Quantum decimation renormalization group method for one dimensional spin-1/2 systems
by Chen Xiyao
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Chen Xiyao (1984)
Abstract:
We have extended both the reliability and the range of application of the decimation renormalization
group method for calculating the thermal and magnetic properties of 1-dimensional quantum spin—1/2
systems. Efforts to improve the accuracy include increasing the spatial rescaling, investigating the
effect of free versus periodic boundary conditions for each renormalization cluster and varying the
iteration procedure. The systems under investigation include (1) spin chains with isotropic Heisenberg
as well as Ising-like and XY-like anisotropic exchange in the presence of a longitudinal field, (2)
chains with uniform antisymmetric exchange in a longitudinal field, (3) chains with alternating
antiferromagnetic interactions in a field, and (4) those with anisotropic interactions in a field with
arbitrary direction. The principal calculated results are magnetic and thermal response functions
(susceptibility, magnetization and specific heat), which are compared (where possible) with previously
published results using other techniques. QUANTUM DECIMATION RENORMALIZATION GROUP METHOD
FOR
ONE DIMENSIONAL SPIN-1/2 SYSTEMS
by
Chen Xiyao
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Physics
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 1984
ii
APPROVAL
of
a
thesis
Chen
submitted
by
Xiyao
This
thesis
has
been
r e a d b y each m
t h e s i s c o m m i t t e e a n d h a s b e e n f o u n d to be
regarding
content,
English
usage,
format
bibliographic
style,
and c o ns i s t e n c y ,
and
s u b m i s s i o n to the C o l l e g e of G r a d u a t e S t u d i e s
ember
of t h e
satisfactory
,
citations,
is r e a d y f o r
Graduate
Approved
ILr L
for
the
Major
C oin m i t t e e
Department
m
Date
Approved
for
the
College
of
Graduate
G r a d u a t e ^Dean
Studies
ii i
S T A T E M E N T OF P ERMISSION
In p r e s e n t i n g
the
requirements
University,
available
further
for
I
to
referred
Zeeb
copies
that
copying
in
the
or
'the
of the
to
exclusive
Signature
—
this
/liL__
shall
of
thesis
Montana
the
State
make
is allowable
'fair
Copyright
Law.
Requests
right
of
this
International,
to
to r e p r o d u c e
in and f r o m
distribute
by
whom
as
for
should
300
I
only
use'
thesis
48106,
it
Library.
with
Michigan
and
at
consistent
dissertation
reproduce
rules
Microfilms
Arbor,
f u l f i l l m e n t of
Library
reproduction
format. '
Date
of
U .S .
to U n i v e r s i t y
Ann
degree
the
under
puposes,
copying
Road,
granted
right
that
in p a r t i a l
a doctoral
agree
scholarly
extensive
for
borrowers
agree
prescribed
this t h e s i s
TO USE
be
North
I
have
and d i s t r i b u t e
microfilm
and
the
abstract
in
any
V
ACKNOWLEDGMENT
I w i s h to express my g r a t i t u d e to P r o f e s s o r Ge o r g e
Tuthil I, who suggested and supported the research reported
here, for his u n l i m i t e d advice and patient d i s c u s s i o n s
during the past three years. He has read and corrected this
manuscript, and given detailed comments which were valuable
for the p r e p a r a t i o n of the final version. I wis h to thank
him sincerely for his help.
Sinc e r e thanks also to P r o f e s s o r Wm. K i n n e r s l e y and
Dr. X u e y u a n Zh u of the MSU P h y s i c s D e p a r t m e n t . It w o u l d
have been impossible to finish so much calculations! work
without the programming assistance which they provided at
the computer facility at Montana State University.
vi
TABLE OF CONTENTS
. page
V I T A ......................................
iv
A C K N O W L E D G M E N T ..................
TABLE OF CONTENTS
LIST OF FIGURES
v
......................
.vi
.......................
vii
LIST OF T A B L E S ................
x
A B S T R A C T .......... * ................... ....................
I.
I N T R O D U C T I O N ........... ............... i . . . . .
xi
I
II.
2.1 QUANTUM DECIMATION RG
2.2 SPIN CHAINS
. . . . .
.........
IN A LONGITUDINAL FIELD
...
10
. . . . . .18
2.2.1 Ant i sy mine trie Exchange ' . ................... 18
2.2.2 Alternating Antiferromagne tic Spin Chains
Decimation for L = 3,5 and 7 Free-End Cluster
Periodic Boundary Condition:
. 30
33
the 6-Spin Ring 42
Thermodynamic P r o p e r t i e s ................ ' . .
52
2.3 SPIN CHAINS WITH ARBITRARY FIELD DIRECTION . . . 73
III.
DISCUSSION AND S U M M A R Y ......... ..
REFERENCES
.
. .............. 92
98
vii
LIST OF FIGURES
page
1.
Decimation
2.
The approximate I —D d e c i m a t i o n ...........
14
3.
Specific heat vs temperature for I-D XY model with
antisymmetric e x c h a n g e ...........
28
Specific heat vs temperature for I—D Heisenberg
chain with various antisymmetric exchanges . . . .
29
4.
5.
in I-D with L=2
. ................... ..
.
Susceptibility vs field at constant temperature
. for copper nitrate . . . . . . . . . .
6.
Two cluster types
for, the
alternating spin chain
7.
The second iteration approach
calculation
11
31
.
34
for free energy
41
8.
Bond-moving
9.
S p e c i f i c heat vs t e m p e r a t u r e for f e r r o m a g n e t i c
Heisenberg chains by decimation using L= 3, 5 and
7 f r e e — end c l u s t e r s ......................... .. . .
59
S p e cific heatvs t e m p e r a t u r e f o r a n t i f e r r o m a g n e t i c
isotropic Heisenberg chains by decimation using
L= 3 , 5 an d 7 f r e e - e n d c l u s t e r s ...................
60
10.
and spin ring decimation . . . . . . .
43
11.
Specific heat vs temperature at constant anisotropy
y
for I s i n g - I i k e spin chains
. . . . . . . . . 61
12.
Specific heat vs temperature
Y for X Y - like spin chains
13.
Susceptibility vs temperature for isotropic spin
chain with ferromagnetic coupling at constant
m a g n e t i c field ........ . . . . . ................
at constant anisotropy
. ................... ..
62
63
14.
Susceptibility vs temperature for antiferromagnetic
i s o t r o p i c spin chain in zero m a g n e t i c f i e l d
.
64
15.
Susceptibility vs temperature for anti ferromagnetic
isotropic spin chain at constant magnetic field
. 65
16.
Specific heat versus temperaturefor
antiferromaghe tic isotropic spin chain at constant
m a g n e t i c f i e l d . ........... ......................
66
vi.ii
17.
Susceptibility vs field for a l t e r n a t i n g chain
at two different temperatures by L=3 decimation RG
61
\
18.
19.
Susceptibility vs field for a l t e r n a t i n g chain
at low temperature by L= 3, 5 and 7 decimation
approaches and the iteration method of Eq.(2-3 2) .
68
S u s c e p t i b i l i t y vs field for alternating chain
at low temperature by L=3 , 5 and? d e c i m a t i o n
approaches and the iteration method of Eq.(2-33)
69
.
20.
Susceptibility vs field for a l t e r n a t i n g c hain
at kgT/J = O .08 by L=3 decimation with iteration
E q . (2— 3 3)and Zeeman terms for the end spins taken
w i t h h 1 = i1 and h2 = 0 .......... ....................
70
21.
Susceptibility vs field for a l t e r n a t i n g chain
at constant temperature by 6-spin ring decimation
71
Energy spectrum of an 6-spin alternating ring as
a f u n c t i o n of the m a g n e t i c field . . . . . . . . .
72
Parallel and perpendicular susceptibility vs
temperature for antiferromagnetic Ising chain by
L= 3 RG . . . . .:...................... .............
84
The x and z direction susceptibilities vs
t e m p e r a t u r e for XY chain by L= 3 R G ............
85
Susceptibility vs magnetic field for uniform XY
chain at temperature tgT/J=0.05 by L - 3 RG . . . .
86
Susceptibility vs magnetic
XY chain at temperature
87
22.
23.
24.
25.
26.
field for alternating
^ RG . . .
27.
Susceptibility vs x-direction field for uniform
XY-chain at different temperature using polynomial
fit to each
10 a d j a c e n t data p o i n t s ...........
88
28.
Eight lowest energy levels of uniform XY cluster
four spins as a function of field in the z
d i r e c t i o n ....................
of
89
29.
Eight lowest energy levels of uniform XY cluster of
four spins as a function of field in the x
d i r e c t i o n .............................................. 90
30.
Schematic energy levels of transverse Ising model
and its b l o c k QRG
.................
96
ix
31.
32.
S c h e m a t i c e n e r g y levels of 2 - s p i n c l u s t e r
for the anisotropic Heisenberg spin chain in
the l o n g i t u d i n a l
field
.....................
Schematic energy level structure of odd- and
even-spin i s o tropic H e i s e n b e r g s p i n c h a i n . .,
96
.
97
I
X
LIST OF TABLES
page
1.
The 8 eigenstates and eigenvalues of a 3-spin
c l u s t e r . . . . .......................................
2.
The 4 eigenstates and eigenvalues of a 2-spin
c l u s t e r ...................... ..........................16
3.
The eigenstates and eigenvalues of spin cluster
with antisymmetric e x c h a n g e ..................... .
4.
The block
sizes
for Hfo f°r L=3 , 5 and
5.
The eigenstates and eigenvalues of a.2— spin
c l u s t e r ......................... ‘................. ..
and eigenvalues of
.
16
23
7 ............... 37
6.
The eigenstates
a3-spin ring
7.
The dimensions of block size of 6-spin alternating
ring Hamiltonian ■.........................
.
37
46
47
ABSTRACT
We have e x t ended both the r e l i a b i l i t y and the range
of a p p l i c a t i o n of the d e c i m a t i o n r e n o r m a l i z a t i o n group
method for calculating the thermal and magnetic properties
of 1 - d i m e n s i o n a l q u a n t u m s p i n — 1/2 systems. E f f o r t s to
i m p r o v e the a c c u r a c y i n c l u d e i n c r e a s i n g the s p a t i a l
rescaling, investigating the effect of free versus periodic
b o u n d a r y c o n d i t i o n s for each r e n o r m a l i z a t i o n c l u s t e r and
v a r y i n g the i t e r a t i o n p r o c e d u r e .
The s y s t e m s u n d e r
i n v e s t i g a t i o n include
(I) spin chains w i t h iso t r o p i c
H e i s e n b e r g as well as I s i n g - I i k e and X Y - I i k e a n i s o t r o p i c
exchange in the pr.esence of a l o n g i t u d i n a l field,
(2)
chains
with' u n i f o r m
antisymmetric
exchange
in
a
longitudinal
field,
(3) c h a i n s
with
alternating
a n t i f e r r o m a g ne tic i n t e r a c t i o n s in a field, and (4) those
w i t h a n i s o t r o p i c i n t e r a c t i o n s in a f ield w i t h a r b i t r a r y
direction. The p r i n c i p a l c a l c u l a t e d resu l t s are m a g n e t i c
and t h e r m a l
response
functions
(susceptibility,
magnetization and specific heat), which are compared (where
possible) w i t h p r e v i o u s l y p u b l i s h e d results using other
techniques.
I
CHAPTER
I
INTRODUCTION
Interest
in one — dimensional
the early 1920's,
reason
and still
for this
tra c t a b l e
than
of
I -D
as
those
problems,
second
the
i n centive
in
an
examination
nature
of
the
of real
popular
TCNQ,
magnetic
can be 'tailor-made'
the
field of
organic
[1,2,3,4,51,
discussion
will
detailing
situations.
briefly
are important
For
review
systems.
have
whose
conductors
exist
recent
often
A
found an
structure
in the laboratory. Also
T here
several
a variety
essentially I —D (quasi —
are naturally quasi-l-D.
theoretical
which
Now
three
solutions
these
p h y sical
systems
which
systems
and
is
real
A central
of
of
of
experimentalists
is effectively ohe dimensional.
I- D ) magnets
world
treatments)
and
arose in
in general , more
familiar
approximate
is that
increasing number
the
are,
there exist a number of exact
accurate
illuminates
continues to advance,
is that I —D m o d e l s
dimensions. In fact
(as well
(I —D ) magnetism
such
as TTF-
r e v i e w s of I -D m a g n e t i c
the curr e n t
this
only
reason
these
experimental
the
following
developments
for the goals of this thesis.
2
Analytic Developments
T h e . effective
anisotropic
I
spin-1/2
Heis.enberg
spin
Hamiltonian
chain may
be
for
the
written
general
in
terms
of Pauli spin operators as
^
=
Y, {"J z<Tiz<Ti+ i2„Jx y (oiXcri+1x_<Tiy 0ri+1y)- H z<yiz-Hx<yix} .
i
(1-1)
We c o n s i d e r b i l i n e a r
spin couplings', and a s s u m e
n e i g h b o r spin i n t e r a c t i o n s
nearest-
only. The e f fective
exchange
c o n s t a n t b e t w e e n the Z - c o m p o n e n t s of the spins is J z , and
that between x and y components,
Jxy“
Ising
obtained by putting J xy= O , and the XY-model
If J=Jz=Jxy we have
putting J z=O.
the
model
is
is obtained by
isotropic
Heisenberg
chain.
The
Ising
classical,
in the
5^- commute
by
Lenz
with
and
phase
in
sense
that
one
solved
solution was
temperature
model
all
another.
in
I -D
essentially
and
the
H^=O
operators
This
by
system
Ising
complete
applied magnetic
case
as
field.
in
is
purely
appearing
was
1925
in
suggested
[6].
The
a function of both
This
model
has
no
t r a n s i t i o n at T> 0, but orders s p o n t a n e o u s l y at T=0.
In 1944,
a very
D ) Ising
model
famous
was
solution
g iven
transition at T>0, but his
on the thermal
complete
properties
solution,
by
of the two-dimensional
Onsager
solution
in zero
including
[7].
He found
provided
magnetic
H z = 0»
(2a
information
field alone. A
Is lacking
to
this
3
day.
Fully
quantum-mechanical
systems,
S=I/2 Heisenberg
m o d e l ■ are m u c h ■more
even
the f e r r o m a g n e t i c
in I-D. For
state
c o n sists
of
all
spins
such
difficult
case,
parallel
as
the
to solve,
J> 0, the ground
and
is
(N+l)-fold
d e g e n e r a t e w h e n H=O. In 1930 Bloch i n t r o d u c e d the concept
of
a
spin
single
wave
and
overturned
deviate
Wortis
gave
spin.
exact
In 1963
[9],
but
difficult
J < 0,
Bethe
the energy
in
three- or more— spin
to
get.
1931
state
eigenfunction
for
chain.
no long-range
order
the
of a wide class
quantum
a variety
spin
ground-state
of
chains.
en e r g y
In
for
I-
and
1938
so,
in this
analytic solutions
as
are
case,
concerning
he
the
Heisenberg
demonstrated
model
remains
even at T=O
the
key to the
of spin chain m o d e l s
2—D
systems
Hulthen
this
states
and
and found the ground-
In- doing
the Be the Ansatz
well
and Fu k u d a
S= I/ 2
that
a
the two-spin
deviate
an Ansatz
exists
for
antiferromagnetic
of the eigenstates,
antiferromagnetic
note
the
formulated
structure
[10] . We
For
of
[8],
the
general
as
eigenstates
state was o b t a i n e d by W o r t i s
more
that
the
other
than
deri v e d
the
antiferromagnetic
m odel
[1 1 ] :
E = -Ni J I (21n2-l/2)
as
compared
with
antiferromagnetic
= -0.886NI J I
— 0.5 N
ground
IJ I
state.
C l o i z e a u x and P e a r s o n derive
for
Not
classical
until
Neel
1962 did
the en e r g y s p e c t r u m
des
of the
4
lowest
excited
relation to be
h(d(q) =
triplet
states,
finding
the
dispersion
[12]:
jt|J
sin qal
where
a is the s e p a r a t i o n b e t w e e n n e a r e s t - n e i g h b o r atoms
along
the. chain.
The I-D spin-1/2
XY model
was solved analytically in
the 1960's i n d e p e n d e n t l y by Lieb, Schultz and M a t t i s
and
Katsura
obtained
[14].
Katsura
exact.results
transverse
magnetic
for
field,
progress
Heisenberg-XY linear
and
the
Pfeuty
Ising
[15]
model
also
in
a
often described as the simplest
solvable quantum mechanical
Real
[14]
[13]
model.
on
the
chain
general
with
s p i n -1/2
symmetry
Ising-
in the XY plane,
dates from 196 6 and the w o r k of Yang and Yang, wh o o b t a i n e d
a detailed zero temperature
solution
in terms
of anisotropy
and m a g n e t i c f i e l d [16] . In 19 80 J o h n s o n and B o n n e r found
the
elementary
thermodynamics
zero
and
very
solution of
excitations
for
and.
low-temperature
the I s i n g - H e i s e n b e r g f e r r o m a g n e t
small
magnetic
this model
field
[17],
A
in
complete
has yet to. be obtained.
Numerical Studies
As
indicated above,
the S= I /2 general
not
available
analytic
analytic
solution for
a n i s o t r o p i c H e i s e n b e r g linear .chain is
even
treatment
a complete
now.
along
m a y be not possible.
For
the
the lines
There
S > I/ 2
of
the
is e x p e r i m e n t a l
spin
chain .an
Bethe
Ansatz
interest
in a
5
variety
sy s t e m
S > I/ 2
spin
(conta i n i n g
ion),
the
of
chains,
the N i z+ ion),
and s p i n - 5 / 2 sys t e m
Bethe
Ansatz
Heisenberg
s.nch
effective, spin-1
s p i n - 3/2 sys t e m
( C r 3+
( M n z+ ion). Eve n for spin 1/2,
approach
alternating
as
cannot
chain,
be
whose
applied
to
Hamiltonian
the
can
be
written as
N/2
:
^
{ Jlcr2i-rof2i + J2 ^ i ,?2i+l
(1-2)
i
This
is
a nearest-neighbor
exchange
model
in which
there
are two unequal e x c h a n g e const a n t s ,
IJ i I > IJ 2 ^ say. It is
convenient
ratio
that
to
a=I
define
an a l t e r n a t i o n
corresponds
corresponds
to
the
to
the
c ase
of
uniform
a = f 2 / ^ I * 80
limit
decoupled
an d
a=0
dimers.
Th e
a l t e r n a t i n g spin c hain is i m p o r t a n t as well to the theory
of
spin-Peierls
an a n a l y t i c
reliable
In
approach
theoretical
the
finite
spin
chains,
b o u n d a r y conditions.
size
the
N.
Bonner
Griffiths
exactly
Numerical
thermodynamic
quasi— I-D
calculations
to as finite
calculating
in
organic s. Since
is not f e a s i b l e 1 at this time,
1 9 6 0 's
independently,
referred
transitions
are.needed.
a n.d
[19]
Fisher
developed
chain e x t r a p o l a t i o n .
the properties
with
both
[18]
a
periodic
This
and
increasing
extrapolations
(N —y- 00).
are
In
and,
technique
of a sequence
and s t e a d i l y
l im it
other
involves
of
small
free-end
the system
t h e n . made
to
way
an
this
6
extensive
for
the
T=O
uniform
By
set of results
for all H and T,
correlation
functions
same
energies
procedure,
[20],
S = I / 2,
I,
Blote
specific
magnetization
heat
isotherms
3/2,
2,
up
to
properties
in
Real.
for
I -D
copper
Sjpa ce
nitrate
to experiment
the
[22]
for
Cu(NO3)2^ . 5 HzO
Renormalization
[18].
and T=O
spin
Blote
calculated
Heisenberg
the
g round-state
Diederix,
and the
alternating
for
susceptibility,
field
5/2.
results
obtained
obtained
[21],
vs.
compared the experimental
chain
were
chain with accuracy comparable
the
and some
values
_e_t _a_l.
thermodynamic
antiferromagnet
[23].
Group.
and
Qu an turn
Renormalization Group (QRG)
The
renormalization
powerful
critical
and
successful
phenomena,
classical
but
s y s terns.
group
approach
method
in
the
[24]
is
modern
the most
theory
of
its use has been confined mainly to
For
example,
th e
real
space
renormalization group method pioneered by Niemeijer and van
Leeuwen
[25]
has
b een
models
[26,27] ,
while
widely
used
s t u dying
c l assical
that
models
their l i m i t a t i o n s
temperature
be
ignored
and
difficulty,
in
[29],
low
however,
in
applied
techniques
which
the
[28]. H o w e v e r ,
as
since magnetic systems
cannot
greater
momentum-space
n - dimensional vectors
spins have
phenomena,
in
successfully
models
have
a
for
to
have
spins
dimensions.
is experienced
at
been
are
classical
magnetic
quantum
especially
Ising
nature
very
low
Considerably
in applying
7
RG
methods
essential
quantum
reason
noncommuting
Very
tested
to
recently
devi s e d
shortly
is
two
first
block
One
by
Most
RG
cases,
and only a very
of
is
to
a
subdivided
int o
and
blocks
have
8=1/2)
to some
is t r u n c a t e d
of
For T=O cases,
been
[30,31]
Kondb
bee n
and
problem.
used for
T=O
of
each
each
The lattice
such
that
the
b l o c k may
be
block
(2Ns levels,
0f levels, and
'
coupling between the blocks
have
transformation
of
sites
number
r'
basis.
of
[33,34,35],
Ns
eigenvectors
The basis
block
is as follows.
of
c a l c u l a t e d exactly.
methods
type
for T>0
spirit of the block RG
eigenvalues
The
problem
theorists
approaches
few
RG
the
field
applied
[32].
is
familiar
kinds
models.
thereafter
The
the
systems.
operators.
on quantum
method
mechanical
the
'
is written within the truncated
the truncated basis need only contain
the ground state and dominant
set of first
excited
states.
This block QRG method has been exhaustively tested on
the
transverse
[30,36], Fields
Ising
model,
e_t a_l.
with
[37,38,39]
on the I-D a l t e r n a t i n g H e i s e n b e r g
in
order
state
as
encouraging
to investigate
used a
T=O
appro a c h
antiferromagnet
the nature
of the lowest
a f u n c t i o n of a l t e r n a t i o n ratio a.
fall
results
QRG
calculations
naturally
odd,
(3, 5, 7 and 9), w i t h N%=2 and N s even,
6 or 8). The N s odd calculation means
into
two
The
chain
excited
b lock
classes:
Ns
w i t h Ni=4 (or
that one N s~spin
8
block
maps
into
calculation
one
means
renormalized
that
the
Ng^^pin
,site.
The
block
maps
renormalized N^/ 2— sp in bloc k. . Fields, ert
method
and found that
unstable
fixed
independent
the uniform
point
dimers
of
(o=0)
this T=O QRG predicts
a
(<z=l), is always an
while
a
system
is a stable fixed point.
an excitation
into
used the N s odd
limit
system,
N g e V en
gap
that
of
Therefore
vanishes
only
in the u n i f o r m l i m i t (no alter n a t i o n ) . They also e x a m i n e d
the q u a n t i t a t i v e
finite
chain
energy,
the Ni=2,
accuracy
of this QRG by
extrapolation
results.
For
N s=9 RG calculation
discrepancy amounting
comparing
the
gives
with
ground-state
a quantitative
to 12% in the uniform limit. However,
the Ni = 2 RG sequence, N s = 3 , 5, 7 and 9, e x t r a p o l a t e d
N s= 00, gives
more
accurate
results.
For
the N s odd calculation gives a value
unreliable
gives
near
results
to
the excitation gap,
that
is quantitatively
the d imer limit. The N g even c a l c u l a t i o n
in very good agreement
with
the extrapolated
curve out to 0 = 0.6 but p r e d i c t s a gap v a n i s h i n g a t q = 0 .9 6 2
rather
than
The
unity.
second
Q RG
method
transformation method first
from
the
its classical
simplest
reviewed
of
approach
in detail
temperature
heat
RG
version
below,
dependence
the
uniform
extended
the
for
quantum
decimation
by Suzuki and Takano
[40,41,42,43].
This
spin
Suzuki _e^ jiJ..
of the
1—D
is
seems to be
systems.
As
calculated
the
internal energy and specific
spin
models
(the
isotropic
9
Heisenberg
couplings,
model
thesis
decimation
readily
ferro-
an d
ant!ferromagnetic
and XY-m ode I)..
In this
the
with
we
HG.
applied
will be
Unlike
at finite
field. F u r t h e r m o r e
concerned exclusively with
the
block
method,
temperature
and
it
can
be
in a m a g n e t i c
the d e c i m a t i o n a p p r o a c h
is correct
in
both the dimer and Ising limits.
The
purpose
reliability
method.
of
this
and the range
thesis
of
the
both
the
decimation
its a c c u r a c y by increasing
the length rescaling factor and
investigating
the effect of
conditions.
We
apply
thermodynamic
we
behavior
(I)
of
uniform,
antisymmetric
Heisenberg
anisotropic
decimation
properties
In p a r t i c u l a r
(2)
to e x t e n d
of application
We a t t e m p t to impr o v e
boundary
is
by
to
various
examine
the
uniform
the
calculation
iteration
thermal
anisotropic
Heisenberg
exchange,
(3)
chains.
and
Finally,
spin chains
in a magnetic
direction.
Wherever
possible,
with those
of other
techniques.
and
we have
magnetic
chains
alternating
we
techniques.
Heisenberg
anisotropic
extend
of
chains,
with
anisotropic
this
field with
compared
QRG
to
arbitrary
our
results
10
CHAPTER
2.1
Su zuk i
decimation
and
RG
[40,41],
have
nearest-neighbor
the
interactions
method
by
the spin-1/2 Heisenberg
anisotropy.
The
a simple
spin
systems
.
illus t r a t e
of
proposed
for I— D quantum
,
we
treatment
(ST)
transformation
with
section
©WANTUK B E CIMATION Kffi
Takano
-
2
only.
reviewing
model
with
In
this
the
ST
uni axial
reduced Hamiltonian (-ener g y/ kB ^
this
model is given by
N-I
H
=
2 ] I K z o i z o i+ 1 z *
+
).
(2-1)
1 .0
where
T is
the
temperature
and kg B o l t z m a n n ' s
Here cfjX, <r^ y , and 0 ^z denote
the
ith
site
parameters
where
of
The
Jz
the
are
e x c hange
components of neighboring
y components,
We
the Pauli spin o p e r a t o r s on
linear
K z an(j K Xy
chain
with
respectively
coupling
spins , Jxy
N
sites.
The
- p J z and, - p J Xy #
constant
between
z
that between x and
and P= I/ kgx.
construct
a decimation
factor L (see Fig. I) according
e xp { A + W
constant.
I = Tr'
e xp { $ } .
transformation
to the
with
scale
procedure
(2-2)
11
Here “/V
is H a m i l t o n i a n on the original
"Sy
is a d e c i m a t e d
new
lattice
spacing
a
with
belonging
trace
to the
(2-1) and
( i.e. r e n o r m a l i z e d ) H a m i l t o n i a n on the
lattice
of original
partial
lattice
chain).
over
new
spacing La,
The
all
(where a is lattice
operation Tr' means
spin
operators
except
to take
those
lattice.
L= 2
&
A e ^yX yX V X yX yV V A yX yV X Z V X yX yX A yV A yX Z ^yV A yV yV yX A yV A A u W X y"
Fig. I. Decimation in one dimension with scale factor L= 2.
Crosses denote
spin o p e r a t o r s to be e l i m i n a t e d (traced
over) by the de c i m a t i o n . Dots denote spin operators to
remain
after
the
decimation.
Lines
denote
the
interactions between neighboring spins.
12
For
a
mode I, this
in
the
I —D
classical
decimation
quantum
case
can be
it
is
system.
say
the
carried out exact ly. How ever
impossible
to
carry
d e c i m a t i o n e x a c t l y even in one d i m e n s i o n be c a u s e
noncommuting
effect
of
operators
First
out
the
of
the
in the H a m i l t o n i a n .
have p r o p o s e d the f o l l o w i n g s i m p l e
one-dimensional
Ising
ST
a p p r o x i m a t i o n for the
decimation.
the full H a m i l t o n i a n is divided
into a sum of
cluster Hamiltonians Hfoj'
/l '
n
Y,
Hbj
•
(2-3)
j= l
Here
L
Hbj = X!
,
i= l
where
®
j,i-1 ,o j ,i) denotes
b e t w e e n cr
^ and aj in
the c o m m u t a t o r s b e t w e e n
clusters,
and make
the
a nearest-neighbor
jth cluster.
operators
the exact
interaction
We. then ignore
referred
to d i f f e r e n t
decimation within one cluster
only:
exp{ Ai.+Hfo ’ (T q ,o l ) } = Tr'
H b ' (^O » ct"l )
is
a
interaction between
exp { Hfo (OOeOl,*" T l ) ) •
renormalized
the remaining
(2-4)
ne a r e s t - n e i g h b o r
spins O q and <tl » and Al
13
I
is a c o n s t a n t
Hbj'
and
Al*
term
we
»w
W#.
H a m iI to n i a n
independent
construct
of T 0 and T L .
an
Then,
approximate
and- an a p p r o x i m a t e
using
transformed
constant
term- A a as
foilows
N/L
“
X!
Hbj'(oj,0.3j,L)
(2-5)
,
j= l
and
Aa
=
(N/L) Al
.
This entire procedure
(see Fig.
2) can be written as
N/L
exp {A+
^
I = Tr'
exp{}^ }
N/L
'= Tr ’ exp{ X Hb j * =
j-1
N/L
Tr' TI Gxpf Hbj I = TT exp{ Al+Hbj' } =
j=l
j=l
(2 - 6 )
J
N/L
= exp { NA1 Z l +
£
Hbj ' } = exp (A + %
} .
j=l
Two
approximations
errors
nature
have
made,
due to the n o n - c o m m u t a t i v i t y
can
be
examined
adjacent clusters only,
exp{
been
by
and
both
introduce
of o p erators. Their
considering
the
case
two
for which we may write
} = exp{Hbl)
e xp { H b 2 ) e xp {I / 2 [Hbl ,Hb2l
C
e xp {Hbl 'I e xp{Hb2'l
of
= exp{ Hbl '+Hb2 '~l/2
+*.'*}
*'
[ H b l ' ,Hb2
■* + '‘ *
(2-7)
14
Roughly
speaking,
truncating
these
the
errors
which
expressions
hav e
are
introduced
opposite
C o n s e q u e n t l y they may be e x p e c t e d to cancel,
part,
since Hb is t r a n s f o r m e d
into H b
signs.
at least,
*n (2-7)
in
after
in
the
decimation.
(a)
#--- O--- #
I
(c)
•\Ay\y\A/»
4
W
AyVfcAAAAA^AAyXA/WXAyXAyV^A/XAAA^V*
Fig. 2.
The a p p r o x i m a t e o n e - d i m e n s i o n a l d e c i m a t i o n . The
original system is divided
into a sum of clusters in (a).
The decimation is carried out
within
one cluster
shown
in (b), to give a new i n t e r a c t i o n w h i c h is d e n o t e d by a
wave
line
in (c ) . F r o m
this n e w
interaction,
the
renormalized system (d) is constructed.
15
Applying
Hamiltonian
the
approximation
(2-1) we
explained
above
to the
can carry out the decimation with L=2.
the Hamiltonian of one cluster is
H b = K z (oOz<yl z+CTl Zor2 z). +
+ K xy (<yOxor1x+<y1x o 2x + 0 O yff1 y+<T1 y<r2 y) .=
= K z (<yOZ01 z+01 Z02 z) + Kx y /2(0O+arl~+<Tl+<T2-+ ll*c') "
(2 - 8 )
Here h.c
means Hermitian conjugate
0+ = 0X+i0y
Th e
and
eigenstates
Im
o~ =
>
an d
and
0*— IoY
the
.
(2-9)
eigenvalues
Em
of
the
H a m i l t o n i a n Hy can be o b t a i n e d a n a l y t i c a l Iy (see Table I)
for the cluster
of L+l=3
^ b ' is of the same for m
sites. The
transformed Hamiltonian
as Eq.(2-8),
e.g.
Hb' = K z fO 0 zOrZ 2 + Kx y '/2 (0O+<72™+® o "®2 + ) *
It
has
2). The
eigenstates
In > and
decimation Eq.
exptAl)
eigenvalues
(see Table
(2-4) can be written in the form
Y t exP t E n fH n X n I
= Tr'
(n}
Y
E n'
^jT
exp{Em > I m X m I
(m)
(2 - 1 0 )
<ilm><mli> e x p (Em ^
(m, i)
where
Ii> 's are all states of the L-I=I
intermediate
sites.
16
Table I. The eight e i g e n s t a t e s and e i g e n v a l u e s of. a ■three
spin cluster
|m>
i
Ittt>
2
!***>
3
a+|tti + It t >
+ b+ It It >
4
a_ I11 I + It t >
+ b _ ItIt >
5
a+ II It + t 11> + b+ IIt l>
E+ = -Kz + G
6
a _ II It + t 1 1> + b-I It l>
E- = - K z - G
Em
2KZ
I
N
W
II
I
N
I
II
W
2KZ
W
*:
m
7
1/^T
Ittl - ltt>
0
8
I/J T
Ilit - t 11>
0
+
G
.
-G
G2 == K 2 2 + BKxy2
a+ 2 — 4Kxy^/(8K%y^+E+^)
a_2 = 4Kxy2/(8Kxy2+E_2)
b + 2 = E + 2 /(8KXy2+ E + 2 )
b_2 = E _ 2 / (8Kxy2+E_2)
T able 2. The
sp in cluster
four
e i g e n s t a t e s and e i g e n v a l u e s
I n>
4
Eu'
I tt>
Kz'
I ll>
KZ
I/J T
I t l + lt>
I/J T
Itl
-
lt>
'
Kz '+2KXy
-K Z '“ 2Kxy
of a two
17
Substituting
these eigenstates
Table I and 2 into Eq. (2 — 1 0),
aI + K z '
we
and eigenvalues
thus
in the
obtain
= InC exp(2KZ ) + b + 2exp(E+ ) + b _ 2exp(E_)
Al - K z ' + 2KXy ' = In 4 + I n { a+2e x p (E+)
}
+ a-2ex p (E - ) )
A1 - K z ' - 2Kx y ' = In 2 .
T h e r e f o r e we
Hamiltonian
find
K z'
parameters
in
the
transformed
Hfo '•
= 1/4
xy
the
ln{ e x p (-Kz ) (cosh G + K z/G sinh G)
}
= 1/2 ln{ ex p (2 K Z)+ e x p (- K z)(cosh G — K zZ^ sinh G)
}
- 1/2 in 2 - Kxy .
( 2-
11)
the magnitude
of
.
Al
= K z ' + 2Kx y ' + In 2
For
the error
this
results
neglected
in
a
Kxy'
show
higher
the
the
by
transformations,
thermodynamic
effects
approximations
the
temperature.
calculated
are found
calculated
A Al ~ O( K 4 )
that
in K z and K x y * Thus „
at
ST also
involved in this approximation:
A K z» ~
These
example
properties
start
at
approximation
Thermodynamic
iterating
as
of n o n c o m m u t a t i v i t y
will
the
be
those found by Bonner and Fisher
order
be t t e r
properties
are
renormalization
from
to be
fourth
becomes
dis c n s s e d
obtained
at high temperature
the
this
below.
The
approximation
in good agreement with
and by Katsura.
18
2.2
SPIN CHAINS IN A LONGITUDINAL FIELD
The
approximate
Eq.(2-6); In
this
quantum
section
decimation
and
the
next
is
summarized
we
will
use
in
this
m e t h o d to e x a m i n e a v a r i e t y of spin systems. We will also
investigate
using
the
periodic
each cluster,
is
used
to
outline
effects
of
rather
increasing, the
than free b o u n d a r y
and by varying
cal c u l a t e
the
of the remainder
the
antisymmetric
antiferromagnetic
thermodynamic
of this
chapter
boundary
exchange,
spin chain.
conditions,
in the a l t e r n a t i n g
and
as
2.2.1
is
Antisynmetric
[44,45]
a three-component
which
The
In
as
alternating
methods
will
factors,
be
tested
section, a general
in a m a g n e t i c
field
of
studied.
Exchange
Antisymmetric,
interactions
for
is as follows:
The effects of scale
iteration
of
w i t h b oth s y m m e t r i c
spin chain. In next
direction
conditions
proper t i e s .
well
d e c i m a t i o n m e t h o d for spin chains
arbitrary
factor,
iteration procedure
this section, we study the I —D model
and
scale
or
are
of
Dzyaloshinskii-Moriya
(DM)
the
D
interaction
form
D.(SjxSj)*
where
is
constant and Sj and Sj are
spins on chaih sites i and j,. T e r m s of this type m a y arise
due
to
spin-orbit
structures.
Among
have
commonly
Teller
been
distorted
coupling
in
those materials
o b s erved
compounds.
low-symmetry
in which DM
are
Another
magnetic
interactions
low-dimensional
recent
example
Jahnis
the
19
chain-like
which
polymer poly (cobalt n-butylphosphinate)[46],
the
symmetric
interaction
isotropic and antiferromagnetic,
weak ferromagnetism.
typically
(S'S)
an order
coupling.
(that
is
Sj.Sj)
in
is
while the DM term produces
In a given system DM interactions are
of
magnitude
Nonetheless
their
functions may be significant,
via series e x p a n s i o n s
weaker than the symmetric
effects
and are known
on
the
response
in detail only
[47] v a l i d in the high
temperature
regime.
Betts
has pointed out
d i r e c t i o n for
lattice
is one which
interaction
rotating
about
all pairs
the
rotation
direction
or
through
plane,
the
can be
the
points
in the same
neighbors,
divided
and
if the
into sub lattice s,
on one
XY)
interaction
t an- 1(ID I/J)
case
sub lattice
of D. If there
a DM
that
there
between
about
is
throtigh 9 0*
is also a s y m m e t r i c a l
D
of the interaction
an
would
the
spins,
eliminate
a
the
[47,48]..
isotropy
in
the
XY-
that is w h e r e the a n t i s y m m e t r i c e x c hange constant
has only a Z component,
in
of nearest
for spins
antisymmetric part
In
if D^j
can be t r a n s f o r m e d into an X Y . i n t e r a c t i o n by
the axes
( H e i s en berg
that
same
direction,
following mapping
D= D z» and that the applied field is
the
relation:
Betts's
statement
The H a m i l t o n i a n
chain with antisymmetric exchange
of
gives
the
the
spin
20
%
=
][
( - K z S i^ S itlZ - K x y ( S iISitlZfSiySit l F) -
i
(2-12)
- D ( S iX S itlF -SiySitlX) - K S iZ }
is
in fact
equivalent
to an a n i s o t r o p i c
spin
chain
with.
Hamiltonian
W - X
t "r Z S 1z S 1H-Iz - * V < S i ’tsl+ ll i s iy s i+1T > " S S 1- )
i
.
(2-13)
where
-yjK xy 2+1)2
Kxy' =
This
In
g. about
(2-14)
can be proved as follows:
the
Ham iltonian(2-12)
the Z axis,
Sifl= =
CO S
one
rotates
the
successive
using
A S i+!=' - sin A SifIy '
s ifly = sin A S i+!='
f
CO S
A SifIy '
(2-15)
s i+1 Z “ ® i+1
If we fix A by the relation
tan A =
D/K^y
s u b s t i t u t i n g Eq. (2-15) and (2-16)
(2-16)
into H a m i l t o n i a n
(2-12)
gives
%
- X
' - k Z s 1z S 1H-Iz'
s I8 s H i 1 '*s i7 s i*iy ’) -
i
- hS.z
}
(2-17)
21
Comparing
E q 1(2-13)
Bot h
with
Eq. (2-17)»
antisymmetric
i s o tropic
and
in the XY-plane.
suitable
rotation
antisymmetric
is
exchange
we
have
symmetric
In the above
made
is changed
Eq.(2-14).
in
exchanges
we
the
see
that
are
if a
X Y - plane,
into a symmetric
the
one.
If
there exists a t r a n s v e r s e field (i.e in the XY-plane), the
isotropy
in the X Y - plane
symmetric
is broken. Thus
the m a p p i n g
exchange holds only for the l o n g i t u d i n a l
to a
field
case.
In this
with
its
s u b s e c t i o n we
uncontrolled
additional
calculate
complexity
the
of
test w h e t h e r
approximations,
antisymmetric
specific heat
can
=
Z
j
the
will
of the Heisenberg, chain by
symmetric
the
the same results
exchange.
The reduced Hamiltonian for this case
^
treat
exchange. . We
decimation R G , and check that we reproduce
as with anisotropic
the QRG method,
is
* ™K zCTiZ<ri+lZ ~ K x y U i X o i+lx+<yiy<ri+ly) "
i
- D ( a ix(ri+1y-ffiy<yi+ 1x) - ha^z
here cr= 2S,
K z= p j z , Kxy = PJxy.
We use decimation
) ,
(2-18)
h=pHz , and P= IZkgT.
with scale, factor L= 2 to. calculate
the t h e r m o d y n a m i c proper t i e s . The H a m i l t o n i a n of a three
spin cluster
is
H b = - K ztff0Za1 ZH-CT1 Zff2Z) - l/2(Kxy+ i D ) (CT0+CTl~+ffl+<T2 ~ ) "
- 1/2 (Kxy_iD) (Ct O-0I ++erl-<T2 + > ~ h(o0 z/2+o1 z+CT2 z/2)
.
22
We have used E q .(2— 9) to write
O 1 x G2 y-®!
and
take
= iZ2 (ffl+tT2_-CTl_<T2 + ^ »
account
the
fact
shared by two n e i g h b o r i n g
should
be
commutes
we
can
included
with
with
that
clusters,
each
solve
analytically,
as
for
shown
its
in
each
This
of the total
eigenstates
Table
Hamiltonian Hfo' is of the same form
end
spin
is
half the Z e e m a n term
cluster.
the Z component
easily
since
3a.
Hamiltonian
spin.
and
The
Therefore
eigenvalues
transformed
as Eq.(2-19), but for a
two-spin cluster:
H b' = - K z 1G 0 zOrZ z " l/2(Kx y *+iD') g o “<*2+ ~
- I/2 (Kxy *-iD')ff0-G2+ ~ h?/2
Its eigenstatei and eigenvalues
Substituting
the Ta b l e 3 a and
of the
are
energy), we also require
.
shown in Table 3b.
these e i g e n s t a t e s
3b into Eq. (2-10)
(G0 Zi-G2 ?)
and e i g e n v a l u e s
(but c h a n g i n g
the
keep
the
sign
that
(2 - 2 0 )
a=d
to
in
two
spin
eigenstates
arising from the decimation.
consistent
4
In> <n Iexpt-En '} =
n=I
those
Therefore we find
4
exp (Al - Hb') =
with
InXnI
n=l
Pn • J
■ (2-21)
•i
23
Table 3 a„ E i g e n s t a t e s .and e i g e n v a l u e s
cluster with antisymmetric exchange
of
a
Im>
m
I
11 tt>
2
U*t>
three
Em
.
-2^ z-2h
-2Kz+2h
3
I/ ■ypl Ittl - a ltt>
—h
4
I/
Ilit - a*tll>
h
5
^l+2lb+ |2 lb+ttl + tit + b+*l?t>
E l+
6
j l +2 Ib _ I2 Ib_t tI + tit + b _ * 4 M >
El-
7
^ I+2 Ic+ I2 lc + |it + It I +
E2+
8
1 1+2 Ic _ I2 |c_ii? + ltl + c _ * t W >
E2-
*:
a
= (Kxy-iD)2/(Kxy2+D2 )
b+
= -2(Kxy+iD)/(h+E1 + )
b_
= -2(Kxy+iD)/(h+E1_)
c+
=
2 (Kxy-ID)/(h-E2 + )
C_
=
2 (K xy- iD) / (h-E2- )
Ei+ = - h /2 + K z +
■y 8 (Kxy2 + D 2)+(h/2 + K z)2
_ = — h/2 + K z -
~\J8 (Kxy2+D 2 ).+ (h/2 + K z)2
E2 + = h/2 + Kz +
^ I 8 (Kxy2+D 2)+(— h/2+Kz>2
E2- = h /2 + K z -
I /8 (K x y 2+D2 )+(-h/2+Kz )2
■
spin
24
T able 3b. The four e i g e n s t a t e s and e i g e n v a l u e s
spin cluster with antisymmetric exchange
In >
I **>
I3 >
=
I / V
I4 >
=
d
it>
K z - + Z - ^ K x y '2 + D ' 2
*t>
K z ' - Z ^ K x y '2 + D ' 2
.
+
-
4-
-»
t 1
If 4
CL
=
N
I2 >
— K z ’- h '
+
If t>
I
=
H
Eu'
Ii>
2"
6
d
=
( K x y - - I D ' ) /
^
IK
Xy *2 - D '2
where
P i = exp {2Kz+2h}
of a
+ l/(l+2|b+|2) exp{-E1 + } +
+ I / (1+ 2 |b_!2 ) Cxpt-E1-I
P2 » exp{2K z- 2h} + l/(l+2 |c+|2 ) exp{-E2 + } +
+ I/ (1 + 2 |c-I2 ) exp{-E2_3
Pg = exp{ h } + e xp {-h }
P4 = 2|b+ l2 /(l+2lb+l2 ) Bxpt-E1+) +
+ 2 lb_|2/(1+2 |b_|2) Oxpt-E1,) +
+ 2| c +I2/(1+2| c +I2 ) e Xpt-E2 +) +
+ 2 Ic_ I2/(1+2 Ic-I2) Bxpt-E2 -)
tW O
25
Solving
Eq s.(2-20)
and
(2-21),
we
find the p a r a m e t e r s
of
the renormalized Hamiltonian:
A1
=
(
In Pl + In ?2 + In Pg + In P4 ) /4
h'
=
(
In Pl
KZ'
=
(
In Pl + In P2 - In Pg - In P4 ) /4
+
I n P2 ) Z2
K x y ' = B(Kxy2- D 2 )/(Kxy2+ D 2)
D'
= 2BKxyD/(Kxy2+D2)
where
B
=(
In P4 - in P 3 )/4
Once we have
done the decimation,
energy
site
per
of
the
we can calculate the free
system
(In
Q)/N
r e n o r m a l i z a t i o n t r a n s f o r m a t i o n s by,using
formula
from
these
the w e l l - k n o w n
[25]:
(1/N)ln Q n S' A1/2 + (1/2) (2/N) In Q n/2
or,
(1/N) In Q n ^ (1/2)
^
(l/-2m )A1_(R(“ ){K,D.h))
(2-22)
m=0
We
can obtain thereby the specific heat
CZ(Nkg)
= ( K 2 92/dK2 + 2Kh B 2ZdKdh.+ h2 B2Zdh2 )(In Q ) /N
(2-23)
26
the magnetization
M = 9/dh (In Q ) /N
(2-24)
and the susceptibility
kB T =
d2/9h2
According
properties
of
Kxy ^ +
decimation
to
results
chain
to those
^xy'=
with
'
static
with parameters
thermal
( 1^z ,Kxy ,D, hz }
of a c hain w i t h p a r a m e t e r s
D'= 0,
ca l c u l a t e
hz
the
anti s y m m e t r i c
confirm
(2-25)
to Betts's mapping relation,
a spin
are e q u i v a l e n t
chains
(In Q ) /N
}.
We
have
specific
used
heat
e x c h a n g e , and
{ Kz'
quantum
for
the
found
spin
that
the
this mapping relation.
In the XY-case,
a chain
with
{ K z=O,
Kxy, D/Kxy=l,
h z= 0 } is e q u i v a l e n t to a chain w i t h { K z^O, K x y '= ■^2* Kxy'
D'=0 , h z = o } . Since k B T / K x y '= (I / ^ 2" )kgT/Kxy° T h e r e f o r e the
thermodynamic
the
D = K xy
and
properties
D'=0
case
as
a function
wil l
t e m p e r a t u r e axis. In Fig. 3
heat
curve
of
D Z K xy=I
have
of
temperature
different
we can see that
is scaled
that with D' ~ 0 .0 5. The Kat sura's
by
exact
units
on
the specific
a factor
result
for
of
from
for symmetric
e x c h a n g e alone is s h o w n in the same figure. C o m p a r e d to the
exact
line,
the same
the L= 2 line has
too hig h a m a x i m u m .
situation as that Takano and Suzuki have
This
got
is
[41].
27
Fig.
4
shows
ferromagnetic
has
the
same
anisotropic
Fig.
12).
chains
that
with
character
chains
the
specific
different
as
without
that
he at
for iso t r o p i c
antisymmetric
of
different
antisymmetric
exchange
XY-Iike
exchange
(cf.
C/NkB
28
OlK-LO
D = O
cN
\ \o.os
I
O
I
l_
0
2
Fig. 3. Temperature dependence
of the specific heat of th e
I-D XY model with antisymmetric
exchange. Solid curves are
the pr es e n t resu l t s and dashed curve is the result of
Katsura [14] for the case of symmetric exchange alone.
C/NK
29
KbTZJ
Fig. 4. Temperature dependence of the
specific heat of I-D
i s o t r o p i c H e i s e n b e r g m o d e l , for vari o u s v a l u e s of the
antisymmetric exchange.
30
2.2.2
Altermatimg Amtiferromagmettic Spio Chaims
The
one-dimensional
magnetic
antiferromagnetic i n t e r a c t i o n s
attention,
materials
stimulated
such
as
and on systems
undergoing
Theoretical
has
largely
copper
chain
attracted
by
nitrate
alternating
much
experimental
Cu(N03 )2 *2.5H2®
spin-Peierls
treatments
with
have
work
on
on
[23,49,50]
transitions
focussed
recent
[4],
a
model
described by a reduced Hamiltonian of the form
e— I
r\/
= 2^ [ -KlTi, iT i, 2 - K2Ti,2Ti +l ,1 - h<cri,iz+ffi,2z) >
(2-26)
As d e s c r i b e d above the chain c o n s i s t s of s p i n s - 1/2 linked
by
isotropic
antiferromagnetic
( K l , K 2 ^®^
exchange
interactions which alternate in size.
For copper nitrate,
A j i r o u_t aj.., D i e d e r i x .ejb a%.
[23],
and B o n n e r
and
T =-1.29K
this m a t e r i a l
e^_t s J . [49]
[49,53]. The
have
found a = K 2 / K 1 = 0.27,
thermodynamic
properties
have b e e n found to be rich.in
s t r u c t u r e at
low t e m p e r a t u r e and in a m a g n e t i c field. For e x a m p l e ,
low— temperature
(shown
peaks
susceptibility
s c h e m a ticallyin Fig.
5)
as
is
at. field H ci and H c2 , which
a
function
characterized
arise from
of
of
the
field
by
twin
an excitation
energy gap. between a nondegenerate
singlet ground state and
a
states.
continuum
magnetic
of
field
excited
depresses
and H > H c2 lies within
the
trip l e t
the Fermi
gap,
level,
Increasing
the
which
for H (HeI
and for Hcl<H<Hc2
is within
31
0.0/90/
O.//SO/
Fig. 5. Sus ceptibility versus field at constant temperature
for copper nitrate, s h o w i n g peak structure near critical
fields H cl and Hc2 (from Ref.[23] ).
\
32
the band. The m a x i m a
and bottom
response.
in the d e n s i t y of states
of the band
then give
rise
to the
double-peaked
With increasing temperature the peaks diminish
size and merge to form a single broad m a x i m u m
No exact analytic approach
the H e i s e n b e r g
alternating
not
possible
on
yet
the
proved
c hain r —
Ansatz.
treatments
are
in p a r t i c u l a r
approximate.
The
z e r o — field,
been
of
numerical
Bonner
Fock
using
approximation
Our
purpose
proper t i e s ,
the
a method
have been
results
and Barr
e^ al.. [23]
for rings of
and specific heat
calculated by Mohan and
to Bulaevski based
on a Hartree-
to
calculate
static
thermodynamic
p a r t i c u l a r l y the m a g n e t i c s u s c e p t i b i l i t y ,
presence
of
it is exact
as
[50].
is
a
decimation method.
that
due
by Duffy
of the t h e r m o d y n a m i c
susceptibility
functions of temperature
recent
alternating
and by D i e d e r i x
up to 12 spins. The field d e p e n d e n c e
the
the
investigated
by B o n n e r .et: Jil.. [52],
and
based
most
antiferromagnet
properties,
technique
it has
the
properties
using extrapolation of exact
[50],
Therefore
finite-temperature
have
in
is presently available for
to find a suitable
well-known Bethe
theoretical
[51],
at the top
magnetic
field
using
the
q u a titurn
Its main advantages for this problem
in the d i m e r l i m i t
in
(as n o t e d above),
are
its
a c c u r a c y i m p r o v e s as the field s t r e n g t h increases, and it
is r a p i d and v e r s a t i l e ,
a l l o w i n g us to vary
H, T
and a
33
over
a
wider
range.
In
order
to
improve
this
■
approximation,
we use a sequence
and 7) w i t h free b o u n d a r y
also
use
a
large
of scale factors
conditions
cluster
(L '= 3, 5
on the cluster,
(N=6) with
periodic
and
boundary
conditons,
D e c i m a t i on for L = 3_, 5 and 7 Fre^z End Cluster
The basic
scale
steps
of the quantum
factor L are first,
decimation method
to solve for
the eigenstates
for
and
eigenvalues, of the H a m i l t o n i a n
of an (L + I)- spin f r e e - e n d
cluster,
renormalized
and
and . then
cluster.
case
of
for
to
We
the
carry
next
two— spin
out
the
describe
alternating
exact
Hamiltonian,
decimation within
this procedure
in detail
for
one
the
interactions.
In order to keep the a n t i f e r r o m a g n e t i c c h a r a c t e r in
the
decimation procedure,
the c a l c u l a t i n g
necessary
thermodynamic properties,
to study
even n u m b e r
(even this may not necessary
the
of spins
critical behavior),
It
is
initial
that
even
i n t e ractions,
viewpoint
may
and eigenvalues
n oted
to allow
Hamiltonian
the
we
and we must
of 4-,
must have
if
we
it
w ill
consider
be
for a n i s o t r o p i c
alternating
spin
solve
6- and 8-spin
mainly
necessary
an
for the
clusters.
isotropic
from
an
RG
interactions. Thus we
also treat easy-axisj and easy-plane
For
a l t h o u g h it is
in each cluster. T h e r e f o r e the scale
factor must be 3, 5 or 7, etc.,
eigenstates
for
systems.
chain
each
cluster
34
---- O
/
AZp-
/4,
2
---- :
----- 1
•-
L
A-/
2
I
I
2
■ Q A / V W ) - - - - - O w A A A /0- - - - - C X A A A / X ) - - - - - C A / W \ 0 -
2
/
2
/
*
'
*
'
2
-cvw
-- O W V V O -
L-f
U
Hbz
kp------------------------- q/ V \/ \/"\/ V X y
f'
Fig.
6.
includes
2-1-2
\,1O------
o'
Two cluster types for the alternating
I interaction pairs,
I or 2-1-2-1
2,
and
is either
as illustrated
of
spin chain.
the
form I-
in Fig. 6 .
and
g ^2 m ay then written as
(L-I)/2
H hi =
( -Klz<T2 iZcr2 i+lz ™
i=0
-K lxy/2 (<y2 i+ff2 i+l“ +CT2 i“ ff2 i+i+)
1 +
(L-I)/2
+
Y,
1 "K 2 za2 i-lz<T2 iz "
i=1
- K 2xy / 2 (0 2 i-l+0 2 i_+<T2 i-l_ a 2 i+)
} “
L-I
-
- hI (a o z+aL?)
(2-27 a)
35
(L-I)/2
x!
Hb 2 -
-K 2z<*2iZor2 i + l Z “
i=0
~K2xy/2 (a2i+a 2 i+ l~+a 2 i~a 2 i+ l + ^ } +
(L-I)/2
+
1 _Klz*2i-lZff2iZ -
Y,
i=1
"K lxy/2 (o2i-l+a2i~+a2i-l~a2i+) 1 ”
L-I
^
HtriZ - h2 ( ct0 z+<t l z)
i=l
(2-27b)
Here
(2-28)
+ H-2 = k
and
^2/
= Kg/Ki
= ct
We have found that
the above
for
e nds
spins
symmetry
at
the
of H bl
of
next
d i v i s i o n of the field
each
and H b 2 under
and 2 , and yields the exact
We
need
the
cluster
results
solve
just
H^i
for
diagonalizing
the
exact Hamiltonian
suitable representation.
[21]
for
Bonner
is
a free-end
[52]
and
7
and D u f f y
for
alternating
Since
I
and
found
written
by
in
a
follows closely that
chain,
cluster.
are
matrix,
The procedure
the
indices
eigenstates
[19], B o n n e r and F i s h e r
the u n i f o r m
and Blote
L= 3,5
of
the
in the dimer limit.
H b l « The
of
terms
preserves
interchange
eigenvalues
used by G r i f f i t h s
(2-29)
(Kz ,KXy = P'Jz ,PJ Xy)
[18],
and B a r r
chain.
and Blote
[51]
The
and
chain
H ^ i and the Z c o m p o n e n t
of
36
total
spin
operator
L
o2 =
Y t
a iZ
(2-30)
i= 0
commute,
a convenient
choice
of basis
again the eigenfunctions of a z which
the
eigenfunctions
2 (L + 1 )x 2(L + I) m a t r i x
of
<r £ z .
in b l o c k
function
|PZ > are
are direct products, of
This
form
yields
in w h i c h
as
the
a
largest
block is
and each block corresponds
A further reduction
introducing
a mirror
interchanges
commutes
the
with
to a single
of block
ctz.
size
can be
reflection operator R
spin
We
vectors
can use
at
obtained by
[21,52],
sites
i
R to introduce
and
which
L - i. R
the simple
symmetric basis states
|P+ Z> = |PZ> + R|PZ>
and antisymmetric states
lP_z> = IP z > - R | P Z>
For L = 3 , 5 and 7,
we list all block sizes for
in Table 4.
We find the eigenfunctions
lm> =
and
^
eigenvalues
ai (m ) Ii>
E ffl numerically
for
the
(L+l)-spin
cluster.
37
Table 4.
The b l o c k sizes for H bl 0f L = 3 , 5 and 7. Since
the
b l o c k size for orz<o is the same as that of & z > 0 , we
just list
that of c z > 0.
L = 3
or
z
4
|P+_>
2
Ip+4 >
Dim.
0
IP+ 2 > IP_ 2 >
I
2
|p+ 0>
|P_0>
4
2
2
L = 5
orz
6
lr+_>
4
IP + 4 > IP _
|p+6>
D im.
2
I
3
8
6
4
> IP
3
>
+ 2
0
| P _ 2 >
9
|p+ 0>
6
|p_0>
10
10
L = T
+
I
NZ
a2
|p+8>
D im .
Table
spin
1
| P + 6>
|p_
4
5 . The f our
cI us ter.
4
6> IP
4
> IP _ 4
16
>
12
eigenstates
IP
+ 2 >
28
= Itt>
I2 > = l w >
I3 > =
I /
Il
ZN
I/
.2 It* + *t>
2 It* - *t>
E 1
' = -Klz'
e2 ' =
E 3
' =
- 2h i '
+ 2hi 1
- K i z '
K i z '
E4 ' = Kiz'
0
IP _ 2
>
|p+ 0>
2 8
and e i g e n v a l u e s
Ein
In)
I1 >
+ 4
2
"
2 K i x y '
+
Z K i z y '
38
|p_o>
32
of a two
38
The
renormalized
Hamiltonian
eigenvalues E n' which we
^ » has
shown
eigenstates
|n>
and
in Table 5.
Next we decimate within one cluster
Tr' ex p { -Hhl
e x p { Ai-Hhl ' }
7 ImXmI
Tr
I
e x p (-Em J
m
( Y, a i (m ) Ii > ) ( Yt a j ( m X j l )
(2-31)
e XpC-Em J
m
Here
Tr'
means
intermediate
to
take
the
partial
trace
spins. Here we list the
all
over
all
(L— I)
situations
that
will be encountered:
Tr'
a2 I tAt) <t At I = a ^ l l X l l
Tr'
a2 I I A l X t A l I = a2 |2X 2 I
Tr'
a2/2
I t A l + l A t > < t A l + l A t I = a2 | 3 X 3 I
Tr'
a2 / 2
ItAl-IAtXtAl-IAtI
Tr'
1/2
= a2 | 4 X 4 I
(a I tAl+tBt>+b I tBl+lAt> ) (a<tAl+lBt I +b < tB 1+ IA t I ) =
= (a+b )2 I3 >< 3 I + (a-b)2 | 4 X 4 l
Tr' 1/2
(ajtAl— lBt>+bltBl— lAt>)(a<tAl-lBtl +b,< tB I- IAt
= (a—b)
2I3 X 3 I +
(a+b)
2|4 X 4 I
A and B are the c o n f i g u r a t i o n s
spins.
We
of the (L-I)- i n t e r m e d i a t e
thereby find
4
4
I
n=l
InXnI
I) =
exp CAl-En'}
E
n=l
InXnI
Pn
39
By
solving
these
equations,
the
parameters
for
the
renormalized Hamiltonian H^l' are found to be as follows :
^l
=(
In
Pi + In Hg + In
Pg + In
P4 ) /4
Kiz'
=(
In
Pi + In P2 - In
Pg - In
P4 )/4
Klxy' = (
In
Pg - In P4 )/4
hi'
In
Pi - In P2
=(
Permuting
K 2 z ',
indices
K 2 x y *»
I
and
and
h2'
Hamiltonian H ^ j The
2,
we
for
the
find
the
parameters
renormalized
Aj,
cluster
renormalized field is thus
given by
h' = h i ' + hj * •
The
free
energy
(In Q)./N = I / (2L)
is calculated by iteration:
Y 4 !/(Lta)
(A 1 +A2 ) (r (m) tK, a ,h })
m=0
We
cases,
ma y
c heck
(2-32)
the p r o c e d u r e
in the
dimer
for w h i c h the d e c i m a t i o n m e t h o d is exact. The free
energy for the isotropically
interacting dimers
(In Q) / N = 1/2 I n { e xp(K+2h)
for the Ising
case
(In Q) / N = I n { exp (K) cosh h +
+
e xp (2K)
sinh%
is
+ e xp(K-2h) + e x p (K) +
+ e x p (- 3 K ) }
While
and Ising
h + exp(-2K)
}
40
From
the
heat,
23,
free
energy,
magnetization
and
that
decimations
calculate
in
properties,
in the
w i t h the a p p e a r a n c e
free ene r g y
method
after
spin chain
every
one
and
(2-
and
needs
two
Since we intend to
do
no t
(i.e. phase
expect
any
transitions
there r e m a i n s
in the choice of r e s c a l i n g steps. An
is
shown
in Fig. 7 . In
one reta i n s
can c o n t i n u e
as before.
time.
specific
formulas
iteration
decimation
iteration
computation
using
of an order p a r a m e t e r )
considerable freedom
scheme,
every
of the a l t e r n a t i n g chain.
singularities
calculate
differentiation.
Eq.2-31
thermal
alternative
can
susceptibility
24 and 25) and numerical
Note
one
needs
just
Using
one
this
this
iteration
an a l t e r n a t i n g
In'
this
decimation,
iteration
way,
reducing
proc e d u r e ,
the
formula for the free energy becomes
(In Q ) /N = I / (L+l)
I / {(L+l)/2)2 Al ( R (m){K.a.h})
Z
(2-33)
m=0
L is still
the
scale
factor,
3, 5 or 7. The r e s u l t s using
this alternative method will be discussed below.
41
----
L s 3
—-SK/VXA/S-
—
---- wv/vw>.
^AAAA#— — —
— —
---- erw\Av*----JjJLa a
—WVWW- —
— — — -♦WVNjW— —
Fig. 7.
Alternative iteration
calculation.
method
fo r
—
f r e e energy-
P.e rjLoj!_i.c Bound a^^ Cond it ions : _thj; 6- Sp in RJ.nx
An
alternative
to using
a
larger
scale
factor
is
to
keep the scale factor c o n stant but increase the n u m b e r of
spins
treated
errors
which
a
cluster
their
once.
occur
with
renormalized
with
at
due
free
on
two
in
important
are
cluster.
which
produce
uniform,
reliable
(Ising)
a ring
symmetry
the
w ill
Periodic
is
also
(for
the
However,
but
vary
boundary
and have been found
classical
for
reduces
operators.
not
in obtaining
process
by
which
the
translational
interactions),
also
conditions
this problem,
dimensional
renormalization
N/2-fold
boundary
position
to be
approach
to noncommuting
interactions
conditions eliminate
example
This
RG
results
s y s tems.
aided
for
by
The
the
alternating
diagonalization
of
the
42
Hamiltonian
time
is
applies
Bonner
and
For
also
others, which
possible
Renormalizing
of
Fig.
reliable
8b , c)
to
can
3
computer
fre e
energy
we
have
in
but
thi s
fac t
ring
be
spins
interactions,
of
is the
decimated.
yi e l d s
uniform
for the purpose
is
found
nonetheless
that
it
gives
results.
We next briefly discuss
arise
a 6- s p i n
which
the
and
in
chain extrapolations
interactions,
than alternating
reasonable,
reduction
customarily employ rings.)
cluster
(see
calculating
(This
to the finite
alternating
smallest
rather
simplified.
in
renormalizing
a
the
ring
technical
of
problems
spins.
The
which
general
decimation formula is
exp{ A 1-Hb , } = Tr'
For
the
free-en d
e x p { -Hb }
cluster
H a m i l t o n i a n H b , has four
E n1. After
decimations,
eigenstates
InXnI
Pn .
unknowns—
the
n=l
This yields
four equations
three p a r a m e t e r s
Kz'.
In > and e i g e n v a l u e s
InXnI
e xp ( Al-En '
n=l
Hb':
decimated
4
I
and
the
decimation we find
4
Ai
(2-34)
K x y' and
h ''
in four
constant
of the r e n o r m a l i z e d H a m i l t o n i a n
43
Fig. 8. Bond-moving and spin ring decimation.
(a)
The e n s e m b l e of s i x -spin rings arises from infinite
spin chain by bond— moving approximation.
(b,c)
The spin ring before and after decimation. Note that
the s i x - s p i n a l t e r n a t i n g ring t r a n s f o r m s into t h r e e - s p i n
uniform ring.
(d)
The e n s e m b l e of spin ring reverts to spin chain by
bond-moving approximation.
44
For
H b '
=
the three
spin ring,
- K z ' { (12) + ( 2 3 ) + ( 1 3 )
}
-
K x y ' { (12)j. + ( 2 3 ) A
+ ( I S )
jl
- h ,{(l)+(2)+(3)l
For
simplicity we have
( ij )
=
(I )
introduced, the symbols
O i Z w j = + d . y * j y
(2- 3 6 )
= O.Z
Using
translation
In >,
eigenstates
the
(2-3 5 a )
= W iZ0TjZ
)x
( i j
}
Table
6 a.
symmetry,
and ei g e n v a l u e s E
We see
characterized
one
by
that
kj
and
w h i c h are listed in
are
two
k2 »
which
are
linear
can use
the mirror
make
the e x p a n s i o n c o e f f i c i e n t s real. The resu l t s of this
six unknowns
additional
a first
symmetric
terms
eigenvalues
i n t e r a c t i o n t erms
and
truncated
6b.
in general
in the decimation procedure,
normal
at
that
the
order
each
bilinear
coefficients.
reflection operator R (see p.3 5) to
shown in Table
course
as
complex
doublets,
One
six d i f f e r e n t
with
eight
of
The
states
the
combinations
procedure are
product
n‘»
there
momenta
finds
RG
produces
If
interaction
in the H a m i l t o n i a n ,
and therefore
two
in H b ', E q. (2-3-5 a). It is of
new
approximation,
step.
determine
we
these
want
and
interaction
the
to
terms,
typically
keep
just
are
the
longitudinal, field
the only i n t e r a c t i o n s w h i c h we
45
can add on tlie u n i f o r m 3- s p i n ring are 3- s p i n terms. Thus
we add two three-spin interaction terms
to Hfo':
Hb' = - K z '{ (12)+(23)+(13) } - K x y '{ (12)A +(23)4. + (IS)4.) - h ' {(l)+(2) + (3)} + T (1 2 3) +
+ S{ (12)x (3) + (23)a.(l) + (13)j.(2) }
Here we use
( ij k)
the same symbols
as those
new
of E q . (:2 — 3 6)
= O iZajZo^Z
(ij)x (k) = ((TiZffjifffiYffjy)
The new
(2-35b )
eigenstates
eigenvalues
are
are the
z
same as
listed
those
of Eq.(2-35a)
the
in Table.6b.
For the 6-spin alternating ring
H b = - K z {.(12)+(34)+(56) +a [(23) + (45)+(16) ] } ^ K xy { (12 )A + (3 4 )4.+ (56 >4.+a [ (23 )x + (45 )* + (I 6)x ] } - h {(I)+(2 )+(3)+(4)+(5)+(6 )}
The original H a m i l t o n i a n m a t r i x is thus 64x64, but can be
block-diagonalized by considering rotational,
and
mirror
symmetries.
largest block which
Phoosing
eigenstates
translational
of o z gives
a.
is 20x20.
Further use
of the operators
and R reduces
the largest
submatrix to 5x5
w i t h all entries real (see T a b l e 7). This
is then r a p i d l y
T (translation)
diagonalized
numerically.
46
T a b l e 6 . E i g e n s t a t e s and e i g e n v a l u e s of 3 - s p i n r ing
Hamiltonian
H': (a) c o m p l e x e i g e n v e c t o r s for Eq. (2-35 a) ,
(b) real
eigenvectors for Eq.(2-35 a) and (2-35b).
(a)
I
In >
Ittt
En
>
" 3kZ " - S b '
I iii>
~ 3 K Z '+ 3 h '
l , k j>
K z ' — h '— 4 K x y ' c o s
l . k p
K z * + b ’— 4 K X y ' c o s
*'
= H-spS
|4tt+ejit*t+ej2tti>
= I/p ^
1
e jI
(
kj
=
=
kj
k J
1I l+e j i it i+e j 2 ^* t >
e x p {i k j },
2 Jtj / 3 t
e j2
and
=
e x p { i 2 k jJ »
j = 0 , I and
2.)
(b )
En
En'*
In >
11> = 1111>
-3Kz'-3h’
-3Kz'+T-3h'
I2 > = I I i I >
“ 3 K z'+3h'
-3Kz'-T+3h'
I 3 > = l / ^ / T | I t t + t i t + t t i>
K z ,- 4 K x y ' - h '
K z '- 4 K x y '- T + 4 S - h '
K z '+2Kxy'-h'
K z ' + 2 K x y ’ — T - 2 S — h.'
\A > = H
p
T
I 2 i t t - t i t - t t i>
I5 y = H p i
|tit-tti>
I6> = l / p 3
11 i i + i t i + i i>
K z ,- 4 K x y , + h '
K z ,- 4 K x y ,+ T - 4 S + h '
11 > = H p e
1 2t i i - i t i - i t t >
K z'+2Kxy'+h'
K z *+ 2 K x y '+ T + 2 S + h '
I8 > = 1 / P ^
|iti-iit>
»»
» :E
*
is
for
Eq. ( 2 - 3 5 a )
and
E n 6*
35b ) ( including 3- spin interactions)
is
for
Eq .(2
-
47
T a b l e 7.
The d i m e n s i o n s of b l o c k m a t r i c e s of 6 - s p i n
a l t e r n a t i n g r ing H a m i l t o n i a n . H e r e w 6 j u s t l i s t t h o s e for
non-negative Z c o mponent of total spin.
oz
6
4
2
O
by <rz
I
6
15
20
by T.R
I
I,I,I,I,I,I
Dec..im a tj. on
I ,4,5,5
Jgrocedureji
4 „4.3.3.3 .3
S u b stitu tin g
the
eigenstates and eigenvalues of both three spin and six spin
ring Hamiltonians into Eq(2- 31) , we get
8
64
Aj - ^
En' I n X n I = In Tr' ^
n=l
exp (-Effl ' } ImXm I
m=l
In the decimation procedure,
situations
For O2 =
Tr'
where
that need to be
(2— 37)
there
are
the following
treated:
3 in the decimated ( 3-spin) ring.
exp{-Em } I t A t B t C X t A t B t C l
I ABC)
is any
state
= exp{-Em ) I l X l I
f u n c t i o n of the
decimated s p i n s .
For crz = I ,
I1+ = Tr'
e x p {-Em I IajlAtBtC + a2 tA4Bt C + a 3 tAtB4C>
<a1 4 A t B t C + a 2 t A 4 B t C + a 3 tAtB*C|
= e x p {-Em }|aji t t + a 2 t i t + a 3 t t 4 > < a j A t t + a 2 t i t + a 3 t t i |
We
|5>
ca n e x p a n d
in Table
this
6b:
state
using basis
states:
I 3), I 4 >
and
48
IaI Itt+a2 t *t+a3 ft 4> = b 3 I 3 > + b4 I 4 > + b 5 I 5 >
and find
■
1I + = ex p {-Em }
The
we
same
b,2
3
b 3b4
b 3b 5
I
b 3b 4
b 42
b 4b5
I
b 3b5
b4b 5
b 52
I
situations obtain for a z = — 3 and -I. As a result,
find from
Eq.(2 — 37)
A 1 “ Hb' = In B
Here B is a s y m m e t r i c b l o c k - d i a g o n a l i z e d m a t r i x that has
the form
I 1x1:
I,
I
iix i
I
I
I
B =
I 3X3 j
I
I
I
We can easily take
0
«
0
I
the logarithm of the matrix B ,
log B = U ( log U -lBU ) U-I
where
U
is
an orthogonal
B ' = U-lBU,
3X3
U-I = U t
We find finally that
,
matrix
(2-38)
that
diagonalizes
and B ' diagonal.
B:
49
8
En ' I n X n l
1I “
(2-39)
= U (log U - 1BU) U-I
n= I
The
left
side
side is not,
in the
the
we
of this
Hamiltonian.
general
the
Fortunately,
situation—
spin chain in large
that
is diagonal.
should have included more
original
more
equation
right
field,
side
the
If the right
interaction terms
we
have
alternating,
examined
anisotropic
a < I , K z = K xy and h> > 0 , and find
of Eq.(2-39)
is
always
diagonal.
In
fact.
8
U (log U-lBU)
U-I
InXnl
Pn
n= I
where
= p 5 and
Therefore
ring
is
constant
P 7 = P g , corresponding
the Hamilton! an
consistent
with
term and five
As a consequence
that
of
of
the
interactions.
I
N
W
CO
-3 h ' = Pl
I
N
W
CO
-
T
+ 3 h ' = P2
K z'
-
T
+
4
S -
+
K z'
-
T
-
2
S +
2KXy
' - h ' = P4
A1 +
Kz'
+
T
-
4
S -
4 K xy
' + h ' = P6
Al +
K z'
+
T
+
2
S +
2KXy
’ + h' = P7
Al +
Al
ring
6-spin
with
a
1
we find:
T
A1
decimated
a 3-spin
+
Al
to E4 '= E 5 ' and
4Kxy '
- h ' = P3
(2-40)
50
Solving
this
set of equations
parameters
of
the
'» T , S ,Kxy ’ and
1.
It
this a p p r o x i m a t e
changes
the
one,
d e c i m a t i o n RG has the
ant!ferromagnetic
since
this
renormalized
S (see Eq.(2-35t>))
one
H b '*
chain
decimation
is
into
for
a
scale
L=2.
2. The
and
Hamiltonian
characteristics:
ferromagnetic
factor
renormalized
an(j ^he
h '.
In s u m m a r y ,
following
gives the constant
order
of
are
magnitude
3-spin
always
than
interaction
smaller
the
one
by
terms , T
more
and
two
than
spin
\
interaction
therefore
From
terms,
reasonable
chain
can be
after
d e c i m a t i o n , we
We
can
turn
bond-moving
repeating
grow
slowly
to truncate
with
field.
It
is
the 3-spin terms.
the viewpoint of bond-moving approximation [54],
a 6-spin ring
spin
and
considered
moving
to come
bonds.
from
When
we
an
complete
one
get an e n s e m b l e of isolated 3- s p i n rings.
it b a c k
into
a spin
chain
a
second
time
procedure
procedure
one
a gain
develops
by
applying
(1/6)
Fig.
8).
the
free
en e r g y
I / (2m) A 1 (R (m ) {K.a,h } )
m= 0
i
the
(see
formula
(In Q ) /N ^
infinite
By
51
Th.e rm.odjrnjim^.c
We have
introduced
above
the approximate
decimation
procedure for L = 3, 5 and 7 free— end spin clusters and for
the
N
=
6
ring.
The
results
of
the
speci f i c
heat
susceptibility calculations
in a variety of cases are
inFig.
appropriate
with
9 t h r o u g h 22,
w here
the finite — chain
In
heats
Fig.
9
are
for u n i f o r m
field.
The
extrapolations
plotted
results
of
L =
3,
are s u c c e s s i v e l y
estimates,
with
respective
5.1% and 3.6%. The
quantitatively
appear
to be
T I / 2.
In
the
calculation
chain
namely
decimation,
closer
the
better
heat
resu l t s
this
in zero
f r e e — end
cluster
to the B o n n e r - F i s h e r
important
region
is not
specific
coupling
at kgT/J = 1.0 of
resu l t s
successive
a s p e cific
but
7
deviations
to an
temperature
gives
and
low-temperature
accurate, but
converging
ferromagnet,
of Ref.[18,5 7],
ferromagnetic
5
shown
they are c o m p a r e d
iso t r o p i c H e i s e n b e r g
decimation
8.2%,
the
and
of the I-D
falling
to
kj}T/J>1.0
than those
case
not
approximations
feature
the
are
zero
the
as
r ing
by L = 7 open
at
the
lower
temperatures.
The
same
coupling,
deviation
case
is
from
larger
comparison,
shown
in
the finite
for
Fig.
10.
antiferromagnetic
At
-
chain extrapolations
than for ferromagnetic
temperature
region
yields
best agreement.
the
isotropic
( k g 1 .0) the
At
exchange.
ring
At
* *®
the
is in every
the
calculation
low-temperatures
higher
again
there
is
52
a p p arent c o n v e r g e n c e
linear
behavior
spectrum
opposed
C(T).
(The
is p r o p o r t i o n a l
to
involving
show
of
of the f r e e - e n d clus t e r results to a
k2
in
finite
the
to m o m e n t u m
ferromagnetic
numbers
exponential
antiferromagnetic
of
spins,
behavior
of
k for
case.)
of
C(T)
small k,
as
Approximations
course,
for
energy
will
always
sufficiently
low
tempera ture.
Fig.
11 s h o w s
the
specific
heat
for
different
a n i s o t r o p y factors y = J xy / j z = i.o, 0 .5 and 0 in the case
of
easy-axis
(Ising-Iike)
antiferromagnetic
used, namely
behavior
of
Heisenberg
chains.
the L = S
ferromagnetic
A single RG tec h n i q u e has b e e A
free — end cluster method. The. thermal
the pure
Ising
chain
is of course
either ferromagnetic and antiferromagnetic
is
known
excited
that
there
exists
an
energy
state for all y <I [17,55 ,56],
in the ferromagnetic
Our approach, however,
Fig.
evidence
12
the
same for
interactions. It
gap
to
the
first
which vanishes as 1-y
case and with an essential
( ~ exp(-cst./*y I -y ) ) for
T to show
and
ant ! f e r r o m a g n e t i c
is not sufficiently
singularity
couplings.
sensitive at low
of these effects.
displays
the
specific
heats
for e a s y - p l a n e
(XY-Iike) chains w i t h y = I, 2 and ” , A gain the L = 5 freeend clus t e r a p p r o a c h has been used, and we r e m a r k that pure
XY chains have
the
coupling.
the same
There
s p e c i f i c heat for either
is no energy
gap for
either
the
sign of
ferro-
53
or
the
zero
no
antiferromagnetic
in p o w e r
clear
fashion.
susceptibility
than those
methods.
Fo r
for
differ
from
3 % , and
those
same
for
coupling
the
finite
for
trend
the
is
finite
are
however,
(see
an d
Fig.
the
RG
gives
general
more
chain
The
close
13),
the
to
the
same
chain
with
zer o
field
at k g T / J = 1.0
(N= 8 ring) p r e d i c t i o n s
heat,
in
isotropic
be
same
by ~
by ~ 8% (see Fig.
the
as m a y
calculations,
surprisingly
obtained by
Heisenberg
Fig.
specific
14.
heat
in
(L=3 o p e n chain)
chain
case,
are
the
apparent
antiferromagnetic
10
too,
specific
s u s c e p t i b i l i t y RG r e s u l t s
Fig.
Here
results
instance,
ferromagnetic
The
so tha t C- s h o u l d a p p r o a c h
indication of the detailed I o w - T behavior.
The
accurate
law
case,
s e e n by
situation
where
the
those
of
the
9).
Heisenberg
comparing
occurs
in
the
susceptibilities
the
infinite
system
[14].
We als o f i n d tha t
for
thermodynamic
accurate
similar
that
the convergence
than for H=O
for
the
values
find
properties
than those
to
in
that
nonzero
the
finite
chain
T his
d e c i m a t i o n RG
field
are
more
s i t u a t i o n is a l s o
extrapolations,
where
w i t h N is apprec i a b l y more rapid in a field
[18].
Su s c e p t i b i l i t y versus
applied
they are
8- spin finite
in
in z e r o field.
ferromagnetic
of
the r e s u l t s of the
ring
Heisenberg
field
in close
[18],
also
are
t emperature
chain
displayed
agreement w i t h
shown.
for
curves
several
in Fig. 13.
We
the results for
1
54
field
Fig.
The
susceptibility
for
the
1 5 . At
uniform
the
versus
antiferromagnetic
critical
field
value
zero-temperature m a g n e t i z a t i o n
appears
t e m pe r a t u r e
to d i v e r g e
saturates)
as T -£> 0, w h i l e
for
both in accordance wit h
[18], Fo r
fields,
temperature
allow
for
limit,
detailed
view
indicates
the
The
magnetic
of
correct
specific
field
H=O
RG
the
included
means
of
for
9-
vs.
For H greater
be
noted
between
agr e e m e n t
the
is poorer.
10-spin
critical
two
In
o ver
t hose
results
the
improvement
for
over
fact
specific
those
that
in
curves
are
the
chain
case.
Th e
in Fig.
at
14
constant
are
The
is e a s i l y seen.
difference
although
the
curves
calculations.)
field little
for
pronounced
H<4
can
the
shoulder
to be s p u r i o u s .
results
h eat,
in zero field,
finite
generally
s h o w n in Fig.
(The d i s p l a y e d
particular
of
not
of R e f . [ 1 8] f o r H/J = 0 a n d
methods,
at T = 0.3 and H = 3 is t h o u g h t
The
it
increasing L.
in the h i g h - f i e l d cas e
than the
the
larger field
provided
temperature
comparison.
improved agreement
in
calculated
these
the a n t i f e r r o m a g n e t
and
the
do
in
situation
16. E x a c t f i n i t e - c h a i n r e s u l t s
5 are
s hown
the k nown behavior
results
trend w i t h
heat
for
the
extrapolation
the
is
should approach a finite zero-
bu t
reliable
constant
H / J = 4 (for w h i c h
approaches zero,
small
chain
at
field
for
susceptibility
and of the f i n i t e — f i e l d
appear
the
to be related
couplings
—
to
both
55
transverse
rapidly
absent.
to
that
for
is
the
than
each
decrease
speaking
the
term,
the
In
the
for
the
coupling
case,
than
the
the
the
the
accurate
the
Zeeman
exchange
fields.
exchange,
and
is
field always
it d e c r e a s e s
of
more
field
ferromagnetic
division
faster
more
when
rather
the r e n o r m a l i z e d
any
smaller
renormalize
cases
iteration
on
e n ds.
—
the
antiferromagnetic
dependent
always
in
other hand,
with
cluster
Zeeman
longitudinal
zero
On the
increases
while
and
chains,
at a rate
terms
at
constants
Relatively
the
larger
the
RG results.
AJ. t^er na_t_inj| antiferrom agnetic chain
We
with
turn
alternating
method
is e x a c t
results
of
next
lead
couplings.
to
dimer
expect
n itrate,
for
we
case
the
the
decimation
a n d the u n i f o r m
reliability
have
chain
in the
case
calculated
0=0.27,
field
antiferromagnet
the q u a n t u m
limit,
greater
the
s canning
to
Since
properties,
susceptibilities
copper
attention
in the
us
magnetic
our
the
corresponding
at
constant
gives
peaks
to
temperature
k g T / J = 0.05.
The
upper
L = 3 open
critical
respectively.
(2.54)
of
chain
fields
While
reference
method
the
49,
t h a n the q u o t e d v a l u e
H
c j_ / J j
latter
the
= 2,1,
agre e s
lower
lower
and
and
closely
critical
(1.675) b y a b o u t
at
25%
with
field
is
( s e e Fig.
result
higher
17).
56
Increasing
resu l t s s h o w n
the b l o c k
10%
in
reaching
excess
extrapolation.
increases
the
We
of
can
peak
a value
tha t
also
structure
We
second
29).
have
also found
iteration
that
becomes
Here
the
method
as
is only
finite
the
obscure
chain
block
and
a
size
third
the origin of which will be
we
still
results
have
of
susceptibility by
(Eq,(2 — 33))
use
for spins at the ends
The
They
field falls
at L = 7 w h i c h
by
the
below.
clusters.
terms
7 gives
critical
given
see
(middle) peak becomes apparent,
discussed
to L = 5 and
in Fig. 18. The l ower
as L increases,
about
size
this
the
based
division
of each cluster
procedure
are
the same character as those
using
on
of
the Z e e m a n
shownin
of Fig. 18,
large-cell
of H cl vs. 1/L,
Fig.
19.
but
the
is clearer
than that by the first. We are able to carry out
[56]
free-end
given by Eq.(2-
peak s t r u c t u r e by the second i t e r a t i o n m e t h o d
extrapolation
the
a linear
giving H ci = 1.61 in the
limit.
We have repeated the above computation with L= 3 freeend
clusters
neighboring
this
approximation
was
dividing
c l u s t e r s --
about 1.8 —
f ield
without
the
a slight
found
temperature,
out
that
Zeeman
is we
l ower
terms
between
let h 1 =% &n d h.2 = 0 . In
critical
field
appears
at
improvement — , but no upper critical
to
H/ J
=
2.8.
At
slightly
^cBT/J=O .0 8, The upper critical
field
higher
appears
57
at
about
2.7
(Fig.
20).
However
magnetization
still
does
not
It is apparent
for
that
the end spins has
the
last
instance
iteration
step
(H / J > 2.3) ,
at
the
ferromagnetic.
field
until
ferromagnetic.
at
the
at
last
the
ef.fect on the
h2 = 0)
and
we
found
the
have
that
iteration
same
of the Zeeman term
the
as
ground
checked
in
high
ground
iterations,
rate
results.
does
state
the upper
critical
field
In
the
field
state
is
the magnetic
the
is
coupling
no
longer
is shifted
or
to a
value.
In Fig.
19 we c o n s i d e r e d
the field terms
k = ^1I + h 2 ,
Hence
first
fields
Thus the magnetization is not saturated,
in other words
larger
treatment
With successive
decreases
constant,
the
a marked
step
at h igh
saturate.
(&i=h,
by
even
the f o l l o w i n g
for the end spins:
and
h 2 / hi = a.
only part of field hi that applies
takes
part
in
d i v i s i o n of
the
iteration.
to the end spins
Therefore
in
the
n ext
is f e r r o m a g n e t i c
state
iteration we will leave a larger field term
Rh = h 2 + h'.
Thus
we
can keep the ground state
during successive iterations ,and the magnetization will be
saturated.
To
improved
summarize,
upper
by taking h2/hi=a (Fig. 19) we find an
c r i tical
f ield
value,
while
by
taking
58
^ 2 / hi =O
(Fig.
2 0)
we ' get
a
better
lowercritical
field
value.
I I
;
entire Z e e m a n
The
problem
course absent with periodic
fact
that
the
critical
field
alternating
are
temperature
of
at
the
broad
low
finite
ring
of
single
iteration)
crossing
field
points
values,
corresponds
finite
to
probable
w hile
the
that
which
the
first
the
middle
the results
number
are
obscured,
correspond to two
only
critical
is of
a c c urate
see
lower
for
the
middle
peak
attribute
the
(since
last
l ower
peak.
and
This
fields.
state
to a
level
critical
level
crossing
only . a s
difficulty
intermediate
limit.
peaks
for the infinite
the
an
upper
to the l a r g e - r i n g
leaving
of
the applied
appe a r s
causes
to
to the use
applies
ground
and
the
again
spectrum
this
intermediate
as
and merge
as a function of
and
of
that
in size
energy
increase with the block size until,
they
can
we
speaking
effect,
the
We
A third
six spins
the
a more
diminish
show
determine
block
extrapolating
21.
maximum.
in Fig. 22. Roughly
spins
susceptibilities
Fig.
peaks
We
end
chain along several different
temperature,
blocks.
alternating
field
gives
The
s h o w n in
rises
a single
appears
ring
value.
the
boundary conditions. We find in
antiferromagnetic
isotherms
form
6 — spin
for
two
peaks
a
for
It is
would
chain,
that
Fig. 9. V a r i a t i o n of specific heat w i t h t e m p e r a t u r e for
ferromagnetic Heisenberg chains by decimation using L= 3, S
and 7 f r e e - e n d c l u s t e r (solid line),
and 6 -spin ring
(dashed line).
C o m p a r i s o n is m ade with the n u m e r i c a l
extrapolations of Ref.[18] (dotted line).
60
KbT/J
Fig. 10. V a r i a t i o n of s p e c i f i c h e a t w i t h t e m p e r a t u r e for
a n t ! f e r r o m a g n e t i c isotropic H e isenberg chains by d e c i m a t i o n
u s i n g L= 3 , 5 and 7 f r e e - e n d c l u s t e r s (s o l i d line), and 6& pin ring
( d a s h e d line).
C o m p a r i s o n is m a d e w i t h
the
predictions of Bonner and Fisher [18] (dotted line).
61
\£ 0.3
K bTZJ z
Fig. 11. Specific heat versus t e m p e r a t u r e at cons ant
a n i s o t r o p y y for I s i n g - I i k e spin chains (L= 5 free end
clusters). F and AF denote the spin chains with ferro- and
antiferromagnetic couplings.
62
F r*i
KbT/ JXY
Fig. 12. S p e c i f i c
a n i s o t r o p y y for
clusters) .
h e a t v e r s u s t e m p e r a t u r e at c o n s t a n t
XY-Iike
s p i n c h a i n s (L= 5 f r e e - e n d
63
KbTZJ
Fig.
13.
S u s c e p t i b i l i t y of i s o t r o p i c
spin chain with
f e r r o m a g n e t i c c o u p l i n g a I c o n s t a n t m a g n e t i c field. S o l i d
lines: L= 3 d e c i m a t i o n R G , d a s h e d lines: e x a c t 8 - s p i n rin g
results of Ref.[18].
64
0.08
0.06
” 0 0.04
0.02
KbTZJ
Fig. 14. Susceptibility of antiferromagnetic isotropic spin
chain in zero magnetic field. Shown are for comparison are
results by L= 3, 5 d e c i m a t i o n RG (solid lines), and from
Ref.[18] the exact 4- and 10-spin ring calculations (dashed
lines) and the e s t i m a t e d l i m i t i n g curve by finite chain
extrapolation (dotted line).
65
Fig. 15.
Susceptibility
versus
temperature
for
a n t iferromagnetic isotropic spin chain at constant magnetic
field.
C/NK
66
KeT/J
Fig. 1 6 .
S p e c i f i c heat ver s u s
temperature
for
antiferromagnetic isotropic spin chain at constant magnetic
field. Solid lines: L= 5 decimation R G , dashed lines: 9- and
1 0 - s p i n ring r e s u l t s for H / J = 5 and the d o t t e d line:
estimated limiting line for H / J=O from Ref.[18].
67
*
X
0.05
0-6
0.24 /I
Hz/T
Fig* 17.
S u s c e p t i b i l i t y of the a l t e r n a t i n g li n e a r chain
wit h a l t e r n a t i n g ratio a = 0 .2 7 as a f u n c t i o n of field for
two different temperatures. Solid lines: L= 3 decimation RG1
dashed lines: of Ref.[50].
68
HzAJ
Fig. 18.
S u s c e p t i b i l i t y of the a l t e r n a t i n g linear chain
with a=0.27, versus
f i e l d at c o n s t a n t
temperature
k g T /J = O .0 5. L= 3 , 5 and 7 d e c i m a t i o n a p p r o a c h e s are shown,
using the iteration method of Eq.(2-32).
69
—
3 0.6
HzAJ
Fig. 19.
Susceptibility of the alternating chain with
a = 0 .2 7, versus field at constant t emperature kgT/ J = O .0 5 .
L = 3 , 5 and 7 d e c i m a t i o n approaches are shown, using the
iteration method of Eq.(2-33).
70
Fig. 20.
S u s c e p t i b i l i t y of the a l t e r n a t i n g linear chain
wit h a = 0.27 versus field for kB T /j = 0 .0 8. L= 3 d e c i m a t i o n
with iteration Eq. (2-3 3) and Zeeman terms for the end spins
taken with hj-jj an(j J^ = O.
71
HzAJ
Fig. 21.
S u s c e p t i b i l i t y of the a l t e r n a t i n g linear chain
with a=0.27 versus field at constant temperature, using 6spin ring decimation.
72
0.27
m
L-ZlzotsWoi-
Fig. 22.
Energy s p e c t r u m of an a l t e r n a t i n g
spins as a function of the magnetic field.
ring
of
six
73
2.3
DECIMATION FOR CHAINS WITH ARBITRARY FIELD DIRECTION
Since
it- may be d i f f i c u l t
in a n 'e xp e r i m e n t to align
the m a g n e t i c field e x a c t l y along a definite crystal axis,
it is i m p o r t a n t
on
the
magnetic
calculations
chains
to learn the effect of a t r a n s v e r s e field
and
have
response.
finite
been
Until
chain
primarily
now,
quantum
extrapolations
concerned
with
for
the
I-D
spin
case
of a
longitudinal
field —
that is, with the field along
axis
case)
perpendicular
(Ising
case).
Bonner
perpendicular
spins,
and
magnetic
using
and
or
and
Fi s h e r
susceptibility
Ta k a n o
exponents
and
systems.
In
have
[41]
longitudinal
easy
an easy
plane
(XY
the
T= 0
calculated
for finite
Suzuki
for
a bond-moving
3-D
[18]
to an
chains
have
and
RG
of up
to 10
discussed
transverse
the
fields
scheme to extend I-D RG m e t h o d s to 2 —
the
following
section
we
present
an
e x t e n s i o n of the d e c i m a t i o n RG m e t h o d w h i c h a l l o w s us to
c a lculate
thermodynamic
field oriented
If there
are both X and Z c o m p o n e n t s
the Hamiltonian.
more
in the
of total
The
spin,
case.
calculation
difficult
Eq .(2-34)
The
of a
and we
of
the
field,
a z, no longer commutes with
for
larger
sized
clusters
therefore extend the quantum
d e c i m a t i o n RG w i t h L = 3 f r e e - e n d c l u sters
field
p r e sence
in an arbitrary direction.
the Z component
becomes
properties
decimation transformation
to the general
is still
that
of
74
exp {
- H b ' } = T r ' e xp t
where Hfe is the Hamiltonian
and
solving
for
(2-34)
of four-spin cluster and Hfo' is
the renormalized Hamiltonian
begin by
}
of
two-spin
the eigenstates
and
cluster.
We again
eigenvalues
of Hfo
H ^ 1^
Z Z£z.s.£AS p_luj:_t_e_r. The
renormalized Hamiltonian
of
two-spins is now
H.b' = - K z 'O0 2ffL z “ Exy/2
~ h z'
(o0 z+crL z)
"
(<70+<yL' +a0”ffL+ > "
h x /2
(®0+ + o 0~ + a L+ + ffL-)
w h e r e E q .(2- 9) has been used. There
we
must
breaks
treat.
the
to allow
though
at
a result
First,
symmetry
from
the beginning
we
expect
are two p r o b l e m s
the viewpoint
of the EG
in XY-plane. As a consequence,
for a n i s o t r o p i c
the
couplings
we have
(2-41 a)
that
the h x
we
need
in the X Y - plane,
even
isotropic
renormalized
interactions.
Hamiltonian
of
As
two
spins to be of the form:
H b ' = - K z 'OQ 2ffL z ~ D x 'crOxc Lx “ Dy' QrOycLy “
V
J
=
(ffOz+cLz> -
hx '
( orox+cLX^
- K 1O q Zo l Z - DS '7 2 (cro + o L- + OQ-oL+) -
- BA' / 2 (cQ +cL++ff 0_CL- ^ " h z' (cyOZ+cL Z) “
“
h x /2
(oo++®
0 *"+ c L + + c L " )
(2-41b)
75
We have
introduced
DS = (Dx+Dy )/2
and
and have used relation
crO xffL x - arO ycrL y
Second,
eigenstates
we
DA = (Dx-Dy )/2
(2-9)
to get
( <r0 +ffL + + ff0 '‘ffL ~
=
cannot
and
in
)/2
general
eigenvalues
analytically. Instead we
(2-42)
choose
of
solve
for
the
Hamiltonian
four
(2 - 4 1 b )
the following basis
set,
II > = It t>
I2 > = U t >
(2-43)
I3 > = I t O
14 > = U O
,
and write Hamiltonian
I -K1- 21V
I
(2- 41b) as
K'
"hX f
Hb ' =
I
I
'hX r
.- 2 D S ’
*
" h x'
-2DA
in
terms
of
six
-Hx'
-Hx'
- 2D S '
K'
-hx'
independent
matrix
-2DA'
I
-HV
I
"hx'
I
K'+2h z '
(2-44)
I
elements.
We
identify the renormalized pa r am eters by comparing
direct Iy with that
of the
this form
decimated Hamiltonian.
-iE-ill cluster. The
spins with anisotropic
will
original
and alternating
Hamiltonian
couplings
is
for
four
76
H b = -Ki(croZffl Z+ <y2 Zor3 Z) “ D S l/2
(Q-Q+<T1 ""+ <T2 +ff3 ~ + h • c } ^
- DAi / 2 (ffO+0'l++CT2+ff3+ + h -c) - K201ZtT2 z ~
-
D S 2/2
(cri+ a'2— + a l -ff2 + )
"
D A 2/2
( (Tl+ » 2 ++<rl""ff2 " >
- h z ( o l z+ a 2 z) - hx /2 ((rl++(T2++ll«c > "
- hzl((TOz+CT3 z) - hxl/2
(<rO++ff3++ h -c)
(2-45)
We have a l l o w e d the fields on the end spins to he dif f e r e n t
from
those
on intermediate
Using
the following basis
>
=
1111>
2>
=
II It >
1
spins.
3> =
III ;>
4> =
t t 44>
5> =
It 11>
6>
=
It lt>
7> =
It 1 1>
8>
It 11>
=
(2-46)
9> =
t It t >
1 0 > =
tilt)
11 > =
t It l>
12 > =
t Il l>
13 >=
1111 >
14 >=
1111 >
1 5 > =
lit l>
16 >=
1 1 1 1>
j
!|
77
we
deve lop the 16x16
computer,
matrix
and diagonalize
of the H a m i l t o n i a n
it to get
(2-45) by-
16 eigenstates
Im > and
eigenenergies E (m) , with
16
Im > =
^
v (m,n) In>
(2-47)
n=l
To decimate
one calculates
the partial
ex p{ .A i - H b '(K',DS',DA ',h z ',hx '> > =
trace
<
(2-48)
e x p { -nt (K ,DS,DA,a,hz ,hx ) }
= Tr'
In the left side of Eq. (2-4 8) , Hfo' is 4x4 s y m m e t r i c m a t r i x
that has the form E q .(2 -44 ) . In the right side of Eq. (2 - 4 8 )
e_ H i s a 16x16 m a t r i x of the for m
16
exp{ -H^ ] =
^
e xp { -Em ) I m X m I
m= I
The
trace
Tr'
operation then has the form
I aj tA t+ a2 ^A t+ 33 tA l + a4 IAi >
<
tA t+a2 I At +a3 tA I.+34 I Al
4
=
(
Z
4
ai I i > ) (
Y.
i=l
I
aj < j I ) =
J= I
a, 2
a I a
aI
»2
a I a
a
22
a2
a
2
a3
a
2
a
2
j
a I ag
1
a I a
4
4
3
a 3
a3
2
a 3 a
4
a I a
4
2
a
4
a3
a
4
a4
2
a
I =
78
the
where
IA > r e p r e s e n t s
I I >,
mi d d l e
spins,
and
vectors
of Eq(2-43).
any
I 2 >,
configuration
I3>
Accordingly
and
I4>
of
are
the
the
we find for right
two
basis
side
of
Eq. (2-48)
a (I ,m)
a (5 ,m) a (6 ,m)
a (7 ,m)
|
a (5 ,m)
a
(2 ,m ) a (8 ,m)a (9 ,m)
I
a (6 ,m)
a
(8 ,m) a (3 ,m)a (I 0 ,m) I
a (7 ,m)
a
(9 ,m) a (I 0 ,m) a (4 ,m ) I
16
T i ' e x p {-Hfo }
e xp{-Em }
m= I
(2-49)
Here
the m a t r i x
expan's ion
elements
coefficients
c a n be e x p r e s s e d
in.terms
v(i.j):
a (I ,m)=v^(l,m)+v^(2,m)+v^(9,m)+v^(10,m)
a (2,m)= V ^ (5,m)+ v ^ (6,m) + v ^(13,m)+ v ^ (14,m)
a(3,m)=:v2(3,m)+v2(4,m)+v2(ll,m)+v2(12,m)
a(4,m)=v2(7,m)+v2(8,m)+v2(15,m)+v2(16,m)
a(5,m)=v(l,m)v(5,m)+v(2,m)v(6,m)+v(9,m)v(13,m)+
+ v (1 0 , m ) v ( 14 ,m)
a(6,m)=v(l,m)v(3,m)+v(2,m)v(4,m)+v(9,m)v(ll,m)+
+v (1 0 ,m)v(1 2 ,m)
a(7,m)=v(l,m)v(7,m)+v(2,m)v(8,m)+v(9,m)v(15,m)+
+ v (10 , m) v ( 1 6 , m )
a(8,m)=v(5,m)v(3,m)+v(6,m)v(4,m)+v(13,m)v(ll,m)+
+v(14,m)v(12,m)
of
the
a(9,m)=v(5,m)v(7»m)+v(6,m)v(8,m)+v(13,m)v(15,m)+
+ v (14 ,m) v( 1 6 ,m)
a(10,m)=v(3,m)v(7,m)+v(4,m)v(8,m)+v(ll,m)v(15,m)+
+v(12,m)v(16,m)
Substituting
E q .(2-49)
I
1 6
A ^ - H b '
=
I n
/
into
Eq.(2-48)
a (I , m )
a (5 , m )
y i e l ds
a ( 6 , m )
a ( 7 , m)
I
e ~ E m
m = I
I
I
a ( 5 , m )
a ( 2 , m)
a ( 8 , m)
a ( 9 , m)
a ( 6 , m)
a ( 8 , m)
a (3 , m )
a ( 1 0 , m)
a (7 , m )
a ( 9 , m )
a ( 1 0 , m)
I
I
I
I
=
B
I
I
I
a (4 , m)
I
( 2 - 5 0 )
We
have
formula
taken
the
logarithm
of
the
symmetric
matrix
by
Eq.(2 — 38).
Since
the
has form Eq.(2-43),
it has zero trace.
and we ma y find d i r e c t l y the s p i n - i n d e p e n d e n t
term w h i c h
arises from renormalization:
A1 =
We then e x a m i n e
matrix)
(2-51)
(1/4) Tr B
the m a t r i x B -(1/4)(T r B )I (I is a 4x4 unit
which has six independent elements
I
A(I)
A M )
AU)
A (5 )
I
B - (1/4)(Tr B )I =
I
I
I
I
I
AM)
A (2 )
A ( 6)
A M )
AU)
A ( 6)
A (2)
AM)
A (5 )
AU)
A (4)
A (3)
I
I
I
I
80
From
Eq s.(2 44,50,51
and 5 2)
we
find
the
renormalized
parameters :
K'
= - A (2)
OS '
=
A ( 6) /2
DA'
=
A (5 ) / 2
hz '
= [A(I)+ A (2)]/2
h x'
Il
(2- 53)
■St
<
The r m o d y n a m i c
calculations:
The
free
energy
c a l c u l a t e d using
the r e s c a l i n g
approach
the
perpendicular
susceptibilities
parallel
25). As
and
a check
on our
susceptibility
Heisenberg
chain
d irections,
direction
for
and
(the
approach,
the
by
isotropic
apply i n g
have
curve
found
is
we
the
the
no
of E q .(2-3 3) and
have
Eq. (2 -
calculated
in
dependence
as
by
in
both
oh
Fig.
Z
the
and
found
that at low
n e a r I y the
same
as the
increas i n g
temperature
temperature
f o u r — spin
14). The
the RG
curve
approaches
bu t
RG
[18].
our L=3 RG result
chain result.
X
field
result was also c o m p a r e d w i t h the finite chain data
We have
the
antiferromagnetic
field
same
is
W
is
ith
the N
extrapolation much more rapidly.
For
those
of
the Ising case,
Katsura
susceptibility
susceptibility,
we have compared our results with
[14, 5 9 ] (see
is of course
i.e.
for the
Fig.
23).
Our parallel
exact. For the p e r p e n d i c u l a r
transverse
Ising
chain,
our
81
result has
the right c h a r a c t e r i s t i c
temperature
is
somewhat
larger
features,
than
but at low
Katsura's
exact
expression.
For
the XY-chain case,
direction
is
calculation.
the correct
good
for
Katsura's
is
the
compared
The
RG
in
k g T / J ) 1.0.
susceptibility
exact results
are
has
chain
in
it
perpendicular and parallel
vs field
been
bur
closely
calculated
approximates
shown
susceptibilities.
shown
in Fig. 24
for
which
in
In the case
( %z
Fig.
25
and
an d
26.
For the
alternating
case
the
maxima
for
there appear as well
or branches,
is evident
shif t e d
from
cases.
Therefore
the gap that
have
a
temperature
we
the
in both responses,
to
should
those
higher
field
of %% • ^ double
exist
susceptibilities
single
see
of the
and
a number of narrow but distinct peaks,
as c o m p a r e d w i t h
arises
should
# x are
of
^x ^* ^iave
the d o u b l e — pe a k e d str u c t u r e
but
no
the susceptibilities
a l t e r n a t i n g XY chain,
iso t r o p i c
is
we must distinguish between
and X - d i r e c t ions,
as
from
comparison.
and alternating XY — chains,
in the Z-
deviation
X-direction,
for
exact
the agreement
result.Also
the
available
Katsura's
substantial
fact
For the anisotropic case,
the uniform
with
temperature,
In
6— spin finite
result for field in the Z-
Fig. 24
result
curve at low
our
peak
that
only.
only
In
a single
only
in alternating
of u n i f o r m
Fig.
peak
27,
peak
chains
at higher
appears.
We
82
conclude
from
that
the
the effect
similar
peaks
of finite
at low
clusters.
temperature
We note
serious
apparently
problem
at
problem
the
for
spurious
the
higher
/^x shows
that
branches
calculation
temperature
of
constitute
at low
tem p e r a t u r e .
kB x=0.08
or
0.4,
functions
of- field.
We
seq u e n c e s
are
except
area,
close
nearly
points
fixed
at
switching
RG.
aech
have
will,
step
that
the
br a n c h
in
space
under
-that
region
The
found
in parameter
points
singular
decimation
render
stable
together
separated
mad e
the
we have checked the iteration sequence for
the free energy calculation and the lowest energy
very
a
still p e r s i s t s to some degree. To shed some light
on this problem,
branch
arise
behavior.
These
Even
small
of
the
flow
approximations
in
the
the
are
branch
area
is
of
a
the
commutators)
the
effect
the system from one RG trajectory, to another,
the results
the
to widely
diagram
have
In
initially
move
(neglected
iteration
iteration
areas.
which
iteration^
is,
the
levels as
of
and
unstable.
If the s i n g u l a r area is small, we may s m o o t h the data
by
fitting
polynomial
0.08
and
a
of low
0.4,
each groups
moderate
we
of 10
number
degree.
show
that this procedure reduces
adjacent
points
to
a
For the t e m p e r a t u r e s kgj= o .0 5,
in Fig.
adjacent
of
27
points
the
the
results
with a cubic.
sizes
of
fitting
We can see
of the. branches,
but
83
cannot remove
The
been
them,
lo w e s t
displayed
energy
as
levels
functions
of
of u n i f o r m
field
W i t h the field along the Z axis (Fig.
linear
point
functions
occurs
correspond
roughly
(see Fig. 29),
second
h.
The
first
at H/ J = O.61 and
susceptibility
first
of
to
the
picture.
the
ground state
at H/J=2.5,
levels
the
first
When
in Fig.
28
have
and
2 8), the levels
ground
state
and
last
maxima
nonlinear functions
the
are
crossing
in
peaks
to higher
the
the X-axis
of h. The
point appears H / J=0.97 and
displacing
29.
second at H / J = I .61. These
the field is along
are
crossing
XY-chain
field.
the
84
0.24
O. 12
Fig. 2 3 .
Susceptibility versus
temperatu refer
antiferromagnetic Ising chain: the parallel susceptibility
l i n e s by L= 3 RG and by K a t s u r a
[14] are s a m e ,
the
perpendicular susceptibility lines by L=3 RG and by Katsura
are different.
85
0.09
KbTZJ
Fig. 24.
S u s c e p t i b i l i t y versus t e m p e r a t u r e for XY chain.
S o l i d line:
line by L= 3 RG in x d i r e c t i o n f i e l d ,
line by L=3 RG in z direction field, dashed line: Kat sura's
perpendicular susceptibility line [14]
a
=
i.
X ,
Fig. 25. Susceptibility vs. magnetic field for uniform XY chain
at temperature k g T = 0 .05. ^fx is susceptibility in x direction
and ^ ^ is that in the z direction, both by L = 3 R G .
87
Fig. 26 . S u s c e p t i b i l i t y v e r s u s m a g n e t i c f i e l d f o r
alternating
XY c h a i n a - 0 .3 at t e m p e r a t u r e k y T / J = 0 . 0 5
X x is susceptibility in x direction, and X z Is that in z
direction, both by L = 3 R G .
88
o.os
~D
X
0
1
2
3
H / J
Fig. 27.
S u s c e p t i b i l i t y versus x - d i r e c t i o n field for
uniform XY-chain at different temperature using polynomial
fit to each 10 adjacent data points.
OO
VO
F i g . 28. Eight
of four spins
lowest energy levels of uniform XY cluster
as a function of field in the z direction.
O
o
Fig. 29.
spins as
Eight lowest energy levels of uniform XY cluster
a function of field in the x direction.
of 4
91
CHAPTER
3
DISCUSSION AMD S U M A R T
We have extended
the quantum
exp{ A1 -Hb I j = Tr'
to
variety
magnetic
point
of
and
three
exp { -H], }
chains,
thermal
of view
carrying
spin
decimation R G :
in
response
types
of
(3-1)
order
to
functions.
situations
c a lculate
From
both
a technical
are encountered
in
out the RG method:
(1) For
simple
cases,
the exact
eigenfunctions
and
e i g e n v a l u e s of both Hj,' and Hfo m a y be found analy t i c a l l y ,
for
instance
in
longitudinal
the
case
field,
of
the
using
a
Heisenberg
3-spin
chain
cluster.
in
The
renormalization transformations may then be found in closed
form.
(2) H jj' may
can be written
cluster
and
have
eigenfunctions
in closed
6-spin
ring
form
(as
cases),
and eigenvalues which
in
but
the
general
the eigenfunctions
eigenvalues of Hfo must be found numerically.
procedure
(3)
cannot
be
f re e-end
The
and
decimation
is carried out by computer.
If
the
found
eigenfunctions
analytically,
ar b i t r a r y
field),
and
eigenvalues
(for example,
we
mus t
of
Hfo'
the Heisenberg
chain
with
choose
a suitable
basis
in w h i c h to r e p r e s e n t H^,. The r.h.s of Eq.(3-1) must
92
then be
expressed
and the
f orm
suit
case,
with
unknowns.
out. We
The
the
same
as that
of
the l.h.s,
can be performed.
sides
of Eg.(3-1)
number
decimation
general
The main
of
must
be
independent
(Eq.(3— I))
can
c l u sters
are
the
increasing
in this
between operators
ignored.
When
approximation
accuracy
at
low
we
approach
as
carried
referred
better,
temperature.
is that non­
to neighboring
the
size
in the
We have
of spatial
in which
comparison with
finite
of
sense
a
of
performed
scale
factors
L= 3 , 5 and 7) and have noted a clear i m p r o v e m e n t
are
the
terms
then be
increase
becomes
RG calculations using a sequence
cases
of
situations.
approximation
commutators
cluster
both
form
can see that the decimation QRG is flexible and can
very
zero
same
decimation procedure
In any
same
in the
(
in those
chain extrapolations
possible.
For the ring d e c i m a t i o n case,
approximation
is applied,
after
a b o n d - m o v i n g type of
which
all
commutators
are
a c c o u n t e d for. C o m p a r i n g the results o b t a i n e d by N= 6 spin
ring
(same
we
decimation and those
number
find
Fig.
the
ring
also
the
gives
decimation
for the temperature region L g T / D O . S
calculations
9 and 10). For
decimation
those
of spins)
found by L= 5 free — end
to
be
alternating
better
chain
critical
found by free-end decimation.
more
a c c urate
case,
f ield
(see
the ring
values
than
93
The free energy
procedure.
We have
(blocking
methods)
The
second
a
used
uniform
iteration.
two different
Its
into
an
iteration
is d e s i g n e d
interactions,
alternating
principal
decimation
techniques
thermodynamic properties.
(Eq.(2— 33))
alternating
chain
iterating the
to c a l culate
approach
for chains with
is found by
a d vantage
since
one
is
specifically
it converts
by
the
redu c e d
second
computer
time.
As
do
all
real-space
renormalization
the decimation QRG attempts to extract
infinite
this
sy s t e m
respect
it
extrapolation
approach
technique
from
is
of
those
of
analogous
the
to date,
process
c a l c ulations.
reasonably
large
its results
as a standard
Nevertheless,
(~1 0-12
it can be
provided
spins)
for
of an
clusters.
is c o n s i d e r e d to be the mos t accurate
available
methods,
the properties
finite-sized
to
finite-chain
group
In
of
direct
The
latter
numerical
it can be extended to
chains,
comparison
size-limited,
and we have
when
such
as
used
appropriate.
in the cases
in w h i c h the field d i r e c t i o n
is not longi t u d i n a l . In such
instances
is particularly useful,
the
in comparing
chain
decimation QRG
results
and L= 3 R G ) we
accurate.
The
account
the
eigenfunctions
for
see
the
same
that the QRG
si m p l e finite
cluster
but
(say 4-spin
in general
chain c a l c u l a t i o n
eigenvalues,
as well.
size
the
QRG
since
is more
takes
into
uses
th e
94
The
other main advantage of the decimation QRG method
is that its accuracy improves
in the presence
field.
QRG
Thus
the
decimation
calculating magnetic properties
The
decimation
QRG
is
a logical
subject
limitations and technical problems,
to
a
for
number
of
which we discuss next.
the scale, factor L i m p roves
the accuracy. At low t e m p e r a t u r e , h o w e v e r ,
of data
choice
in nonzero field.
is
In g e n e r a l , i ncreasing
of a magnetic
(L= 3, 5 and 7) are not
the thr.ee sets
s u f f i c i e n t even for a rough
extrapolation.
A more
numerical
appear
serious
derivatives
to
be
difficulty
is
the
fact
of the free energy at.low
unstable
in
certain
temperature
circumstances
these regimes
the results depend strongly on the successive
this
behavior
trajectories
'closely
has
near
iteration,
estimate,
effect
At
is evaluated.
to the
points
fact
in
so that any small
on
the
exact
although we
shifts
due
(the neglect
whose
temperature.
energy
such
effect
approximation
net
as being
packed',
a large
sy s t e m
the free
each
from
results.
of certain
size
know
at high
and
that
change
The
then,
the
involves
are
an
at each-
difficult
the lower
to
the
renormalized
one RG t r a j e c t o r y to another,
can be very irregular at low
RG
in p a r a m e t e r s
QRG
is
true
space
commutators)
effect
In
interpret
the
parameter
that it is larger
iteration,
We
field.
—
in the
at which
chain
the
particularly
points
alternating
that
temperature.
and the
95
Since
fact
tha t
these
the
approximation,
difficulties
decimation
we
next
appear
QRG
discuss
is
to be
a
suitable
for
the
zero
the possibility
of
applying
The block QRG
temperature
case,
since
p r e s e r v e s the ground and the l o w e s t e x c i t e d states
system. However
extending
problem.
simplest quantum
is
it
of the
the block QRG to high field cases
other than the Ising chain in t r a n s v e r s e field
difficult
to the
high— temperature
a block QRG (see Chapter I) at high field.
more
related
The
transverse
model and is the
Ising
only
is still a
chain
spin
chain
is
the
that has
b een t r e a t e d by the b l o c k QRG in a m a g n e t i c field. This is
beca u s e at any field s t r e n g t h the lo w e s t en e r g y levels
two-spin
cluster
spin (see Fig.
of
the
energy
uses
the
will
Heisenberg
level
levels
each
structure
of
even and
cross
to the
energy
30).On the other hand,
anisotropic
has an energy
the
can map
a single
odd
say).
Therefore
to build
a suitable
On the contrary,
the
as
the
(see Fig.
field
the
is
the energy levels
mapping
a single
longitudinal
is difficult
spin b l o c k QRG,
other
Fig. 32,
spin
of
the t w o - s p i n cluster
chain , in
which
levels
of
field
to map
to
31).
If one
en e r g y
levels
increased
(see
are too complex
relation for all
field ranges.
d e c i m a t i o n QRG can eas i l y
treat
the
f ield cases, but it does not w o r k wel l at low t e m p e r a t u r e
since
the
it does not preserve,
system
[42].
in general,
the ground state
of
96
E
E
truncation
Fig. 30.
Schematic energy levels of transverse Ising model
and its b l o c k QRG, (a) for 2- s p i n cluster, (b) for one
spin.
Fig. 31.
Schematic energy levels of 2-spin cluster for the
a n i s o t r o p i c H e i s e n b e r g spin chain in the l o n g i t u d i n a l
field: I. for the i s otropic H e i s e n b e r g spin chain, 2 . for
the XY-Iike spin chain, 3. for the Ising-Iike spin chain.
(/> w Ti
lO t e t M - H - OQ
B e> '
to
O
B
w
B
to
*
CO
O
M-CJ-W
to
O
1/1
E
£■
* -Cl fcr
H-
O
9 B
to
H*
5 =2
9 O
- p
«
W OQ
M>
O Hi
M n>
<
o>
I H"d M
H - HB M
. B
Vi
(o)
O
Hh «■*
O M
H O
►d
4X H -
IO
(Z))
REFERENCES
99
REFERENCES
[ 1]
M . Steiner, J.Villain, and C.G.Windsor,
A d v . in Physics, 25., 87 (197 6) .
[ 2]
R. J.B ir g e ne au and
(1978),
[ 3]
J . C.Bonner,
[ 4]
J.C.Bonner, H.W.J.Blote, H.Beck, and G. M n I I e r ,Phy^ j.jc s^
AS SSS B i JSSSSiSS» Eds. J.Bernasconi and T.Schneider,
S p r i n g e r Series in S o l i d - s t a t e S c i e n c e s 23, 115
(Springer-Ver lag Berlin Heidelberg New York 1981).
[ 5]
E.H.Lieb and D.C.Mattis, .MsiilESiiSSi B B Z S i S S IS One
Dimation (Academic Press, New York 19 6 6) .
[ 6]
E.Ising,
[ 7]
L.Onsager,
[ 8]
M.Wortis, P h y s . Rev. 1.3.2 , 85
[ 9]
N.F u k u d a and
1675 (1963).
[10]
H.Bethe,
[1 1 ]
L.Hul then. Ark. Mat. Astron. Fys. 2 6 A, no.Il^ (1 93 8).
Here the exchange H a m i l t o n i a n used by Hul then is
-O t T- T- -l -i • T h e r e f o r e the J is twice as that in
G.Shirane,
Phy s. Today
J. A p p l . P h y s . 49(3),
Z. Phy s. 31 , 2 53
Phy s. Rev.
1299
1^2 , 32
(1978).
(1 925).
6.^* 117 (1944).
M.Wortis,
Z. P h y s . 71, 205
(1963) .
P h y s 1 Chem.
Solids,
24,
(1931).
BqrMyV
[12]
J.des C l o i z e a u x and J.J.Pearson, Phy s.R e v . 128, 2131
(1 962) (see [11]).
[13]
E.Lieb, T.Schultz,
(1961) .
[14]
S.Kat sura, Phy s . Rev . 1.2 7 , 1508
[15]
P.Pfeuty,
[16]
C.N 1Yang and C.P.Yang, Phy s. Rev. 14 7 , 3 03 (1 96 6),
ibid. 150 , 3 21 , 3 27 (1 966), ibid. 151 , 25 8 (1 966).
[17]
J.D.Johnson
(1980).
and D.Mattis-, Ann. Phy s. 1.6, 4 0 7
Ann. P h y s . 5.7 , 79
and
J.C.Bonner,
(1962).
(1970).
Phy s.Rev.Lett.
44,
616
100
[18]
J.C.Bonner
(1964).
[19]
R.B.Griffiths,
[2 0 ]
H.W.J.Blote,
Physica
93B , 93 (1978).
[21]
H .W.J.Blote,
Phy sica
7 9 B , 427
and
M.E.Fisher,
Phys.
Rev.
135,
A640
Phys . Rev. 135, A65 9 (1964) .
(1975 ) .
[2 2 ] J . J . S m i t , L. J .d e Joiugh, D.K l e r k an d H . W . J . B l o t e ,
C o l l o q u e s I n t e r n e s C N R S 242, P h-^_sj. C^uj? .S.0u.s C h_a m £ js
Ma^netiques
I s i e n q e q , 253 (1974).
[23]
K. M .D i e de r i x , H.W.J.Blote, J.P.Groen, T. 0. K I a a s s e n ,
and N.J.Poulis,
Phys. Rev. BI 9, 420 (1979).
[24]
K.G.Wilson
(1974).
[25]
Th.Niemeijer
and J.M.J.
van L ee uwe n,
Phaqq
I r a n s i t i q n q and C r i i i q q i P h e n o m e n a , Eds. C.Domb and
M.S.Green (Academic Press. London) V o l .6 , 427-50 5
(1976).
[26]
D.R.Nelson
(1973).
[27]
H. J.Hil hors t , M . Schick and J.M.J. van L e e u w e n ,
Rev. Lett. 40 , 1605 (1 97 8 ).
[28]
S .M a , M o d e r n T h e o r y of C r i i i q q I
(W.A.Benj am in New York 1976).
[29]
D.C.Mat tis, The Theory of Maqnetj s m 1 , 19
verlag Berlin Heidelberg New York 1981).
(Springer-
[30]
S.D.Drell, M.Weinstein
D14 , 4 87 (1 97 6).
Phys.
[31]
P.Pfeuty,
R.J u l l i e n an d K. A.Pen son,
R i S i z Sqqqq
5.®£.2ZffiJL1 1
£ > Eds. T.W.B u r k h a r d t and J.M.J. van
Leeuwen, 119 (Springer-verl ag Berlin Heidelberg New
Y o r k 19 82).
[32]
R.Jul lien, J.N.Fields
4 889 (1 977).
[33]
S-T.Chui
[34]
J.E.Hirsch and J.V.Jose,
and
and
J.Kogut,
Phys.
M.E.Fisher,
Reports,
12q,
No.2
A. I.P. C on f .P r o c i i<B , 888
Phenomena,
and S.Yankielowicz,
and S.Doniach,
Phys.
Phys.
Rev.
Rev. B 6 ,
and J.W.Bray, Phys. Rev. B18, 2426 (1978).
J.Phys.
C13,
L53
(1980).
101
[35]
A.J.Berl inski
(1981) .
and
C.Kal lin,
Phy s.
Rev.
B23 ,
2247
[36]
R.Jtill-ieri, P.Plenty, J.N.Fields and S.Doniach,
Rev. BI 8 , 3568 (1978).
[37]
J.N.Fields,
[38]
J.N.Fields,
H.W.J.B lote
P h y s. 50, 1808 (1979).
J.C.Bonner,
J.
Appl.
[39]
H.W.J.Blote,
J.C.Bonner and J.N.Fields,
Magn. Mater. I..Sn1.(5, 405 (1980).
J.
Magn.
[40]
M . Suzuki and H.Takano,
[41]
H.Takano and M.Suzuki,
[42]
C .C a steI I a n i , C.DI Castro
Phys. B200, 45 (19 82).
[43]
M.B a r m a, D.K u m a r
(1979).
and R.B.Pandey,
[44]
I .D z y aloshinsky,
I.Phys.Chem.Solids,
[45]
T.Moriya,
[46]
J.C.Scott, C.B.Zimm
53 , 2768 (1 982).
[47]
P.L.Seifert and J .E. D rum he I Ier , P h y s . Rev.
(1978).
[48]
D.D.Betts, Phas^ Transitions, and C r i M c a J Phenomena,
Eds. C.Domb and M.S.Green, (Academic Press. L o n d o n
1 974), Vo I .3 , Chap. 6 , 57 9 , 5 81 .
[49]
J.C.Bonner, S.A.Friedberg, H.Kobayashi, D.L.Meier
H.W.J.Blote,
Phys. Rev. B2 7 , 248 (1 983 ).
[50]
M.M o h a n and
(1982) .
[51]
W . Duffy and K.P.Barr,
[52]
J.C.Bonner
(19 82 ).
Phy s.
P h y s. R ev . BI 9 , '2637 (1 979).
Phys. Rev.
P h y s. Lett. 6 9 A, 426 (1979).
J. Stat. Phys.
120,
and
J.C.Bonner,
and
and
2(J, 635
and J.Ranninger,
J. Phys.
(1981).
Nuclear
C 1^2, L909
4, 241
(1958).
91 (1960).
A.Sar hang i,
J.
J.Appl.Phys.
Phys.
H.W.J.Blote,
Appl.
Phys.
B18 , 3 703
5.3.X.11.2.,
Rev. 1.6i, 647
Rev.
Phys.
and
8035
(1968).
JB2 5., 6 95 9
102
[53]
The Hamiltonian in [49]
W
«1<
is
S21- ^ 2 ^ o T 21T 2 i+i)
The y found J i /k g = - 2.58K.C om p a ring this H a m i l t o n i a n
w i t h ours E q( 2 - 2 6) we find Kj = / (2 kuT), e.g.
K 1 T = - I .2 9 K .
'
[54]
T. W.Bur k h a r dt , .Re a_l - Sg a c e R e n o r m a l i z a t i o n , Eds .
T.W.B u r k h a r d t and J.M.J. van L e e u w e n , 33 ( S pringerVerlag Berlin Heidelberg New York 1982).
[55]
J.D.Johnson
(1980),;
[56]
J.des C l o i z e a u x
(1966).
[57]
In order to compare our results with Bonner's results
we should change the unit as follows:
The Hamiltonian that Bonner used is
and
J.C.Bonner,
and
Phys.
Rev.
B 2^2 , 251
M .G an d in , J.Math.Phys.
%,
1384
and the Hamiltonian used by us is
here S = (1/2)e. Comparing
this, we got
JB = 2J, X e / (g2|i2) =(1/4)X,
g pHB /JB = H / J ,
here X is susceptibility and the quantities
sub B
are used by Bonner;
[58]
that have
We use the linear extrapolation method that was
used by B o n n e r and B lote [52] to e s t i m a t e the L = <»
lower critical field values. We assume that the he I
is smooth function of L :
he I= ao+ai/L+a2^k^*
Thus
a 0 ls th e estimated L=00 lower critical
feild.
103
[59]
In order to compare our results with Katsura's
results, we should change the unit as follows:
The Hamiltonian that Katsura used is
^K
= -(1/2)
Compared with the Hamiltonian that Bonner used
we got
1 K=J b ,
2m=gp
and
where the quantities
Kat s u r a .
[57],
J K X B / (4Nm2 >=J b X B / [ (g P )2 n ^ >
that have
sub K are used by
J
MONTANA STATE UNIVERSITY LIBRARIES
3 1762 10005433 5
D378
Ck2
cop.2