.
spin-1/2 quasi-particle (SHQP)
by
binding
These
=
1/3.
«
simple picture
a
The
spin-charge separation,
singlet.
hopping
(J')
materials,
requires breaking
and that has been confirmed
breaking
leg
rest of the notation
J/t
terms t
in the real
couplings.
spin, creating
not show
is the
double occupancy of
terms
hopping
there is
as
superposition
a
•
of the ladder system in
exchange
isotropic
of additional holes leads to pairwise
boson model with
Jnj,lnj,2
In cuprate systems, it is estimated that the
consider isotropic
expected
picture does
isotropic couplings
~
the kinetic energy, and the J
larger
three times
we
minimize the cost J' of
core
system
and the
temperature DMRG
we
ni,a,tni,a,i) prohibits
—
(rungs).
the ladder
strong exchange couplings
charge +|e|.
even
finite
Cv,
spin index, and o(= 1,2)
denotes the
HiaC*
The hole is then bound to the unpaired
the
The
graphical representation
J, t, t'. Doping this system with
3>
new
heat
we
3
many of its properties remain valid at isotropic
system is
the
specific
strong on-site Coulomb repulsion. The
the
We have shown
term is
Although
%> the
J + J' 2_, (^ Sj,lSj#
'
exchange couplings along
mentioned
thermodynamic
given by
jn3>an,j+ita
The first two terms of the Hamiltonian
hopping
of small
3,"
projection operator
site and
on
length.
limit of infinite
3A,a
where the
the
on
a
E V Ow**-W + B.c) T-t'^V {c\Xac]Xa + H.c) V
*
-
based
are
t-J ladder model, its Hamiltonian is
two-leg
the first exact results
have discussed in the previous chapter. With this method
we
systems in the
diagonalization [112]
exact
from
correlated fermions and for the first time
strongly
entropy 5. These numerical investigations
method that
are
fact, apart
a
fact that remains valid
many numerical and
of holes
hole-pairs
parameter t* and
a
on
can
with
analytical
the rungs in order to
be
mapped
to
a
hard¬
weak attraction of the hole
95
pairs
V*
in the
[56,107].
that the
sense
Remarkably,
the
the rungs,
on
This
the
excitations
spin
in the limit of
chapter is organized
universality
to the
same
isotropic couplings J'
regime by considering
results
can
on
the
density nh(n,T)
the hole
magnetic susceptibility
be described
by
we
strong couplings
pairs only, and
on
will
we
we
x a* fixed
The
investigate
this
and the
we
of the
magnon-excitations
will discuss the entropy 5 and the
will
and show how these
particle density
SHQP
briefly
will
the rungs.
for fixed chemical potential /i. Then
combination of the
a
undoped ladder system. Further
specific
Cy.
heat
Numerical methods
5.2
have used the novel finite temperature DMRG
In this
study,
of the
thermodynamic properties
namic
we
quantities
are
of the
obtained from the
two-leg
merical methods has been limited to
time-steps M.
creasing
This
problem
time-step At
T/M.
algorithm
and
unbiased, the only
truncation
errors
an
kept
m
by extrapolating
calculations,
is
to
T and
a
negligible compared
0
the truncation
errors
fifty.
to the
but its
a
application
for
nu¬
small number of Trotter
algorithm by
constant value of
a
single
limit of infinite system size. The results
These truncation
and the
by fitting
to
error
a
errors are
in¬
Trotter
method,
are
usually
exact
very small if
in the Trotter time steps
polynomial
the size of the Trotter time steps ranges from At ft
number of Trotter steps from ten to
are
[26],
property of the virtual transfer matrix
large enough,
->
By this method, the thermody¬
from the finite size of the Trotter time steps and the
algorithm.
At2
for
investigation
of the virtual transfer-matrix. The
time since
iteratively
outstanding
originate
for the
algorithm
the finite temperature DMRG
by
thermodynamic
errors
long
high temperatures
is solved
of the DMRG
the number of states
eliminated
As
works in the
this
At2
a
the number of Trotter time steps
=
t-J ladder.
largest eigenvalue
virtual transfer matrix method is known
in
the hole
by
is determined
regime
low temperature
gapless.
couplings
important aspects of
of the finite-T DMRG method to the t-J ladder model. Then
review the excitations of the t-J ladder in the limit of
concentrate
to weaker
J.
=
will discuss the most
we
liquids [7],
excitations remain
universality class by going
follows: First
as
class of Luther-Emery
charge
while the
gapped
are
remains in the
system
even
application
belongs
This model
=
in
0.01 to
error
from the DMRG method.
interpolation
to fixed
be
At2. In the present
Ar/t
=
0.2 and the
For these values of the Trotter number the
of the
can
particle
errors
number and
96
biggest problem
The
for
complex
a
implementation
in the
system such
of the finite temperature DMRG
Hilbert space grows much faster than for
comparable
a
Since the virtual transfer matrix needs to be
on
a
site and
single
m
of the system
possible.
as
Our calculations
of
we
with
an
kept for
to converge for
m
Since the particle
to evaluate thermal
the transfer matrix
our
calculations ranges from
m
m
for
symmetries
analogue
of the spin
subspace
the rungs. The
40 to
=
only m2d2
many
as
have used the
along
at each
3 is the number of
contrast to
exploit
to
performed
=
is also useful in order to alleviate numerical instabilities
number of states
=
(see
60,
of
zero
exploitation
Sec.
4.4).
The
and the results
50.
=
density fluctuates strongly
expectation
with the chemical
values of local observables
potential
directly
/x, it is necessary
from the eigenvectors of
by Eq. (4.2.37).
Results
5.3
Let
important
number and the reflection symmetry of the ladder
symmetries
seem
are
(see Fig. 4.2.2)
where d
kept, in
(Eq. (4.2.41)),
conservation for the virtual transfer matrix
winding
very
cluster
a
m2d4,
is the number of states
zero-temperature method. It is therefore
the
temperature DMRG calculation.
zero
enlarged by
DMRG step, the size of the Hilbert space scales like
states
algorithm
the t-J ladders stems from the fact that the dimension of the
as
start with
us
Refs.
[56,57].
t-J ladders
As
are
a
we
brief review of the low energy excitations of the t-J ladder
have mentioned in the
the collective
The matrix element for the
charge
hopping
introduction,
modes created
is
given
by
the
only gapless
the coherent
in second order
following
excitations of the
hopping
of bound
pairs.
perturbation theory by
2i2
There is
a
weak attraction among the hole
1
Phase
region
separation
that
of these
we
occurs
only
consider in this
gapless singlet charge
only gapless
excitations
even
at
at
pairs
neighboring
on
3J2
rungs, to second order it is
4t2
unphysically large parameters
study.
In the strong
excitations is
coupling
quadratic.
isotropic couplings J'
=
These
J,
far
beyond
the parameter
limit J' > J, t, t' the dispersion
charge excitations
but the
remain the
dispersion relation
is linear
97
in this
1]
Ecs[Nh
—
=
Nh holes. In
the
bonding
the cost of
t'
on
binding
The
case.
0]
the
or
-
energy of the hole
Eas[Nh
antibonding
breaking
the rungs. Two holes
binding
quasiparticles
essentially
are
behave
as
the lower
of
a
J'
=
more
bonding
lying bonding
complicated
lowest
are
band of the
lying
excitations.
bonding
kx
SHQP
spin excitations
are
given by
are
perturbation theory
=
SHQP
=
applies
couplings,
a
it is
is
=
1/2.
These
SHQP
antibonding
coupling
separated by
due to
the energy
2t'. The energy gap from the groundstate
Aqp
for the
pair
orbital and two separate
it is reduced at weaker
are
a
=
Eb/2
and the
coupling
bonding band,
dispersion
limit. At
and the
to
has the form
isotropic couplings
dispersion relation is
created
by breaking
remnants of the
triplet
state
the magnons
SHQP
riQp, is limited
by
a
hole-pair
are
in
a
triplet
are
the
state. The
undoped Heisenberg ladder, the "magnon"
on a
larger
fully occupied
rung and
than that of the
SHQP,
move
along
the
to second order
given by [113]
and the magnon gap Am
difference among the
SHQP,
By breaking
2.6.3.
antibonding
or
in the strong
-k
\-k
Am
At isotropic
but their kinetic energy is
(5.3.3)
Pig.
in
The two bands
states
ladder. The excitation gap of the magnons is
in
the ladder and
spin excitations in the system. The above discussed SHQP
ones, since the
They
along
given by
[57].
two kinds of
second kind of
orbital is
given by
presented
coupling limit,
antibonding
the minimum shifts to kx
There
bond,
one
energy is therefore
excited states.
cosine with the minimum at
J,
lying bonding
free fermions and the bandwidth of both the bonding and
and
system with
a
J'-2t-2t>+j)_4t>2/jr
=
=
into the ladder will stay in either
in the kinetic energy of t
of these states is
higher
2Eos[Nh
=
energy of
created, each of them carrying charge +|e| and spin S
with the
difference of the
by Eb
At2
orbital bands is 2t in the strong
hybridization
gain
either go into the
can
groundstate
doped
rung also break
single
on a
graphical representation
of holes, the holes
is the
orbital. The energy of the lower
EB
A
be estimated
can
hole
single
a
bond J' and the
a
Eos[Nh]
where
strong coupling limit,
t*. An estimate of the
only
2])
—
pairs
1J2
=
are
J'-J+2J7-
spatially
remains
larger
more
(5.3.4)
extended,
than the
SHQP
but the
gap
same
Aqp.
picture still
An important
excitations and the magnon excitations is that the number of
the total number of holes
doped
into the ladder, uqp < nu, where
98
Exactly
rth is the concentration of holes.
doubly occupied
solely by
determined
i'/2
hard
=
J'/IO
singlet
for the rest of this
boson gas with
core
neighbor
the
attraction V*
in this ID
of
t-J
they require
As
chapter.
(Eq. (5.3.2)),
ladder in the limit J' S> J,t
two-leg
pairs [56], and
hole
will consider the
we
have discussed
we
case
pair fluid
can
be
i/2
=
above, the hole pairs form
(c.f. Eq. (5.3.1)). Neglecting
the hole
of J
are
mapped
to
=
a
the weak nearest
an
ideal Fermi gas
geometry with
eHP
As the calculations
hole
doped
a
bandwidth of 4t*
a
as
n^.
—
properties
properties
The low temperature
to 1
proportional
rungs and their number is thus
Low temperature
5.3.1
holds for the magnons,
opposite
the
were
=
performed
eHP +
in the
2t*
cos
grand
k +
(5.3.5)
2/j,nh.
canonical ensemble
first
we
investigate
the
density
«*(/*,T)
A selection of
Fitting
our
presented
results is
energy to add
a
hole
pair
diagonalization
to
from considering
5.3.1.
an
=
4.82(6)*
undoped ladder
validity
=
Fig.
seen
by
the liard
a
/
+
is enp
boson model for the hole
boson model in this
the fit to the hole
figure
involves
no
—
(ejj-p
obtain
2t*
~
density
=
estimate for the center
1.5(2)*.
4.1(l)i,
=
4.71*,
core
an
4t*
fa
in
The minimum
good agreement
1.494*
[56]).
boson model for the hole
and the energy
pairs. We
density nh(ji,T) (Eq. (5.3.6)).
density
doping (nh
uass
<
note that the
fit since all the
pairs
per site
(1 -n#P)ln(l -nHp)],
5.3.2 the entropy at T < 0.3* and low
core
we
bandwidth 4t*
of this hard
-lnHP^nHp
determined from the free energy density
be
and
the low temperature entropy
shp
in
Fig.
results for finite clusters
A further confirmation of the
comes
in
(5.3.6)
the data for rih < 0.1 at temperatures T < 0.54
of the band for hole pairs at enp
with exact
\ f'dk /
=
0.1)
(5.3.7)
=
(w-/)/T.
As
can
is also well described
entropy of the hard
core
parameters have been obtained from
99
/ A
/
0.08
/
A
r
/
/A
A
0.06
nji=-i.5t
V
A
A
A
On=-1.81
/'
?
1
A|l=-1.92t
—
.,.
-
A
0.04
9
4?
0.02
O,'''
<?
0.3
T/t
FIG. 5.3.1: Hole
density
strong coupling regime J
hole
n^asa
=
t/2
—
function of electron chemical potential p and temperature T in the
J'/IO.
The dashed lines
are
pairs. Note, the hole pair chemical potential is —2/i. The
extrapolated
fits to
a
hard
core
boson model for the
size of the Trotter time steps has been
to At —> 0.
0.35
1
1
1
1
1
1
n=-i.st
0.30
*•'
O,,-'''
On=-1.8t
0.25
0.20
1
An
=
-1.92t
«H
=
-2t
«'
fit
.
_«--'"'
0.15
A
A
,--'
.'-
j»'
A,--'
A--
A,--''
A-'"
°
„
,--
.•
9--''''
p--'''
A^
0.10
S>'
~-9'
0.05
a'
0«°
s--a
0.00
0.20
T/t
FIG. 5.3.2:
Entropy density
coupling regime
as
in
J
—
t/2
Fig. 5.3.1, using
extrapolated
=
5 as a
J'/10.
function of chemical potential \i and temperature T in the strong
The dashed lines
are
the values for the
same
hard
core
boson model
the parameters obtained in that fit. The size of the Trotter time steps has been
to Ar —> 0.
100
Susceptibility
5.3.2
At
and the thermal excitation of magnons from rung
processes show up in the
spin susceptibility x(T),
ensemble with fixed hole density n^. Therefore
The values of x(^i
of
a
At
h/t
=
5
x
govern the
which is easiest to
we use
n/j(/i,T)
interpret in the canonical
x(/i,T)
to remap
x(nh,T).
—¥
in the presence
10~3 (Eq. 4.2.39). The results for x{nh,T)
high temperatures
/,
T »
x follows
Curie-law for free spins x
a
it decreases when the temperature is lowered below the magnon-gap
peak
The maximum of the
a
thermodynamics. These
by measuring the magnetization (SZ(T))
calculated
were
singlets
presented
are
5.3.3.
Fig.
in
T)
small external field
independent SHQPs
the thermal dissociation of hole pairs into two
higher temperatures
is shifted towards lower T with
by the holes. At
of x with
a
indicating
that the energy of
very low temperatures of T < 0.5*
smaller gap, which
we
a
Aqp
=
Eg/2.
~
ft)/4T,
~~
4.13*
(Eq. 5.3.4).
localized magnon is not much
we can see a
Note the
and
Simultaneously the magnon
second
attribute to the recombination of
temperatures below the SHQP-gap,
Am
(1
increasing doping, indicating
reduction of the magnon gap due to interactions with holes.
bandwidth is enhanced,
=
changed
exponential decrease
SHQPs into hole pairs
at
of this contribution
magnitude
increases with n/,.
A quantitative
of the
description of x(nh, T)
SHQPs xqp and of the
can
be
given by adding separately the contributions
magnons xm, i.e.:
X(nh,T)
=
XQp(.nh,T)+xM(nh,T).
the value for free spins XQP
Xqp is approximated by
dependent density
we
parameterize
as
of the
£qP
—
SHQPs determined by
Aqp
+
a,QP(l
+
n
The
in hole
density
pairs
of rungs
is 1
—
n&
occupied by
but
that the rung
two
density
so
simply
to scale the form for
is then 1
undoped
nQp{T)/4:T
with
dispersion
of the
the energy
fc)/2
J-it
spins
exciting SHQPs
SHQP
cos
=
with
ePeQP
at low
/3
(5.3.8)
=
raj,
ladders
—
SHQPs which
1/T:
+ 1
temperatures where all holes
reduces the number of such rungs
—
temperature
a
uqp.
Our approach to
proposed by Troyer
et al.
a
by
are
one
bound
for each
model for xm is
[27] by
this
two-spin
101
0.04
0.03
0.02
0.01
0.00
Z~~j
(b)
0 20
s*
0.15
0.10
n„=0.0
n„=0.05
on„=0.1
Anh=0.15
x
n„=0.2
n
o
0.05
fit
noo
U-—
1
i
1
10
15
ii
—
20
T/t
FIG. 5.3.3:
Uniform magnetic
different hole-densities
and the solid lines
in Tab. 5.3.1.
(b)
shows the
temperatures.
are
ni,.
The
susceptibility per
symbols
the fitted
In the upper
high-temperature region
\t of the t-J ladder for J
=
t/2
=
denote the results of the finite temperature DMRG
curves
figure (a)
site
according
we
to
show the
and the
Eq. (5.3.8).
The
crossover
and
are
listed
while the lower
figure
fitting parameters
low-temperature region,
J'/10
algorithm,
to the Curie-behavior for free
spins
at
high
102
TABLE 5.3.1:
Gap of the spin S
1/2 quasi-particles AqP
=
and magnon gap
Am,
well
as
as
the
parameters uqp (am) which determine the bandwidth of the quasi-particles (magnons) obtained by
fitting Eq. (5.3.8)
to
our
finite temperature DMRG data for different hole densities
0.0
rung
density leading
[A|f
z((3)
+
=
4aM(l
0.7(1)
3.4(1)
0.9(2)
1.6(2)
0.05
0.8(1)
3.3(1)
0.9(2)
1.8(2)
0.1
1.0(1)
3.3(1)
0.6(3)
1.7(2)
0.15
0.9(1)
3.2(1)
1.3(2)
2.0(2)
0.2
0.9(1)
3.2(1)
1.4(2)
2.0(2)
to
=
(l-nh-rIQP)/3r-^^,
and the magnon
we
obtain excellent
the whole temperature range
(see Fig. 5.3.3).
is the decrease of the magnon gap
magnons and
SHQPs.
Due to
o,qp is also reduced from the
is in reasonable
Aqp
increase of
effective
Aqp
5.3.3
Let
us
leading
are
The parameters obtained
order
by
shown in Tab. 5.3.1. The main
Am [31,81,114,115]
hybridization
as
ekM
=
with
fit of this
change
upon
due to interactions between the
higher lying
perturbation
a
bands the
result oqp
=
SHQP
bandwidth
2t, but the SHQP gap
agreement with the second order perturbative estimate of 0.98i. The
Aqp (or equivalently
repulsion
was
dispersion is parameterized
agreement with the finite temperature DMRG data
model to the finite temperature DMRG data
doping
(5.3.10)
cosfc)]1/2 [109].
With this model
over
0.7(1)
-
0.025
/^r?rdfc(27r)_1 exp(—PekM),
+
a-M
a,Qp
4.1(1)
-
XM
where
AM
Aqp
nh
n*..
between the
found in Ref.
[115].
the
binding
SHQPs
This is
energy
and hole
an
Eb)
with n^,
can
be attributed to
pairs. A similar increase of the SHQP
issue which warrants further
an
gap
investigations.
Entropy and specific heat
start with the discussion of the
entropy-density
entropy of the doped ladder is increased compared
the additional
degrees
s
to the
first. At
high temperatures T,
undoped Heisenberg ladder
of freedom created in the presence of holes. In the limit of T
the
due to
-> co
the
103
FIG. 5.3.4:
Entropy density
of the doped t-J ladder for J
a
t/2
=
t'/2
=
-
J'/10
and different
hole-densities n&.
entropy density is given by
is at rift
=
1/3.
In
concentrations n^.
Uh
At
acquired
entropy has
=
Fig.
s^
5.3.3
=
we
(1
shows
liquids.
n^
—
uncertainties
us
fits
are
arising
entropy-density
(1
—
n/J ln(l
—
a
s
=
0.
Am,
the entropy decreases
In the presence of hole
however better
above
specific
can
performed
remapping
heat
also be
core
on
in the
Cv-
as
is
the maximum
s(fi,T),
The
=
as
the
T/t
=
20 the
0.025 and 99.7% for
for the
undoped
exponential decrease
expected for Luther-Emery
boson model
to constant hole
CV finally.
seen
n/,),
exponentially,
doping,
linear decrease at low temperatures,
from the
—
t-J ladder for various hole
doped
between 99.4% of its maximal value s<x> for «a
consider the
s
lnn^
the entropy is almost constant, at
This behavior is consistent with the hard
Quantitative
Let
to
In 2
high temperatures,
ladder down to
a crossover
rift)
show the entropy of the
0.2. Below the magnon gap
Heisenberg
—
we
proposed for the magnons.
did earlier, due to added
doping.
same
excitations
as
discussed for the
specific heat
as
Because of the additional numerical derivative involved in the calculation of the
(5.3.11)
specific heat,
104
FIG. 5.3.5:
Specific heat of
calculated
the
by
the t-J ladder for J
more
seen
that at
Soo, there is
same
pronounced peak
a
to the
undoped
be found below the
X, the
t'/2
=
J'/IO
and different hole-densities m,
case.
Nevertheless,
specific
system due to the
SHQP
by
now, followed
gap
AqP
an
Fig.
5.3.3 it
drops from its
exponential decrease due
doped system
heat of the
SHQP,
s
in
and
a
is
T
can
—> oo
be
value
to the magnon
markedly
increased
weak second exponential decrease
at T < 0.5.
Discussion
5.4
In this
=
temperature T where the entropy-density
gap Am- At low temperatures T, the
can
t/2
difficult to get accurate results in this
it is
compared
=
finite-temperature DMRG algorithm.
chapter
we
have
entropy density
temperatures. This
presented
for the first time exact results for the
problem
specific
heat
Cy
has been attacked
by
s, and the
of the
doped
magnetic susceptibility
t-J ladder down to very low
number of numerical
a
algorithms
without
success.
Based
low-lying
of the
on
an
intuitive
picture
in the strong
excitations of the t-J ladder.
pairs of bound holes.
carrying both charge
and
spin
The
By breaking
are
coupling limit J'
>
only gapless excitations
such
a
created. These
pair
are
of bound
the lowest
J,t,t'
are
holes,
we
the
two
discuss the
charge-modes
quasi-particles
lying spin-excitations.
The
105
third kind of excitations is inherited from the
It is
energy-gap of all these excitations.
rung.
Despite
the
largest
magnetic susceptibility
is
proportional
density
these
are
the t-J
proportional
picture
we
be discussed
and
and the magnons
a
are
more
detailed
to
investigate
=
investigate time-dependent
finite-size version of the DMRG
size version of the
algorithm
are
results
by
the
for the bulk of the
as
their number
two kinds of
x and the
given
is
we
SHQP
later.
a
wider parameter range,
have discussed here.
correlations
However,
where the interactions among the
finite-temperature DMRG algorithm
we
entropy-
finite-temperature DMRG method for
[72].
Basically
i.e., the
it is also
it is also
case
possible
possible
to
Matsubara-frequencies by analytic
For this purpose
one
has to
use
the
have discussed in section 4.2.1. With the finite-
obtain results
only
for
by
infinite-system
fully occupied
a
agreement with the DMRG results.
in contrast to the whole series of temperatures obtained
method.
responsible
magnetic susceptibility
investigations
algorithm
one can
on
low-doping regime,
correlation functions for the
imaginary-time
triplet-state
system, while for the other
investigation,
J. With the
other quantities than
continuation of the
good
in
taken into account will be
isotropic couplings J'
local
to the hole-concentration.
preliminary
Another issue that is left for later
of
a
heat in the
have shown how the
quantitatively
the very first and
ladder,
specific
to the number of electrons in the
this
on
s can
given by
energy-gap, these excitations
and of the
excitations the number is
Based
undoped Heisenberg ladder, and has the largest
a
single
the
value of the temperature,
enlargement procedure
of the
106
107
6.
Conclusions
In this PhD work
powerful
advanced and
strongly
algorithms
numerical
and the finite temperature
QMC loop algorithm
algorithm
be
can
generalized
density
quantum systems with
to convention local
at much lower
The second
algorithm.
from the
a
simple six-vertex
sign problem.
slowing
currently
most
An
new
more
development
This allows the
by
complicated fifteen-
contribution that
of
improved
can
estimators for
orders of magnitude
investigation
also be
QMC loop algorithm
Our results prove that the
updating algorithms.
algorithm
we
have discussed and
developed
unstable otherwise for
more
algorithms
for the simulation of
large
a
of much
are
large
compared
larger systems
complementary
to
or
some
The
a
belongs
to this
chapter.
sign problem
class, especially
can
and is the
issue of modern scientific
is
numerically
computing
that
to low
can
half-filling.
is weak
is best
or
not
many unfrustrated spin
only
on
be applied
efficiently
the other hand does
only method capable
sign problem down
a
We have demon¬
QMC loop algorithm
extent. The
finite-temperature DMRG algorithm
a
the inclusion of
fermionic systems away from
sign problem, the algorithm
negative sign problem
by
finite-temperature DMRG
systems in any dimension where the
low-dimensional fermion systems with
important
the
complicated systems
systems. For simulations with
not suffer from the
is the finite temperature DMRG
which is discussed in the fourth
number of systems
high temperatures.
improved
stabilized version of this method
re-biorthogonalization,
strated that without the
present. A
We show how the loop
model to the
important
loop
algorithm.
temperatures than before.
We have
These two
chapter.
down" and the simulation times
re-biorthogonalization algorithm,
An
matrix renormalization group
to other models than the t-J model is the
reduces the "critical
at
two
for the numerical simulation of low-dimensional
is discussed in the second
vertex model and thus to the t-J model.
applied
implemented efficiently the
and
correlated fermion systems at finite temperatures: the quantum Monte Carlo
algorithm
The
investigated
have
we
to simulate
large
temperatures.
not be shown in this thesis
108
here is the
development
become
efficient
that
ing
as
not be
can
methods.
by object-oriented design. Object
complex algorithms due
into FORTRAN programs. One
methods
The first system
we
are
have
chapter
chains in
investigated systems beyond
three.
In order to capture the
each with its
Curie-constant, separated by anomalies
system of this type, and
data
not
are
we
have
chapter
specific
in the
investigated
a
investigated
have
we
a
magnetic susceptibility
numbers of
legs.
Three
pair of bound holes and the
these excitations
x> the
are
seems
The
a
into the
find three
clearly
x and its
own
alloy SrsCuPti-ajIr^Oe
of the t-J ladder
long
special
time and
interest
low-temperature
s
in
we
find
s
as
they
a
specific
allow to
can
be
on
analytical
study
created
undoped Heisenberg
of
in this direction is needed
gapless
by breaking
ladder.
doping,
binding
a
Based
x and
energy of the
pairs of bound
the other hand.
are
among the
strongly
on
the
We find that the gap of the
numerous
examples
where the fruitful combi¬
results with numerical methods is needed in order to
complex physics
of ladders with
quantitative description of the magnetic susceptibility
while the
to low
the dimensional
distinguished:
1/2 quasiparticles
good agreement with the numerical results.
to be enhanced
exact results
Cv down
heat
spin-liquid groundstate
excitations
=
present the first
we
and the
magnon excitations of the
systems discussed here
nation of
down to very
susceptibility
heat. The novel
out of
thermodynamical properties
entropy-density
of
magnon excitations is reduced upon
holes
Curie-like
we
=
realistic model of SrsCuPti-jJr^Oe in order
from ID to 2D systems, and because of the
the entropy
gradual freezing
S
low-temperature
modes of the pairs of bound holes, the spin S
on
ferromagnetic spin
accurate
the
great challenge for
temperatures. Ladder systems
even
own
previous
yet available for this material.
systems. This has been
crossover
has
portable paralleliz¬
the scope of
predictions
comparison with experimental results. Unfortunately,
In the last
for the
the
design
data structures
large systems
in these systems, the simulation of very
spin segments
to allow the
and
antiferromagnetic
the random
distinguishable temperature regimes,
a
optimized
example is
low temperatures is necessary. In agreement with theoretical
is
to
oriented
library used for the implementation of the QMC loop algorithm [33,34].
new
1/2 Heisenberg
effective
FORTRAN code for
incorporated
Monte Carlo
With these
as
of reusable code
correlated fermion systems.
the way to the full
understanding
gain deeper insight
Definitely
much
more
of the 2D cuprates.
work
109
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Curriculum Vitae
Last
name:
AMMON.
First
name:
Beat.
Nationality:
Swiss.
Date of birth:
August 27th,
Marital status:
Single.
First
1969.
German.
language:
Foreign languages: English, French, Spanish.
April 1976
March 1982:
-
Primary
April
1982
-
school: Primarschule
September
Secondary
November 1988
Hohfuri, Biilach,
school: Kantonschule Ziircher
-
Switzerland.
1988:
February
Unterland, Biilach, Switzerland.
1989:
Internship: Swiss Register of Commerce, Zurich, Switzerland.
February
1989
Military
October 1989
-
June 1989:
service:
-
Isone, Switzerland.
October 1994:
University: Diploma
Switzerland.
in
physics
at the
Eidgenossische
Technische
Hochschule, Zurich,
Diploma thesis: "A comparison of the Properties of the One-Dimensional
Hubbard and t-J Models"; supervisors: Prof. Dr. T.M. Rice and P.D. Dr. D. Wiirtz.
October 1994
-
today:
PhD work at the
Eidgenossische
Methods for the
Thermodynamics
Technische
Hochschule, Zurich, Switzerland. "Numerical
of Low-Dimensional Fermion
Prof. Dr. T.M. Rice and P.D. Dr. D. Wiirtz.
Systems"; supervisors: