Renormalization-Group Studies of Disordered
Magnetic Systems, Strongly Correlated Electronic
Systems, and Polymeric Systems
by
Gabriele Migliorini
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
MASSACHUSETTS INSTITUTE
Doctor in Philosophy
MAL23u
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOG
LIBRARIES
May 1999
@ Massachusetts Institute o
99
echnology 1999. All rights reserved.
Author ........
Department of Physics
March 18, 1999
Certified by.
A. Nihat Berker
Professor of Physics
Thesis Supervisor
Accepted by........
Prof. Patrick Lee
Chairman, Department Comnitee on Graduate Students
Renormalization-Group Studies of Disordered Magnetic
Systems, Strongly Correlated Electronic Systems, and
Polymeric Systems
by
Gabriele Migliorini
Submitted to the Department of Physics
on March 18, 1999, in partial fulfillment of the
requirements for the degree of
Doctor in Philosophy
Abstract
In the first part of this thesis, the properties of physical systems in the presence of
quenched disorder are studied. The phase diagrams and the statistical mechanics of
classical Ising systems that go undergo phase transitions are obtained for the case of a
random external field and a random exchange (spin-glass) interaction between spins.
Uniting phase diagrams, for the general case of a system that include both spin-glass
exchange interaction and the external fields, are obtained via renormalization-group
theory in spatial dimensions d = 2 and d = 3. The strong violation of critical phenomena universality, previously found at random-bond tricriticality in d = 3 is now
seen in the case of the spin-glass in d = 2. Renormalization-group theory is again
applied in the context of electronic conduction models. Specifically, we constructed
the renormalization-group recursion relations for the Hubbard model, and the d = 2
and d = 3 phase diagrams are evaluated, at arbitrary filling. In particular, the problem of phase separation, a crucial concept to understand the conducting properties of
the Hubbard model, are discussed in d = 2. The results are in very good agreement
with numerical results found in the literature. In the rest of the thesis we consider
the collapse transition for a polymer in d = 2, via renormalization-group theory. The
presence of randomness and the properties of directed polymers in random media are
also discussed. The problem of random self-interacting heteropolymers is considered
in chapter 5, and new results are obtained in this context.
Thesis Supervisor: A. Nihat Berker
Title: Professor of Physics
2
Acknowledgments
I would like to thank my thesis supervisor A. Nihat Berker, who introduced me to
Renormalization-group theory and to the physics of quantum electronic systems. I
also wish to thank him for making it clear that research and teaching are closely
related matters, a lesson I will keep in mind in the future. I wish to thank Prof.
Mehran Kardar, not only for the example he gives by his own figure and character,
but also for fashinating discussions in the context of polymers in random media and
heteropolymers in random self-interaction. I thank Prof. Yavuz Nutku and the Feza
Giirsey Research Institute for the kind hospitality I received. I wish to thank also
Prof. John D. Joannopoulos and Prof. Patrick Lee for their teaching ability and
kindness during these years. Many thanks also to my parents and my sister Matilde,
and also Sohrab, Gabor, Alkan and Dicle for all the nice moments and all the good
advises they gave me during my years in Boston. This research was supported by
the Italian Istituto Nazionale di Fisica Nucleare (INFN), the Scientific and Technical
Research Council of Turkey (TUBITAK), the US Department of Energy under Grant
No. DE-FG02-92ER45473, and the US National Science Foundation Grant No. DMR94-00334.
3
Contents
1
2
Introduction: Phase Transitions and Critical Phenomena
8
1.1
Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Microscopic Theory and Thermodynamics of Phase Transitions . . . .
11
1.3
Critical Exponents
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.4
Renormalization-Group Theory . . . . . . . . . . . . . . . . . . . . .
13
The Random-Field Ising Spin-Glass
18
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Global Random-Field Spin-Glass Phase
Diagrams in Two and Three Dimensions . . . . . . . . . . . . . . . .
2.3
Renormalization-Group Mapping of the Quenched
Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . .
3
20
28
Phase Transitions in Electronic Conduction Models
42
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.2
Finite-Temperature Phase Diagram
of the Hubbard Model in d = 3
. . . . . . . . . . . . . . . . . . . . .
47
3.3
The Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.4
Phase Separation in Two Dimensions . . . . . . . . . . . . . . . . . .
62
4 Polymers in Two Dimensions
4.1
70
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
70
4.2
Collapsing Transition of a Polymer in Two Dimensions:
Grand-Canonical Renormalization-Group Theory
. . . . . . . . . . .
..
. .
77
. . .
4.3
Conclusion . . . . . . . . . . . . . . . . . . . . . .. .
4.4
Appendix: Polynomials in the Recursion Relations . . . . . . . . . . .
77
86
5 Heteropolymers
6
71
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.2
Mean-Field Theory of Heteropolymers
. . . . . . . . . . . . . . . . .
88
5.3
The Bose Formulation of Heteropolymer Chains . . . . . . . . . . . .
96
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
100
Conclusions and Future Prospects
5
103
List of Figures
2-1
Global random-field spin-glass phase diagram in d = 3. . . . . . . . .
2-2
Constant p cross-sections of the ferromagnetic phase boundary in d
2-3
The zero random-field cross-section of the phase diagram in d = 3..
2-4
The zero temperature cross-section of the phase diagram in d
2-5
Fixed distributions of the quenched probability in d = 3. . . . . . . .
35
2-6
Spin-glass phase diagram in d = 2.
. . . . . . . . . . . . . . . . . . .
36
2-7
Fixed distributions of the quenched probability in d = 2. . . . . . . .
37
2-8
Asymmetric spin-glass phase diagram in d = 2 .. . . . . . . . . . . . .
38
3-1
Calculated phase diagram of the d
3-2
Calculated phase diagram of the d
3-3
Calculated phase diagram of the d
4-1
412 local polymer configurations contributing to ii
. . . .
79
4-2
439 local polymer configurations contributing to v'.
. . . .
80
4-3
534 local polymer configurations contributing to i-v.
. . . .
81
4-4
Calculated phase diagram for the lattice polymer.
. . . .
82
4-5
Calculated bending density of successive segments.
. . . .
83
4-6
Calculated bending ratio of successive segments. .
. . . .
84
6
=
=
=
3.
31
3. 32
33
34
3 Hubbard model for U/t = 20.
64
3 Hubbard model for U/t
65
=
4.44.
3 Hubbard model for U/t = 0.8.
66
List of Tables
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
. . . . . . . . . . . . .
77
The polynomials for ii, v', i. .. . . . . . . . . . . . . . . . . . . . . . .
78
1.1
Critical exponents.
4.1
Polymer configurations and critical exponents
4.2
7
Chapter 1
Introduction: Phase Transitions
and Critical Phenomena
A physical system is said to go through a phase transition if, changing some control
parameters (e.g., temperature T, pressure p, or magnetic field h), it goes from one
phase to another [1].
Let us consider a physical system that goes through a phase transition when
temperature is changed: for T < T, the system is in phase A and in phase B for
T > T,. We can distinguish between two different cases: a) in order to go from phase
A to phase B the system requires a certain amount of energy or latent heat. This
is manifestly the case when observing a cube of ice into a glass. About 80 calories
are required indeed to melt one gram of ice; the internal energy U of the system
is a discontinuous function of temperature and the phase transition is a first order phase transition; b) if instead the internal energy U is a continuous function of
temperature, so that the latent heat is zero, the phase transition is not of first-order
nature. Typical examples of second - order phase transitions are the ferromagnetparamagnet transition and the superconductor-conductor transitions.
If the transition is governed by another parameter than temperature, consider the
free energy F = U - TS, where U is the internal energy and S is the entropy of the
system. In this case the derivative of the free energy F with respect to the parameter
governing the transition would play the same role as the internal energy U in the
8
previous case [2].
The same system, as stated above, might be in different phases, and the number
of parameters governing the transition is, in many cases, larger than one. It is always
convenient to describe the mutual structure of the different phases of the system, for
the possible values of the parameters, in terms of a diagram, i.e., the phase diagram.
As an example one might represent the temperature on the abscissa of the diagram
and one or more control parameters on the other axes. For each phase where the
system might physically be, the corresponding region for the control parameters are
indeed represented by the phase diagram. For example, the behavior of a conductor
entering the superconducting phase of the II type can be understood in the lowtemperature region, in the temperature-magnetic field phase diagram. Three different
phases are present and bordered by second-order phase transition lines. Similarly, the
phase diagram of water, in the pressure-temperature plane represents the liquid to
gas transition. The transition line terminates at the critical point; it is possible to go
from the liquid to the gas phase without going through a phase transition, considering
the system at values of pressure and temperature somewhat larger than the critical
values (e.g., 200 atm and 400'C for water).
The phase diagram of a typical ferromagnet is also an important example. If temperature is changed and the external magnetic field is zero, a second-order transition
is observed at the critical temperature value Tc. In the region of temperature below
the critical value, when the external field changes its sign, a first-order transition
is seen and the magnetization changes from positive to negatives values. As in the
previous case, the phase transition line terminates in a second-order phase transition.
1.1
Phase Transitions
The main feature of first-order phase transitions is the presence of metastable states.
Typical example of a metastable state is a mixture of oxygen (02) and hydrogen (H 2 )
at room temperature. This mixture is a dangerous one since a little perturbing spark
would produce an explosion, to generate water (H 2 0). Similarly, eventhough the con-
9
sequences are somewhat safer, cooling down a very clean sample of water, the liquid
state is seen to resist down to temperatures as low as -25"C. The presence of any
impurities or "little icebergs" in the sample would immediately start the crystallization process. The inverse metastable state of ice resisting at temperatures just above
the 00C temperature is harder to see in experiments. The presence of a little layer of
liquid around the sample makes the experiment harder, eventhough superheated ice
has been recently observed, using a laser heating technique, at few degrees Celsius
above the zero. The perturbation that makes a metastable state into a stable one
might also be produced by thermal fluctuations of the system itself. In the case of
supercooled water it is essential, on the one side, to deal with very clean samples,
since any single impurity would start the crystallization process. On the other side,
even in absence of impurities in the sample, crystallization would start after some
time, due to internal fluctuations of the system, at the molecular level. The mean
life of a metastable state is in many cases very long, as in the very interesting case of
real glasses. The radius of the ice crystal able to move the entire system towards the
stable frozen state is, in the vicinity of the critical point, proportional to (T - Tc)-and the mean life of a metastable state is proportional to
exp{-C(T - Tc)},
(1.1)
where C is a positive constant.
Metastable phenomena are also common in magnetic systems. In a real system
the presence of long-ranged dipolar forces tends to reduce the global magnetization
of the system; vice versa, the local ordering of parallel spins tends to increase the
global magnetization.
This results in the formation of magnetic domains, of the
order of micrometers pm, separated by thin regions of the order of 100A in size,
where magnetization does not have a definite sign. Inside each domain the spins are
parallel to each other, while domains are oriented in different directions.
10
1.2
Microscopic Theory and Thermodynamics of
Phase Transitions
For a simple Ising system, neglecting fluctuations around the critical point, it is found
that the free energy of N spins is given by
F(p) = U(p) - TS(p),
(1.2)
U(p) = N(-hp - -Jp2),
(1.3)
2
S~) N -k2
2
)k2
2
where the density of magnetization m is proportional to the order parameter p, T
is temperature, h is the magnetic field, and J is the exchange interaction between
spins. The free energy per spin, (f = F(p)/N) can be differentiated with respect to
the order parameter,
Of
-=
0
Op
(1.5)
so that one obtains
p = tanh(oJp + 3h),
(1.6)
where # = 1/kBT and kB is the Boltzmann constant. In a similar way, a simple
microscopic description of the liquid-gas transition can be obtained, neglecting fluctuations around the critical point. In this case the density p plays the same role as
the order parameter p and the order parameter satisfies, in this case
(P + p 2 )(P- 1 - b) = RT,
(1.7)
where P is pressure, T is the temperature and a,b are constants that depend on the
material one considers. The critical properties of this system can be characterized
quantitatively. In the case of first-order phase transitions, the order parameter has a
11
discontinuity at the transition. This leads to observables, such as the latent heat, that
quantify the magnitude of this jump. Differently, for second-order phase transitions,
the order parameter is continuous, while its first derivatives, corresponding to second
derivatives of the free energy, are discontinuous.
1.3
Critical Exponents
The properties of a second-order phase transition can be characterized by introducing
the critical exponents. Close to a critical point (second-order phase transition) , e.g.,
the one seen in the case of a simple Ising magnet, the magnetization m is continuous across the transition. Below the critical temperature, but very close to it, the
magnetization grows as
m (x (T, - T)",
(1.8)
where 3, not an integer, is one of a large class of quantities, known collectively as
the critical exponents, that fully characterize the system. These quantities, along
with the phase diagram, can be used to compare the theoretical predictions with the
experimental results. Another important critical exponent is the one associated with
the specific heat c, which is singular in the neighborhood of the critical temperature
T, as
c cx IT - Tc-'.
(1.9)
The other two major critical exponents are related to the two-point correlation function G2(r). In the case of spin models, this quantity is defined as
G 2 (r) = ( (O) - (r)) - ( (0)) - ((r)).
(1.10)
When r is large and T = Tc, the asymptotic form of G2 (r) is given by
G 2 (r) cc
12
1
(1.11)
Exponent
Definition
Quantity in fluid (magnetic) systems
a
C ~ti-o
Specific heat at const. volume (magnetic field)
3
PL -PG
7
x
v
(M)
~ (-t),
Density difference (zero-field magnetization)*
~ It|-7
Isothermal compressibility (susceptibility)
~tI-"
Correlation length
r7
F (r)
6
M ~ H 1 /5
~ Ir-(d-2+)
Pair correlation function (t=O)
Critical isotherm (t=O)
*Valid only for T < T, by definition of order parameter.
Table 1.1: Critical exponents and related thermodynamic quantities; t = (T - Tc)/T.
Similarly, if r is large and 0 : IT - TcI/T << 1,
G 2 (r) c exp(-r/ ),
where
(1.12)
, known as the correlation length, diverges as,
cx IT - Tc-".
(1.13)
r1 and v are the exponents that describe the long-ranged correlation within the system.
The list of the exponents just given is not complete but we will return many time on
the concept of critical exponents along this thesis.
1.4
Renormalization-Group Theory
The main difficulty in dealing with second-order phase transitions might be explained
as follows. In the case of ferromagnetic systems, as explained above, large regions
where the spins are parallel are formed. A careful analysis shows that each region
can be decomposed in subregions with similar characteristics. The argument can
13
be iterated down to scales comparable with the lattice spacing and the radius of
the larger region ( ) diverges at the critical temperature. The number of degrees of
freedom tends also to infinity at the transition point making the description of the
system more difficult. Renormalization-group theory avoids this difficulty in a natural
way. If Ising spins are considered on a cubic lattice of spacing a, we can introduce a
second lattice of spacing L (L/a is an integer). The local order parameter, relative to
the scale L being PL, is defined on this new lattice and proportional to the sum of the
spins in a box of size L, where the proportionality constant is fixed by the condition
(P2)=1
(1.14)
For L = a, PL coincides with a single spin and in the limit of L becoming infinite, PL
corresponds to the usual global order parameter. In the high-temperature region, p',
is the sum of an infinite number of independent variables and its distribution will be
gaussian; indeed there is no spontaneous magnetization and spins at distances larger
are not correlated. In the low-temperature region p. is simply going to be 1
than
according to the sign of the spontaneous magnetization), since no fluctuation
is present around the mean value of the magnetization. If the probability distribution
(or -1
of the variables PL is, for large enough values of L, the same as that of the variables
P2L,
the system is said to be invariant under a scale transformation and, if PL is the
probability distribution for the variables PL,
P2L = R(PL).
(1-15)
where R is a non-linear operator. The transformation above is the scale transformation, while R is the operator of the renormalization-group transformation. Indeed
the R, operators obtained applying the R operator n times (P2nL = Rn(PL) ) form a
group, usually referred as renormalizationgroup [3]. The hypothesis of scale invariance translates in this language into the existence of the probability distribution of
the variables PL in the limit of L going to infinity, satisfying the non-linear equation
14
(1.16)
Po= R(Po).
In the next chapter of this thesis we consider renormalization-group theory to
study the properties of two models: the random-field Ising model and the EdwardsAnderson model for a spin-glass in a magnetic field. The random-field random-bond
model, that naturally connect these two models within renormalization-group theory
is also introduced. The Ising model including field randomness and +J randomness is
studied by renormalization-group theory in spatial dimensions d = 2 and d = 3. With
field randomness but no ±J randomness, in d = 3 ferromagnetic and paramagnetic
phases occur with no intervening spin-glass phase. Also in d = 3, at sufficient +J
randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for
any nonzero random field, also implying its disappearance for any nonzero uniform
field. In d = 2, no ferromagnetic phase under random fields and no spin-glass phase
occur. Global phase diagrams uniting the random-field and spin-glass problems are
evaluated. The strong violation of universality, previously found adjoining randombond tricriticality in d = 3 is seen here in d
=
2.
In the third chapter of the thesis we consider renormalization-group theory to
investigate the properties of the Hubbard model, the simplest model to describe
strongly interacting electronic systems. We obtained, we believe for the first time,
the finite-temperature phase diagram of the Hubbard model of electronic conduction
in spatial dimension d = 3. This phase diagram exhibits, around half filling, an
antiferromagnetic phase that is completely due to electron hopping and, between
30 - 40 percent electron or hole doping from half filling, a new
T
phase in which the
electron hopping strength t asymptotically becomes infinite under repeated rescalings;
electron hopping asymptotically vanishes under rescaling in all other regions of the
phase diagram. Next to the T phase, a first-order phase boundary with a very narrow
phase separation (2 percent jump in electron density) occurs. Depending on the
strength of the on-site Coulomb repulsion U, this narrow phase separation can be on
the low electron density side of the T phase (for strong U), directly abutting the
15
T
boundary (for intermediate U), or on the high density side of the T phase (for weak
U). At temperatures above the r phase, an incommensurate frozen spin density phase
occurs. In d = 2, we find that the Hubbard model has no phase separation (or other
phase transition) at finite temperature.
In the fourth chapter of the thesis the collapsing transition of a polymer is considered in two dimensions, by constructing a renormalization-group theory by an
exhaustive enumeration of polymeric paths on a lattice. The phase diagram of a
linear polymer is obtained by application of renormalization-group theory to a lattice model. The phase diagram is in the variables of bending energy, monomer-unit
chemical potential, and temperature. At low chemical potential and favorable bending energy, a collapsed phase occurs in which the end-to-end separation R grows as
the power v = 0.5 of the chain length N. At high chemical potential and unfavorable
bending energy, a rod phase occurs in which the long-range behavior is v
=
1, but the
local behavior changes in different regions, from thick rod (bending energy even higher
than in the collapsed phase) to thin rod (negligible bending density). A second-order
phase boundary, where v = 0.817, separates the two phases. In the fifth chapter of
the thesis we consider the problem of a random heteropolymer chain and the statistical properties of this system are studied. In particular, the independent interaction
model for a random heteropolymer chain is considered in detail. Conclusions are
given in the last chapter.
16
References
[1] E. Stanley, Introduction to Phase Transitions and Critical Phenomena,
(Oxford University Press, New York, 1971).
[2] K. Huang, Statistical Mechanics (Wiley, New York, 1963).
[3] K.G. Wilson, Phys. Rev. B 4, 3174, 3184 (1971).
17
Chapter 2
The Random-Field Ising
Spin-Glass
2.1
Introduction
The random-field Ising model and the spin-glass Edwards Anderson model, where
randomness is present in the exchange interaction between neighboring spins, are
two central problems in the physics of systems with quenched randomness. Indeed
the two problems are closely related and it has been recently suggested that in the
random-field problem a spin-glass phase might occur between the ferromagnetic and
paramagnetic phases.
Considering this two problems from the point of view of renormalization-group
theory naturally yields to consider a uniting random-field random-bond model. Under scale transformation it is seen in the present chapter that randomness in the fields
generates randomness in the bonds, and the close connection between the two original
problems becomes manifest in our approach. The effect of weak bond randomness in
the random-field model, and similarly, the effect of a random field in the EdwardsAnderson model for an Ising spin-glass are carefully studied, writing the recursion relations that govern the renormalization-group trajectories under scale transformation
for the system. A highly detailed renormalization-group study is done, with analysis of the renormalization-group flows of over than 10' quantities that determine the
18
quenched probability distribution. Both the zero random-field and the zero antiferromagnetic bond concentration cross sections of the global phase diagram are discussed
in the first part of the next section, and the intermediate uniting phase diagrams
that connect the two problems are discussed in detail afterwards. The unstable fixed
distributions attracting, in d = 3, the phase transition between the ferromagnetic
and the spin-glass phases, the multicritical distribution where all the three phases
meet, and the phase transition between the spin-glass and the paramagnetic phases
are obtained and discussed in detail. Finally, in d = 2 the strong violation of critical
phenomena universality, previously seen in the presence of random-bond tricriticality,
is observed and discussed.
19
2.2
Global Random-Field Spin-Glass Phase
Diagrams in Two and Three Dimensions
Gabriele Migliorini and A. Nihat Berker
Department of Physics, Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, U.S.A.
Abstract
An Ising model including field randomness and ±J bond
randomness is studied by renormalization-group theory in
spatial dimensions d = 2 and 3. In d = 3, with field randomness but no tJ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase.
Also in d = 3, at sufficient ±J randomness, a spin-glass
phase occurs, but is replaced by the paramagnetic phase for
any nonzero random field, also implying its disappearance
for any nonzero uniform field. In d = 2, no ferromagnetic
phase under random fields and no spin-glass phase occur.
Global phase diagrams uniting the random-field and spinglass problems are evaluated. The strong violation of critical
phenomena universality, previously found adjoining randombond tricriticality in d = 3, is seen here in d = 2.
PACS Numbers: 75.10.Nr, 05.70.Fh, 64.70.Pf, 81.05.Rm
Two central problems of the statistical mechanics of systems with quenched randomness have been (1) the survival of conventional (e.g., ferromagnetic) order under
the disruption of frozen random fields locally coupling to the order parameter [1] and
(2) the onset of unconventional (spin-glass) order in systems dominated by random
competing (frustrated) interactions [2]. These two problems are in fact connected.
Firstly, recent work [3] on the random-field problem has suggested that a spin-glass
phase may occur between the ferromagnetic and paramagnetic phases. Secondly, the
application of random fields on a spin-glass system can simply be shown (see below)
to be equivalent to the application of a uniform field; the survival of the spin-glass
phase under the latter condition has been discussed [4, 5].
We have addressed and connected the random-field and spin-glass problems in
a global renormalization-group study of a system that incorporates both types of
randomness, with Hamiltonian
Z=[Pij sisj + Hij(si + sj) + Ht(si
-
si)
,
(2.1)
<ii>
where si = ±1 at each site i of a hypercubic lattice, < ij > indicates summmation
over nearest-neighbor pairs of sites, with j > i along each hypercubic axis. The fields
Hij and H1. are independently and randomly +H or -H (i.e., with equal probability
of 0.5). The interactions Jij are +J or -J with respective probabilities of I-p and p,
the elementary squares of the lattice with odd numbers of each sign being frustrated
[6]. The limits p = 1 or H = 0 respectively correspond to the purely random-field or
purely spin-glass problems.
The renormalization-group study of a quenched random system is via the recursion
of the quenched probability distribution, [7, 8] given by
P1
[ii
IKlt
t(y3
diiP(ij
-
R({ Ki 1)),
(2.2)
where Kij = (Jjj, Hij, Hh ) represents the set of interactions at locality ij, the primes
refer to the renormalized system, R({Kij}) is a local recursion relation, and the prod-
21
uct index ij runs through the localities of the unrenormalized system that effectively
influence the renormalized interactions at the renormalized locality i'j'. Our basic
premise [8, 9, 10, 11] has been that the crux of the quenched randomness problem
lies in the convolution in Eq.(2.2), whereas the recursion relation is a smooth local
relation that can straigthforwardly be approximated. The smoothness, locality, and
approximability are related characteristics of recursion relations that underpin the
general success of the renormalization-group approach.
We used the Migdal-Kadanoff approximation [12, 13], due to its proven effectiveness and simplicity, yielding the recursion relations from the "decimation"
j,,,
=
ln[R(+1, +1)R(-1, -1)/R(+1, -1)R(-1, +1)]/4,
Hj,j,=
ln[R(+1, +1)/R(-1, -1)]/4,
H ~ilj,=
ln[R(+1, -1)/R(-1, +1)]/4,
R(si, 84)
=
(2.3)
where
exp[J 1 2 ss
2
+ J 2 3 s 2s 3 + J 34 3s
4
S2 83=±1
+H 12 (s1
+ s 2 ) + H 23 (s 2 + 83) + H 34 (s 3 + 84)
+t (81 - 82) + Ht3 (s2
-
83)
+ H34 (s 3 - 84)],
where the "bond-moved" interactions are
3 d-1
= E
SijJinin
(2.4)
n=1
and similarly for lij and Ht.
factor of b
=
These are recursion relations for a length rescaling
3. An odd number is necessary for b, in order to treat the ferromagnetic
and antiferromagnetic correlations of the spin-glass problem on equal footing. These
local recursion relations are approximate for the hypercubic lattices and exact for
d-dimensional hierarchical lattices [14].
Equation (2.2) complicates, after a few rescalings, even a simple initial quenched
22
probability distribution. The initial distribution has a double symmetry, P(Jij, Hi, Hg)
P(Ji,, Hij, -HY)
=
P(Jij, -Hij, HY ), which is preserved under the renormalization-
group transformation and which is calculationally exploited in the steps described
below.
The distribution P(Jjj, Hjj, H/j) is represented by histograms.
Each his-
togram is characterized by four quantities, Ji, Hij, H1, and p, where the latter is the
associated probability. Equation (2.2) is the convolution of b distributions. This is
numerically constructed from pairwise convolutions of distributions. To achieve the
overall convolution of Eq.(2.2), for d = 2 and 3, respectively, three and five pairwise
convolutions are needed.
A pairwise convolution is achieved as follows: (1) The histograms are placed
on a grid in the space of interactions Jij, Hij, Ht. All histograms that fall within
the same grid cell are combined in such a way as to preserve the averages of the
interactions. The histograms that fall outside the grid, representing a very small
probability, are similarly combined into a single histogram. In this procedure, a
histogram that falls within a narrow band of a grid boundary is proportionately
shared by each side of this boundary. This gridding is done separately for J > 0
and J < 0.
(2) Two gridded distributions are convoluted as in Eq.(2.2) with a
bond-moving R(Ki,j, Ki2 j 2 ) = Kj1 j1 + Ki 2 j 2 or with the decimation R of Eq.(2.3),
regenerating the original number of histograms.
The convolution of
3d
distributions is achieved, from pairwise convolutions, as
follows: (1) The pairwise bond-moving convolution is cycled, once in d = 2 and three
times in d
=
3. (2) The resulting distribution is bond-moving convoluted with the
beginning distribution, which completes the implementation of bond-moving of 3 d-1
quenched random interactions, as prescribed in Eq.(2.4). (3) The resulting distribution is pairwise decimation convoluted. This completes the entire renormalizationgroup transformation, yielding the histograms for the renormalized quenched probability distribution P'(J,,,, Hj,, H/,'j,). Most of our calculations have used 63,504
independent (Hij, H!. > 0) histograms, corresponding to the renormalization-group
flows of 254,016 quantities.
The flows of the probability distribution P, under renormalization-group trans23
=
formations, determine the phase diagram of the system.
Stability analysis of the
unstable fixed distribution onto which a phase boundary collapses determines the
order of the corresponding phase transitions and the values of the critical exponents
in the case of second-order phase transitions.
The calculated global phase diagram of this random-field spin-glass system in
d = 3 is shown in Fig. 2.1, in the variables of temperature 1/J, random-field strength
H/J, and antiferromagnetic bond concentration p. All of the phase boundaries seen in
this figure are second order. The phase diagram is of course symmetric about p = 1/2,
with a mapping between ferromagnetism and antiferromagnetism. At low temperature, low random-field strength, and p in the neighborhood of 0 (1), a ferromagnetic
(antiferromagnetic) phase occurs. At low temperature, p in the neighborhood of 2'
and zero random field, a spin-glass phase occurs. The remainder of the phase diagram
is taken by the disordered, namely paramagnetic, phase. We now analyze different
aspects of this global phase diagram for d = 3.
Constant p cross-sections of the ferromagnetic phase boundary are shown in Fig.
2.2. The outermost curve is the phase boundary of the purely random-field problem
(p = 0) in d
=
3 [8, 10, 15]. Each point on these curves is a second-order phase
transition point, for the corresponding value of p, between the ferromagnetic and
paramagnetic phases. The arrow drawn on the axis indicates the multicritical point,
occurring at p
=
0.378 and zero random field, between the paramagnetic, ferromag-
netic, and spin-glass phases.
The zero random-field cross-section of the phase diagram, corresponding to the
purely spin- glass problem, is shown in Fig. 2.3. The boundary between the spin-glass
and paramagnetic phases agrees with a previous renormalization-group calculation,
also using the Migdal-Kadanoff procedure, on the purely spin-glass system [5]. The
phase boundary between the ferromagnetic and spin-glass phases is second order (as
in fact are all phase boundaries of the global phase diagram) and has curvature, as
seen in the inset of Fig. 2.3. Thus, the so-called "reentrant" [16] spin-glass phase
is obtained, in the sense that in a given quenched random system characterized by
a fixed p, as temperature is lowered, ferromagnetic disorder reappears at the lowest
24
temperatures. The multicritical point where the three phase boundaries meet occurs
essentially on the Nishimori [17, 18, 19, 5] symmetry line:
1
J = - ln(p2
1
(2.5)
- 1).
In the numerical procedure, this multicritical point moves more precisely onto the
Nishimori line as our binning of the quenched probability distribution is made more
detailed.
The zero-temperature cross-section of the phase diagram is shown in Fig. 2.4. As
seen in this figure and in the global phase diagram at finite temperature (Fig. 2.1),
even an infinitesimal random field destroys spin-glass ordering. In fact, let us suppose
that this result is also valid for the random- field problem where Hi
i. Redefine the spins (a local gauge transformation) as si
-+
=
±H at each site
-si at all sites i where
the random field is originally Hi = -H. This flips the sign of these fields Hi, so that
the transformed system is under a uniform field. Each local gauge transformation at
i also flips the signs of all bonds Jij connected to site i. At p
= 2'
the latter sign
flips clearly leave the bond randomness invariant, so we conclude that an infinitesimal
uniform field as well as an infinitesimal random field destroys spin-glass order. For
general p, only one of the two connected sites is gauge transformed for a random half
of the total number of bonds, so that the bond flips sign. Thus, the transformed
system has j = (I)p + (}) (1 - p) = 1, but with the frustration of the original system,
namely of a fraction 4p(l -p)
3 +4p 3 (1
-p) of the number of elementary squares, since
the gauge transformation flips two bonds in an elementary square, leaving the oddnumbered antiferromagnetic-bond squares as such. Accordingly, a statement cannot
be made directly on the effect of a uniform field for p 0 1, although continuity from
would suggest the disappearance of the spin-glass phase under any uniform
field or any random field. This result, for the uniform-field case, was deduced in the
P=
previous renormalization-group study, [5] from the instability of the spin-glass phase
sink to a small uniform field.
The singly unstable fixed distributions corresponding to the phase transitions be-
25
tween the ferromagnetic and spin-glass phases and between the spin-glass and paramagnetic phases are shown in Fig. 2.5(a,c). The former is a strong coupling distribution: at each renormalization-group iteration, the average bond strength increases
by
'
bo 4 5 J, but the scaled distribution remains fixed. The eigenvalue exponent
=
y for the deviation of the scaled standard deviation from its fixed-distribution value,
[(u/i)'
(a/j)*] = b[(a/J) - (o/J)*], being y
-
=
0.756 different from d, shows
[20] that the phase transitions between the ferromagnetic and spin-glass phases are
second order. This is consistent with the general arguments [21, 22] indicating that
symmetry-breaking occurs as a second-order phase transition, rather than a firstorder phase transition, under strong randomness in d = 3. The doubly unstable fixed
distribution for the multicritical point is shown in Fig. 2.5(b).
At d
=
2, an infinitesimal random field destroys the ferromagnetic and antiferro-
magnetic phases [1, 8]. Thus, the phase diagram occurs at H/J = 0, as shown in Fig.
2.6. This phase diagram also has no spin-glass phase [5]. However, a multicritical
point on the Nishimori line divides [18, 5] into two segments the phase boundary
between the ferromagnetic (antiferromagnetic) and paramagnetic phases. These segments are attracted to different unstable fixed points. The facts that these segments
are both second order (again y 5 d) and have different critical exponents indicate
a strong violation of the universality principle of critical phenomena [23]. The lowtemperature segment of second-order transitions is attracted to a strong-coupling,
strong-randomnes (both J and a renormalize to infinity) fixed distribution, shown
in Fig.
2.7(a).
The high-temperature segment of second-order transitions is not
attracted to strong coupling. This signifies that the critical exponents of the phase
transitions between the ferromagnetic and paramagnetic phases, under bond randomness, are the same within each segment, but differ from one segment to the other. (In
fact, the low-temperature segment, but not the high-temperature segment, flowing
to strong-coupling, has critical exponents that violate hyperscaling [24, 25].) This
amounts to the strong violation [11, 26] of the empirical principle of critical phenomena universality which states that identical critical exponents should occur along the
entire second-order phase boundary between two given phases [27]. This phenomenon
26
was first seen [11] under the effect of bond randomness on tricritical phase diagrams in
d = 3, where no symmetry line such as the Nishimori line is apparent. For the current
system, we have also studied the asymmetric spin-glass case, with different ferromagnetic and antiferromagnetic bond strengths, where there is no Nishimori symmetry
line. The same strong violation of critical phenomena universality is seen, as shown
in Fig. 2.8.
This research was supported by the Italian Istituto Nazionale di Fisica Nucleare
(INFN), by the US Department of Energy under Grant No. DE-FG02-92ER45473,
and by the US National Science Foundation Grant No. DMR-94-00334.
27
2.3
Renormalizat ion-Group Mapping of the Quenched
Probability Distributions
The local recursion relations for a length rescaling factor of b = 3 have been given
in the previous section. The distribution P(Jij, Hij, H) is represented by properly
normalized histograms for given values of the three variables Jij, Hij, H!t. After each
binary operation, which is either a pairwise bond moving convolution or a pairwise
decimation convolution, the overall probability is conserved. The binning grid, after
each binary operation, is constructed according to the symmetry properties of the
probability distribution: the overall symmetry of the probability distribution under
reflections on the (Hij, Hh ) plane suggested to focus our binning procedure into one
of the four quadrants. The entire domain of the probability distribution is then reproduced by unfolding the bins into the four quadrants, after each binary operation.
Along the third interaction direction Jij, the grid is constructed by placing two independent grids in the positive and negative regions for the exchange interaction Jij. It
is very important to note that the local recursion relations will generate bins in the
positive and negative region for the exchange interaction. Indeed, the general case in
which the two grids, constructed in the positive and negative regions for Jij, are coupled, has been investigated carefully. Within each grid the probability of histograms
located near a intrabin boundary are proportionally shared between neighboring bins.
Similarly, one should couple the two grids in the positive J > 0 and negative J < 0
regions, so that bins falling in the vicinity of J = 0 would be proportionally shared
between the two grids. Indeed it has been shown that, from the numerical point of
view, the proper sharing of the probability between bins is important in order to obtain correct results and smooth probability distribution profiles. On the other hand,
sharing the probabilities between the two grids in the positive and negative regions for
J does not affect the renormalization-group trajectories. This is also understood by
considering the analytical form of the recursion relations for the renormalized bond
variables derived in the previous section.
28
L
j
Fig. 2.9: A 3 x 3 binning grid with a given treshold
for common probability sharing within neighboring bins.
Any histogram falling within the dashed boundaries in the previous figure will
contribute also to neighboring bins. The probability contribution to neighboring bins
is proportional to the distance from the intra-bin boundary. The fractional size of the
regions of common sharing probabilities has been fixed to the optimal value of 0.05L,
where L is the size of the bin. A fractional threshold exceeding this value would
make the numerical effort increase without gaining any accuracy in the calculation,
while values below this threshold would cause the mentioned problem of probabilities
profiles and, consequently, phase boundaries being non smooth. The convolution of
3
d--
distributions is achieved, from pairwise convolutions, cycling, once in d = 2 and
thrice in d = 3, the pairwise bond-moving convolution. After this binary operation,
the probability distribution is again a properly normalized one. Presence of bins that
might fall outside the grid, are properly included considering a weighted average, to
be included in the distribution itself. The resulting distribution is bond-moving convoluted with the beginning distribution, and finally pairwise decimation convoluted,
to complete the entire renormalization-group transformation. As noticed in the previous section, most of our calculation have used 63, 504 independent (Hi, H'. > 0)
bins. The number of bins obviously depends on the grid one decides to use. In
particular, for the phase diagram calculations in the non-zero external fields region
(Hi, H!
0), the grids for the negative and positive regions J > 0, J < 0 were each
chosen of size 53. On the other side, calculation of the pure spin-glass problem, where
29
no external field is included, has simple one-dimensional grids, and the number of
bins along the J direction can be increased up to 49 bins, symmetrically, for both the
negative and positive region. It is also important to notice that for the case in which
one is computing the exact location of the multicritical point and of the second-order
boundaries, together with numerical estimates of the critical exponents, as reported
in the previous section, the number of bins has been increased. Specifically, the number of bins is increased, typically considering grids of size 7 x 7 x 7 and 9 x 9 x 9 . The
phase diagram shown above, both in the d = 2 and d = 3 case, are computed considering a grid of size 5 x 5 x 5, for both positive and negative J regions, corresponding
to the renormalization-group flows of 254, 016 quantities.
30
1.5
Paramagnet
20
0:
0
30
F'erromiagiwl
e-,,
20
*i110
0.5
0
Figure 2-1: Global phase diagram of the random-field spin-glass in d = 3, calculated
with 254,016 renormalization-group flow variables. All of the phase transitions are
second order. A mirror image of the portion shown here occurs at p = 0.5 to 1, with
the antiferromagnetic phase replacing the ferromagnetic phase.
31
5i
1
0-D
Paramagnet
1
.- '
"0
p=()
0.51
0.12
0.24
36
0
I
10
20
30
Temperature 1/A
Figure 2-2: Constant p cross-sections of the ferromagnetic phase boundary in d = 3.
The outermost curve is the phase boundary of the purely random-field problem (p = 0,
no antiferromagnetic bonds). The other curves are, consecutively, for equally spaced
values of p up to 0.36. Each point on all of these curves is a second-order phase
transition point between the ferromagnetic and paramagnetic phases. The arrow on
the axis indicates the multicritical point, occurring at p = 0.378 and H = 0, between
the paramagnetic, ferromagnetic, and spin-glass phases.
32
V%.
Para
4
30
-z
2
Ferro
0.36
-
SI
0 8
20
3.-
Paramagnet
C)
'-4
C)
10
Ferromnagnet
.
Spl
01
C
A ntiferromnagnct
Glss
0.5
1
Antiferromagnetic Bond Concentration p
Figure 2-3: The zero random-field cross-section of the phase diagram, corresponding
to the purely spin-glass problem. The inset shows the curvature of the phase boundary
between the ferromagnetic and spin-glass phases and, thus, the "reentrance" of the
spin-glass phase. All of the phase boundaries (full curves) are second order. The
Nishimori symmetry lines [Eq.(2.5)] are shown with the dotted curves.
33
1.5
I
Zero-Tem perature Ph ise Diagram
1
Paramagnet
0.5
C
-o
Arntiferromagnet
Ferromagnet
Spin Glass
01
0
1
--
7
*
0.5
1
Antiferromagnetic Bond Concentration p
Figure 2-4: The zero temperature cross-section of the phase diagram. As seen in this
figure and in the global phase diagram at finite temperature (Fig. 2.1), even an infinitesimal random field destroys spin-glass ordering. It is argued that an infinitesimal
uniform field has the same effect.
34
I
I
I
I
0 0r-
0.0 5
T
I
I
I
- -
-
-
I
I
(c) SG-Para
(b) Multicritical
(a) Ferro-SG
-
B
Z3-
0S
0.0003
-
0.0006
Of
-1
0
0.00015
I
-1
0
W-
9.000t1
-3
U
1
1
Exchange Interaction
Figure 2-5: Histograms (lower figures) of the unstable fixed distributions attracting,
in d = 3, (a) the phase transitions between the ferromagnetic and spin-glass phases,
(b) the multicritical points where all three phases meet, (c) the phase transitions
between the spin-glass and paramagnetic phases. Respectively 6,724, 2,500, and
14,884 histograms are shown in these lower figures. In the corresponding upper figures,
these histograms are shown combined for better visibility in the figure. The circles
and crosses are probabilities before and after a renormalization-group transformation;
by directly falling on top of each other, they exhibit the fixed-point character of the
distributions. Note that (a) is a strong coupling distribution: at each renormalizationgroup iteration, the average bond strength increases by ' = b0 4 5 J, but the scaled
distribution remains fixed. All fixed distributions in this figure occur at zero field
and are singly (a,c) or doubly (b) unstable, with eigenvalue exponents y less than d,
which indicates all second-order phase transitions. The fixed distributions (b,c) were
also found in the previous renormalization-group study (Ref. 5).
35
4
C
d=2
4
2
C11
0
p
0.5
2
Paramagnet
Ferro
0
Antiferro
I
0.5
1
Antiferronagnetic Bond Concentration p
Figure 2-6: Phase diagram in d = 2, at H = 0 (only the paramagnetic phase occurs at
H 0 0). This phase diagram was calculated with 26,896 renormalization-group flow
variables. All phase transitions are second order and are shown with the full curves.
The Nishimori symmetry lines are shown with the dotted curves. The inset shows the
projections of renormalization-group flows (here using 18,496 variables) and, thereby,
the mechanism for the strong violation of critical phenomena universality: Shown
are actual renormalization-group trajectories that are all initiated very close to each
other within the open circle in the inset. The horizontal axis of the inset is the total
weight of antiferromagnetic bonds (which acquire a distribution of magnitudes as
a trajectory progresses) and the vertical axis is the inverse average bond strength.
A double crossover is exhibited by the thick curves, with flows going first to the
multicritical fixed point M, then crossing over to different critical fixed points C and
C', then crossing over to the ferromagnetic and paramagnetic phase sinks. Thus, two
segments of the same phase boundary renormalize respectively to second-order fixed
points C and C' with different critical exponents. The thin curves are trajectories
from initial conditions further away from the phase boundary (still within the initial
open circle in the inset), so that the close neighborhoods of C and C' are not accessed.
36
(a) Ferro-Para
0
(b) Multicritical
0.1
A--
os
C0
r..
0.05
OA]
411
I
1W
T
I
T
Be
0.0006-
0.0009
0.0004 -
0.0003
0.0002-
0.0003
0
-4
0
4
-L
U
z
0
Ji/
Exchange Interaction
Figure 2-7: Histograms (lower figures) of the unstable fixed distributions attracting, in d = 2, (a) the low- temperature segment of the phase boundary between the
ferromagnetic and paramagnetic phases, C' in Fig. 2.6, (b) the multicritical point
between the two segments of this phase boundary, M in Fig. 2.6. Respectively 6,724
and 4,624 histograms are shown in these lower figures. In the corresponding upper
figures, these histograms are shown combined for better visibility in the figure. The
circles and crosses are probabilities before and after a renormalization-group transformation; by directly falling on top of each other, they exhibit the fixed-point character
of the distributions. (a) is a strong coupling distribution: at each renormalizationgroup iteration, the average bond strength increases by ' = b0 4 7 J, but the scaled
distribution remains fixed. These fixed distributions are singly (a) or doubly (b)
unstable, with eigenvalue exponents y less than d, which indicates all second-order
phase transitions. Specifically, y = 0.541 < d at C'.
37
d=2
2
M
4
Sf
a
0.3
0.1
L
2-
Paramagnet
a
Ferro
AF
0.51
0
Antiferromagnetic Bond Concentration p
Figure 2-8: Phase diagram in d
=
2 for the asymmetric system, where the ferromag-
netic and antiferromagnetic bond strengths are respectively J and -J/2.4, at H = 0
(only the paramagnetic phase occurs at H # 0). This phase diagram was calculated
with 26,896 renormalization-group flow variables. All phase transitions are second
order and are shown with the full curves. For this asymmetric system, there is no
Nishimori symmetry line. The inset (using 18,496 flow variables) shows the strong
violation of critical phenomena universality for asymmetric as well as symmetric systems: At each of f (0) and a (K), referring to the main (asymmetric system) Fig.
2.8 here, and at s (0) referring to the multicritical region of (symmetric system) Fig.
2.6, a pair of renormalization-group trajectories are started very close to each other,
both on the phase boundary. Projections of these renormalization-group trajectories
are shown [in the case of the trajectory from a (0), the horizontal axis is the total
weight of ferromagnetic bonds]. It is seen that all trajectories flow to the multicritical fixed point M as predicted in Refs. 18 and 19, then the two trajectories in each
pair, which were indistinguishable at the resolution of the inset until M, split to cross
over to the second-order fixed points C and C', which have different critical exponents. This shows that both the ferromagnetic-paramagnetic and antiferromagneticparamagnetic phase boundaries, both for symmetric and asymmetric systems, have
the strong violation of critical phenomena universality. (For visibility, every fifth
renormalization-group iteration is shown in the crossovers from M.)
38
References
[1] Y. Imry and S.-k. Ma, Phys. Rev. Lett. 35, 1399 (1975).
[2] S.F. Edwards and P.W. Anderson, J. Phys. F 5, 965 (1975).
[3] C. de Dominicis, H. Orland, and T. Temesvari, J. Phys. I 5, 987 (1995).
[4] A.J. Bray and M.A. Moore, J. Phys. C 17, L613 (1984).
[5] E.J. Hartford and S.R. McKay, J. Appl. Phys. 70, 6068 (1991).
[6] G. Toulouse, Commun. Phys. 2, 115 (1977).
[7] D. Andelman and A.N. Berker, Phys. Rev. B 29, 2630 (1984).
[8] S.R. McKay and A.N. Berker, J. Appl. Phys. 64, 5785 (1988).
[9] A.N. Berker and A. Falicov, Tr. J. Phys. 18, 347 (1994).
[10] A. Falicov, A.N. Berker, and S.R. McKay, Phys. Rev. B 51, 8266 (1995).
[11] A. Falicov and A.N. Berker, Phys. Rev. Lett. 76, 4380 (1996).
[12] A.A. Migdal, Zh. Eksp. Teor. Fiz. 69, 1457 (1975) [Sov. Phys. JETP 42, 743
(1976)].
[13] L.P. Kadanoff, Ann. Phys. (N.Y.) 100, 359 (1976).
[14] A.N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1979).
[15] M.S. Cao and J. Machta, Phys. Rev. B 48, 3177 (1993).
39
[16] The terminology of "reentrant" spin-glass phase used in the literature is, strictly
speaking, a misnomer, since on each side of the ferromagnetic phase, the hightemperature (paramagnetic) and low-temperature (spin-glass) phases are different, themselves being separated by a phase transition elsewhere in the global
phase diagram, unlike the case of the true reentrant phases [P. Cladis, Phys.
Rev. Lett. 35, 48 (1975); Mol. Cryst. Liq. Cryst. 165, 85 (1988)]. However,
reentrant correlations, that increase but do eventually weaken as temperature
is decreased, because of frustration effects, have been shown [S.R. McKay, A.N.
Berker, and S. Kirkpatrick, Phys. Rev. Lett. 48, 767 (1982)] to be essential to
the chaotic rescaling behavior characteristic of the spin-glass phase.
[17] H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981).
[18] P. Le Doussal and A. Georges, Yale University Report No. YCTP-P1-88 (1988).
[19] P. Le Doussal and A.B. Harris, Phys. Rev. Lett. 61, 625 (1988).
[20] B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35, 477 (1975).
[21] A.N. Berker, J. Appl. Phys. 70, 5941 (1991).
[22] A.N. Berker, Physica A 194, 72 (1993).
[23] The flow structure imposed by the unstable fixed point at the Nishimori line,
per se, does not imply a violation of universality until the terminus fixed points
of the outgoing flows are calculated, as done here. Thus, a standard tricritical
phase boundary (first- and second-order segments meeting at the unstable fixed
point) or a redundant crossover phenomenon (two second-order segments with
the same exponents meeting at the unstable fixed point that also has the same
exponents), with no violation of universality, are also a priori possible with this
flow structure.
[24] A.J. Bray and M.A. Moore, J. Phys. C 18, L927 (1985).
[25] A.N. Berker and S.R. McKay, Phys. Rev. B 33, 4712 (1986).
40
[26] A. Falicov and A.N. Berker, Tr. J. Phys. 21, 59 (1997).
[27] We qualified this phenomenon (Ref. 11), where along a single critical line, segmentwise uniform critical exponents change from one value to another at a
multicritical point, as the "strong" violation of universality, to distinguish it
from the previously known phenomenon of continuously varying critical exponents along a critical line, as occurs in certain models in d = 2 [e.g., R.J. Baxter,
Phys. Rev. Lett. 26, 832 (1971)].
41
Chapter 3
Phase Transitions in Electronic
Conduction Models
3.1
Introduction
Motion of "holes" in an antiferromagnet is of crucial importance for the theory of hightemperature superconductors. Insulating materials, like the La 2 CuO 4 and Nd 2 CuO 4
present antiferromagnetic order (AF) at half-filling. In the high-Tc superconductors
AF order is rapidly suppressed both in electron and hole doped materials. For example in La 2 -,SrCuO
4
[1] the Neel temperature is reduced from 300K at half-filling
to 10K for a hole doping fraction x ~ 0.02 while for the electron-doped materials
Nd2 _xCexCuO4_y [2] the antiferromagnetic order disappears at an electronic doping
of x ~_0.14. The recent discovery of electron-doped superconductors is an important
challenge for the theory of high-temperature superconductors.
We have performed the finite-temperature statistical mechanics of the Hubbard
model of electronic conduction in d dimensions, obtaining the phase diagrams and
electron densities using the Migdal-Kadanoff renormalization-group procedure. No
finite-temperature phase transition in d
=
1 is seen. In d = 2 no phase separation is
seen [3]. In d = 3 dimension a remarkably complex phase diagram, with a new phase
and multiple reentrances at different temperature scales and the overall electronhole doping symmetry is found. Our renormalization-group calculation automatically
42
yields the global finite-temperature phase diagram and statistical mechanics of a
generalized Hubbard model, in which each site i of the lattice has one spherically
symmetric orbital and can be occupied by one or two electrons of opposite spin. The
Hubbard hamiltonian
=
E(c!,,c,
-t
-U
ni,fn, 4 +
IIZEni,
a
i
reduces, in the strong coupling limit U/t
(3.1)
+ c ci,a)
>>
1, to the tJ model where each site can
be occupied by at most one electron,
-p3+
=P[-t
(cac, + ct cia)
-
Si S
J
(i,j)
(i,j),a
+V 1: nini + /-t
(i,j)
nil P,
z
(3.2)
where
#
=
1/kBT, c,
and ci,a are the creation and annihilation operators for an
electron in a Wannier state at the ith site of the lattice with spin o- =T, 4; ni,
at site i. The
Ci,,,Ci,, is the occupation number at site i and Si is the spin operator
projection operator P = rHi(1 - niinit) projects out all states with doubly occupied
sites. The traditional tJ hamiltonian is a special case of eq.(3.2), obtained for V/J =
0.25. Similarly, the traditional Hubbard model can be obtained from the generalized
Hubbard hamiltonian we will introduce in the next section. On bipartite lattices
(i.e., lattices that can be separated into two sublattices such that any two nearestneighbors are on different sublattices), the sign of the hopping term t can be reversed
by a simple redefinition of the Wannier states on one sublattice. Thus, with no loss
of generality, we restrict to t > 0.
43
While zero-temperature properties of both the tJ and Hubbard models have been
studied by mean-field theory [4, 5], small-cluster calculation [6] and the Bethe ansatz
[7], the finite-temperature behavior of the Hubbard model is largely unexplored
[8], especially in d = 3 where we now obtain a rich structure. The position-space
renormalization-group method is well suited for the latter task. Our approach starts
with an approximate decimation in d = 1, which is then developed onto higher dimensions by the Migdal-Kadanoff procedure. Determination of the global connectivity of
the flows also determines the global phase diagram, cross-sections of which apply to
the traditional tJ and Hubbard models. Summation over entire renormalization-group
trajectories yields the finite-temperature electronic densities.
The starting point of our study is the dimensionless Hubbard hamiltonian [9].
tij = t represents the hopping integral between neighboring sites, p and U are the
chemical potential and intrasite electron-electron interaction. The following symmetry properties, for the hamiltonian (3.1) have been discussed in the past [10]:
1) The grand partition function associated with (3.1) is invariant under a phase
change of the Wannier representation
#i(Y)
-
#i(z) exp(ixi)
ZI,(t) = Tre~-P(H-ILN)
-Z
_t
2) The hamiltonian is invariant under the particle-hole exchange, ct
33
* ci-, so
that 3H(t) - pN transforms into ,3H(-t) - (U - p)N + (U - 2p)Ns, where N and
Ns are the number of electrons and number of sites respectively in the system and
3H(t) includes the hopping and interaction terms of the hamiltonian,
Z,(t) = exp{-3(U - 21 )Ns} Zu-, (t).
(3.4)
The particle-hole symmetry line is given by p = U/2; this condition will be extended
to the case of the closed renormalization-group hamiltonian, introduced at the end of
this section.
3) The charge and spin operators are defined in terms of the creation and annihilation operators according to the following relations:
44
p
ni, + niT
p
pgi
-
1
Si
4,ct,
-
St
Ct~,
cic
S
ciC,,
ci, Cif,1 S
(3.5)
,
-
i
ci,T.
The charge operators are nonzero only when acting on nonmagnetic sites
while the spin operators are nonzero when acting on magnetic sites
(o),
t))
(I t), I 4)).
Magnetic and nonmagnetic sites are described in analogy with the Blume-EmeryGriffiths model of He3 - He4 mixtures along the A line and near the critical mixing
point [11, 12]; the He3 concentration of the BEG model is analogous in this case to
the nonmagnetic impurities that compete with the magnetic ordering due to single
occupied states; the fermionic constraint for the c's operators also imply:
(Sf)2 + (p4')2 = 1,
v, v' = x,y,z,
Sp = 0,
SZ = St + SZ-,
(3.6)
S=-i(S
pA = p + A , Pi = -i" P
- S),
- Ag)
The two terms in Eq. (3.1) corresponding to the chemical potential and the intrasite
repulsion in the hamiltonian are invariant under a local rotation in "spin space" and
"charge space"; magnetic (non-magnetic) sites can be locally affected changing the
quantization axis of the spin (charge) operators of an angle ac
('yi) where i is the
lattice index via
Us = flexp(iaiKi - Si),
U, = fexp(iyiq - pj),
45
(3.7)
where Ki and ji are arbitrary unit vectors and ai,-yj the local parameters of the
transformation. The hopping term in (3.1) is instead globally invariant.
46
3.2
Finite-Temperature Phase Diagram
of the Hubbard Model in d
=
3
1
2
Gabriele Migliorini', and A. Nihat Berker ,2,3
'Department of Physics, Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, U.S.A.
2 Feza
Giirsey Research Center for Basic Sciences
Qengelk6y, Istanbul 81220, TURKEY
3 Department
of Physics, Istanbul Technical University
Maslak, Istanbul 80626, TURKEY
Abstract
The finite-temperature phase diagram of the Hubbard model
in d = 3 is obtained from renormalization-group analysis. It
exhibits, around half filling, an antiferromagnetic phase and,
between 30%-40% electron or hole doping from half-filling, a
new
T
phase in which the electron hopping strength t asymp-
totically becomes infinite under repeated rescalings. Next to
the r phase, a first-order phase boundary with very narrow
phase separation (less than 2% jump in electron density)
occurs. At temperatures above the
T
phase, an incommen-
surate spin modulation phase is indicated. In d = 2, we find
that the Hubbard model has no phase transition at finite
temperature.
PACS Numbers: 82.70.Gg, 64.70.Fx, 36.20.-r, 5.70.Ln
The Hubbard model [9] is the bare-essentials realistic model of electronic conduction, yet essentially no knowledge has existed even phenomenologically on its most
frontal macroscopic feature, namely its phase diagram at finite temperatures. In this
research, we obtain a finite-temperature phase diagram for the Hubbard model in spatial dimension d = 3, from an approximate renormalization-group calculation with
flows in a 10-dimensional hamiltonian space. This rich phase diagram, in the variables
of temperature, electron density, and on-site repulsion, exhibits, around half-filling,
an antiferromagnetic phase completely due to electron hopping. At 30-40% electron
or hole doping from half-filling, a new
property. In the neighborhood of the
T
T
phase occurs with distinctive conduction
phase, a phase separation so narrow that
the jump in electron density is less than 2% occurs. At temperatures above the
ir
phase, an incommensurate frozen spin modulation phase is indicated. In d = 2, no
phase separation or other phase transition occurs at finite temperature in the Hubbard model, in contrast[13, 14] to the closely related, but less realistic, tJ model of
electronic conduction.
The Hubbard model is defined by the hamiltonian
-t
YWc,
+c
ci
(3.8)
where ct and ci, are the electron creation and annihilation operators with spin or 4 at site i of a cubic lattice, < ij > indicates summation over all nearest-neighbor
pairs of sites, and
ni, = c.,cij.
and ni = nit + ni
(3.9)
are the electron number operators. The terms in the hamiltonian of (3.8) are, respectively, the kinetic energy term, the on-site repulsion (U > 0) term, and the chemical
potential term included in order to study the system over its entire density range
48
from zero to two electrons per site.
The renormalization-group transformation is formulated [14, 15] by first considering a d = 1 system. An exact renormalization group transformation can be formally
written,
I v 1 v 3 v 5 ...
< uiu3 u 5 ...
2
W4W
=
I v 1 w2 v 3w 4v 5 w 6 ...
< uiw2u 3w 4 U5 W6 ... I
E
W
>
6
>
(3.10)
,
...
where U1, w 2, v 3 , etc. represent the single-site states. Primes indicate the renormalized
system. The transformation given in equation (3.10) conserves the partition function,
Z = Z', but cannot be implemented due to the non-commutativity of the operators
in the hamiltonian. An approximation is used:
Treven sites
exp (-- 3)
- 1, i) - OW(i, i + 1)
= Treven sites exp (i/n(i
even
~
fi Tr,
exp (-,3h(i - 1, i) - 37(i, i + 1))
even
=
(
fi exp (-3'W'(i -
1,+i
1))
even
exp
-'3"h'(i
- 1, i + 1))
exp(-#'$t')
(3.11)
,
where
-#7 (i, j)
-(U/2d)
=
-t (c%0cy, + c"ci)
5 (nitnq
+ njtnj ) + (p/2d)
(3.12)
5 (ni + nj) .
Thus, the approximation consists in neglecting the commutation relations beyond
segments of three consecutive unrenormalized sites. This approximation is effected
twice (~-) in (3.11), in opposing directions, hopefully with compensatory effect.
49
The crux of the calculation is extracted from the third step in (3.11),
Tr
R-p1,2)-
W(2,3)
=e-O'W'(1,3)
-J~we
(3.13)
.(.3
When written in terms of three-site (on the left) and two-site (on the right) matrix
elements, this equation amounts to contracting a 64 x 64 matrix into a 16 x 16 matrix. This operation is facilitated by block diagonalization of the matrices, using the
conservations of particles, total spin magnitude, total spin z-component, and parity,
so that the largest blocks are 4 x 4 and 2 x 2 for the unrenormalized and renormalized
is extracted. The
systems, respectively. Thus, a renormalized hamiltonian -O'7R'
is more general than equation (3.8), namely
closed form of -#'7-'
=
S
-
[tohihj, + t1 (n_,hj_, + hi- nj._,)+ t 2 ni-onj_,] (cIOcy, + fcCi,)
<ij>,el
+
+ cjtcitc> c+i)
(C
-tX
5:
[J
-
U
-
5 mtnj 1
+ p1
(3.14)
ri
+ V2 riinj + V3 (nmtnj~nj + njnjtnj ) + V4rnrlj~tnJj,
<ii>
where the hole operator is hi, = 1 - ni, and the electron spin operator at site i is
i=
where 7
5 c ,--,O'ci&,
(3.15)
,
is the vector of Pauli spin matrices. The four hopping terms in the flow
hamiltonian (3.14) correspond to one electron hopping with or without the opposite
spin electron present at the initial and final sites (two of these processes are related
by hermitivity and therefore have the same hopping strength tj) and to two electrons
simultaneously hopping from one site to a neighboring site. These four processes can
be called vacancy hopping (to), pair breaking (ti), pair hopping (t 2 ), and vacancy-pair
interchange (tx). For
to = t = t 2
tx=J=V
50
2
=V 3 =V 4 =0
,
(3.16)
the flow hamiltonian (3.14), reduces to the Hubbard hamiltonian (3.8). Thus, equations (3.16) are the initial conditions of our renormalization-group flows. However,
in general, the hopping strengths renormalize differently and the new interactions
are generated under rescaling, so that the renormalization-group flows are in the
10-dimensional, )
=
(to, t 1, t 2 , tG, U, P, J,V2 , V3, V4), hamiltonian space.
The transformation is implemented in d > 1 by using the Migdal-Kadanoff procedure, so that
=
(bd1/f)7(fA)
where b = 2 is the length-rescaling factor, the
function 7l is the contraction process specified in the previous paragraph, and
f
is
an arbitrary bond-moving factor, set to yield the correct transition temperature of
the Ising model (f = 1.2279 and 1.4024 in d = 3 and 2). This renormalization-group
transformation yields known information about quantum systems, such as, in d = 1,
the absence finite-temperature phase transitions; in d = 2, a conventional phase transition for the Ising model, a Kosterlitz-Thouless transition for the XY model[16, 17],
no phase transition for the Heisenberg model; in d = 3, ferromagnetic and antiferromagnetic phase transitions for the Heisenberg model, the antiferromagnetic transition
occurring at a 22% higher temperature than the ferromagnetic transition, a purely
quantum mechanical effect[18]. The 10-dimensional renormalization-group flows also
conserve the particle-hole symmetry, given by the map:
o
~
V2
(3.17)
=J ,
=
t2 ,
=
-p
=
V2 - 2V 3 +V 4 , V 3 =-V+V 4 , V 4 =14.
= ti , i2 = to , i
il
U + 2dV-
= t ,
2dV4 , U=U+4dV-
2dV4 ,
The global analysis of the renormalization-group flows yields the phase diagram of the
system. We have thus obtained the global phase diagram of Hubbard model, presented
here in Figures (3.1),(3.2),(3.3), where first- and second-order phase boundaries are
respectively shown by dotted and full curves. The particle-hole symmetry (3.17)
dictates that the Hubbard model [Eq. (3.8)] phase diagrams be symmetric about
p/U = 0.5, which is seen in all of our results.
Figures (3.1) are for U/t
=
20. Figure (3.1, right panel) shows the full phase
51
diagram in temperature versus chemical potential. Figures (3.1, left, middle panels) show the details in temperature versus electron density and chemical potential,
respectively. It is seen that an antiferromagnetic phase occurs around half-filling,
purely due to electron hopping, since the Hubbard hamiltonian (3.8) does not contain an explicit antiferromagnetic coupling. In fact, we traced the occurrence of this
antiferromagnetic phase to the non-zero value of the pair-breaking strength ti. The
antiferromagnetic phase is unstable to at most 10% hole or electron doping from halffilling. Between 30 to 40% hole or electron doping, a
T
phase occurs in which the
vacancy hopping strength to or the pair hopping strength t 2 [see equation (3.14)], respectively, renormalizes to infinity under repeated renormalization-group transformations. Thus, for hole doping, under repeated renormalization-group transformations,
to -+ oo, J/to
=
2, V2 /to = 3/2, p/to
=
6, tipo
=
0, U
-
oo, ti/U = 0, V/U = 0. Sym-
metrically, for electron doping, the overbarred variables of (3.17) have this behavior.
In all other regions of the phase diagram, all hopping strengths renormalize to zero under repeated renormalization-group transformations. Near the r phase, a first-order
phase transition (dotted curves) occurs, seen as a single curve in Fig. (3.1 center)
in terms of electron chemical potential and opening up into a coexistence region in
Fig. (3.1, left) in terms of electron density. The latter shows the distinctive feature
of this first-order transition, namely that it involves a very narrow phase separation,
e.g., a discontinuity in electron density of less than 2%. This is similar to what is
seen experimentally in lanthanide compounds.[19] At temperatures above the T phase,
a sequence of antiferromagnetic and disordered phases is seen, at many temperature
scales Figs. (3.1, 3.2, center). We interpret this as the presence of an incommensurate
spin modulation phase, with a temperature- and (less strongly, by the alignment of
the sequencing) density-dependent periodicity. Our renormalization-group transformation, with a commensurate rescaling factor and a built-in approximation, acts as a
spurious substrate potential which, at small incommensuration, registers the incommensurate phase and, at large incommensuration, disorders it. The incommensurate
phase that we thus deduce is indicated in Figs. (3.1, 3.2, left). The features described
above were also seen in the simpler, less realistic, tJ model.[14, 15]
52
As U/t is decreased, the first-order phase boundary moves with respect to the
T
phase. It is seen that, for U/t = 4.44 [Figs. (3.2)], it actually abuts the boundary of
the T phase and, for U/t = 0.8 [Figs. (3.3)], it is on the other side of the T phase.
We have thus calculated a finite-temperature phase diagram for the d = 3 Hubbard
model that is rich in phase transition phenomena. We have also repeated the same
calculation for d = 2. We find that no phase separation (in contrast to the tJ model
[14, 15]) or other phase transition occurs at finite temperature for the Hubbard model
(3.8) in d = 2.
This research was supported by the Italian Istituto Nazionale di Fisica Nucleare
(INFN), U.S. Department of Energy under Grant No. DE-FG02-92ER45473, and by
the Scientific and Technical Research Council of Turkey (TUBITAK). We gratefully
acknowledge the hospitality of the Feza Girsey Research Center for Basic Sciences
and of the Istanbul Technical University.
53
3.3
The Recursion Relations
In the first part of the work we derived closed-form recursion relations for the Hubbard
model, by constructing the properly symmetrized wavefunctions in order to reduce
the algebraic difficulty of the original problem. It is shown that the proper form of
the wavefunctions, that satisfies the requirements of conserving parity, total spin, and
total spin along the z direction, reduces the problem to analytical recursion relations.
Indeed the hamiltonian is block diagonal and the size of the largest block is 4 x 4. The
statistical trace of the Boltzmann factor is then computed, neglecting commutation
relations beyond blocks of three sites, consistent with the algebra of the relativistic -y
matrices. In the tJ limit, sites of the lattice can only be occupied by single electrons;
this is reminiscent of the non-relativistic limit in the Dirac theory of electrons and
positrons. In this limit, blocks of the hamiltonian reduce their size, and the largest
block size is 2 x 2. In this case the trace is simply related to the algebra of the
Pauli - matrices and the recursion relations for the tJ model [14] are recovered. As
seen in the equation (3.11) of the previous section, the crux of the calculation can be
performed. Indeed
Trw2 e "((1,2)-N
7-t(2)
=
-
''(3.18)
traces and eliminates the degrees of freedom of one sublattice, once the proper choice
of eigenfunctions is made. In this case,
e--O'W'('k) -- A
-r
,
- OW(ij) --OW(j,k),(.
)
(3.19)
where i, j, k are consecutive sites, or in terms of the matrix elements,
Wi
where ui, ii,
Wj, Vk, Vk
are states of the sites, lead to the recursion relations. This
equation amounts to contracting a 64 x 64 matrix into a 16 x 16 matrix.
54
This
operation is facilitated by block diagonalization of the matrices, using, as mentioned
above, the conservations of particles, total spin magnitude, total spin z-component,
and parity. In the two-site system, this corresponds to the following choice, for the
wavefunctions, indicated below with the corresponding set of particle number, total
spin, and spin z-component, and parity quantum numbers (n, s, in, p):
(0,0, 0, +) :
(3.21)
q$1) =-oo,
o) - 1o)},
(1,1/2,1/2, -) : 102) = 1 11
1
{111
0) + 10 M),
16) =
{W
0 ) - 10 W},
107) =
{11.
(1, 1/2, 1/2, +) : 104) =
(2, 0, 0, -) :
108
(2, 0, 0, +)
1
109) -TT?),
(2, 1, 1, +)
1 10
(2, 1, 0, +) :
10)
=
{|1
(3, 1/2,1/2, -) :
t1i)
T) },
t)+ | t)},
(3, 1/2, 1/2, +)
1014)
=
(4, 0, 0, +) :
|11)
-- | $$
1
|
(3.22)
The wavefunctions
10 2 ), 1
4
) ,q 9 ),
112)
10 3 ), 1$ 5 ),
,114),
q11),
10 13 ), 015)
are obtained by a spin-flip operation of
respectively. The hamiltonian is indeed symmetric under
an up-down transformations of all spins.
The unrenormalized block-diagonalizing wavefunctions for the three sites are
(0, 0, 0, +) :
IV5i) = 1000),
55
(3.23)
(1,1/2,1/2,-) 1'2) =
t
1f
oo) - 100 T)}
(1,1/2,1/2, +) : 144) = 11 t oo) + 1oo t)}
105) = 10
(2,0 0, , -) :
L'8)
=
2
0),
)+
) -1
{
lo td) - lo4T)},
1
00) -
=10
(2, 0, 0,+) :
(2, 1, 1, -)
(2, 1,,
-):
(2, 1, 1, +)
loo W,
111) = o1 o),
1012)
=
21{lTto)
1013 )
=
113
v_
101l4)
=
105)
IV)18)
101)
1
1
I4o)
-
o
o1)
+ lo ),
S00) + loo W,
TT 0) -
0) + |
{11
=
-
o t)},
o) - 1o T) -
o
t)},
{tt),
=I
2 ft
=/
T) + 10 M17,
(211, 07±):
1/20) =
(3,1/2,1/2, -) : LP23)
1{1t 0) +
=
t) -
0) + ot W + 1ot)}
IlTt)},
24102 4 )
V
11
V OI
0 ) -o 1ot}
T tol1025)
1 fl
56
1~j o)
10~/26) =
(3, 1/2, 1/2, +) :031)
=
1032)
=
(3, 3/2, 1/2, +) :
040)
Again, 13), 106),L07), 1'16),
f{ IT 0 ) + I o ),
f V o) + 1o T),
| tt),
{ ItT) + I T) + I Tf) },
=
k121),
o) + 10 t)}
ftI
34)=
1039) =
10 MI,}
{21 T4t) - I tiT) - I 4tt,)
033) =
(3, 3/2, 3/2, +):
-
10 22),
V)27-30),
kb35 - 38 ),
4141),
1042)
are obtained
by spin-flip operation. Finally all unrenormalized wavefunction with four, five, or
six electrons are obtained in terms of
11)...022)
under a particle-hole symmetry
operation. This yields all wavefunctions through
1064) =
I
"
),
(3.24)
which is indeed the particle-hole symmetric counterpart of 11) = 1ooo).
Let us define
(Ti 1 ex r (-
(3.25)
7)1 I T;)
ai = (0iI exp(-'7V)I)
(#iI0).
(3.26)
The quantum numbers above define the blocks of the renormalized and unrenormalized hamiltonians. The blocks are 2 x 2 for the unrenormalized matrix elements
corresponding to wavefunctions with at most one electron per site and 4 x 4 for
57
the matrix elements of wavefunctions with two electron per site. The mapping of
Eq. (3.20) is indeed expressed in terms of matrix elements of the renormalized and
unrenormalized system, according to the following relations:
ai = (01|1H1) + 2(
'|k
H5 )+ (01111011),
3
2
3
a2 = (041104) + - (b181101
8)
1
2
1
+ -(013101
(3.27)
3
)+
(b 3 2 110
3 2),
a4 = (451105) + - (41411014) + -(91109) + (02411024),
2
2
a5 = (1011H0o) + 2(02611 $26) + ( 56 110 56),
a6
('6111061)
+
31
2(0r4311043)
-±-1 (
3
1
+ -(053110 53) + (0 3 3 110
a7 = (46211K62) + - (0 47 110
2
47)
49 110 49)
+ (025110 25),
2
a8
=
a9
= (41211012) + 2(02311023) - (0 5o110 5o),
33 )-
a3,
(054 1154) - 2(0631163) + (0/
64 1064 ),
aio= (01611016) + 2(03411034)
+ (V)52||b 52 ).
In the previous equations (3.27), the different terms corresponding to matrix elements of the block-diagonal unrenormalized exponentiated hamiltonian are explicitly
computed in terms of the parameters appearing in the three-site hamiltonian. The
diagonal elements from the block of the unrenormalized exponentiated hamiltonian
are obtained by diagonalization of the unrenormalized hamiltonian. For example, in
the (0
2 1 ..24 110 2 1 ..24 )
block,
(02111021)
Z
vf |2exp(Ai)
(3.28)
i=1,4
where the Ai are the eigenvalues of the subblock and v29 is the 21st component
of the corresponding normalized eigenstate. The eigenvalues are computed directly
considering the quartic solution of the fourth order polynomial relative to the 4 x 4
hamiltonian subblock.
Once the different polynomials are computed, one can immediately get the renor58
malized values for the ten parameters entering the closed-form Hubbard hamiltonian
introduced in the previous section,
t' =
2
log(a4
a2
log(),
2
1+Ig~s
, 1
2s
t2 =
JL'=ILlog(I - Ig~s,
a214),
A =pL+ log( al2),U
J'=log(-), V
a3
(
1
,
70
(3.29)
a3
=I18(
a5
tI'-log(2
)
all
a
Ul=2lgala5,
2g(
,
a 2 a4
2
log( aa
2 asr
),
a
a=
a 5a 1 0 a 2 a 4
3
a2a4
1
,
74' = -lo(
2
aia8
a 5a 10
-
The recursion relation for the hopping parameter t 2 , corresponding to the breaking
of a singlet state, depends, via IgI and s on the off-diagonal polynomial a 6 _7
(#0
6 exp(-#'H')1# 7 ) according to
g = 2a-7
a 5 + a6
s= 1 +m 2, m = a5 - a.
2a 6 _7
(3.30)
A helpful discussion about the symmetry properites of the wavefunctions in the
regime of sites with two electrons of opposite spin and in the regime of where at most
one electron can be found on a site of the lattice, according to general group theory,
is given below. The largest blocks are 4 x 4 and 2 x 2 then, for the unrenormalized
and renormalized systems, respectively. For the latter one we can easily trace the
exponential matrix using the algebra of Pauli spin a- matrices, while for the former
we recall the fundamental commutation relations between matrices in the two spinor
representations of the relativistic group SU(3),
[#wl S] = 2M,
(3.31)
where M has two non-diagonal 2 x 2, equal to t1 and where I is the identity matrix.
59
In expression (3.31), 0' and Os correspond to the same spinor in the Standard and
Weyl representations. according to
[aw, as] = [Ow/7s]
(3.32)
and
[ax Is
1
] = 0.
(3.33)
In the Standard representation the non-relativistic limit reduces to two independent
Pauli submatrices, while in the non-inertial Weyl representation the two Pauli representations remain coupled in the non-relativistic limit, corresponding in our case, to
the limit of singly occupied states that dominate the partition function. Indeed, the
hamiltonian, up to a constant value, can be expressed as
-dI + H = e/w + fdw + h6s + gS + kZ,
(3.34)
where the matrices Ei couple the two representations and as in the previous expression
(3.33), aw,
s are the three spinors along the three cartesian directions ax, a , a.
Both the spectrum of the hamiltonian as a function of the initial condition values in
the renormalization-group transformations, and the single components of its eigenvectors are expressed in terms of the coefficients appearing in the expression (3.34).
This can be done because the quartic equation corresponding to the regime where
doubly occupied states are present, can be analytically solved. The equations admit
an explicit solution, that also reduces to quadratic equations in the limit where sites
of the lattice are occupied by single electrons. It is also important to remark that, in
order to construct the phase diagram for the Hubbard model, in different regimes for
the initial Hubbard condition U/t
=
const., we had to follow the trajectories dictated
by the recursion relations in a numerical renormalization-group study. The previous
expression for the hamiltonian is indeed conserved when the trace has to be numerically inferred. Simply enough, any power of the hamiltonian, and the entire Taylor
series still preserve the same form of eq.(3.34), so that we consider it, together with
the commutation relations (3.31), as an operative method to compute exponentials of
60
a matrix, once one notes that simple recursion relations for the coefficients of integer
powers of any order take the form
d'= fi(e, f, , g,),
...
k=fio(ef,hg,k),
(3.35)
where fi,..,fio are simple linear combinations of the single coefficients, and are uniquely
dictated by the commutation relations (3.31). The method can be extended to the
case of larger matrices, even tough beyond blocks of size 4 x 4 the analytical solution
one obtains in the current case and uses as a crucial operative check, would not be
available. In the current case instead we check carefully that our numerical flows correspond to the ones dictated by the quartic solution for the three-body hamiltonian
in the proper wavefunction representation. This method has been carefully checked
and compared with several standard algorithms for tracing matrices, e.g., the Jacobi
method, and turn out to be the most effective one, as one would expect since all the
symmetries of the original Hubbard hamiltonian are reflected in expression (3.34).
61
3.4
Phase Separation in Two Dimensions
During the last years particular attention has been devoted to the properties of
strongly correlated electronic systems. It has recently been suggested that electronic
correlation effects in narrow energy band materials [21], the metal-insulator transition [22], metallic magnetism [23], and high-Tc superconductivity can be understood
in terms of a single orbital level approximation [4], even though intense debate has
been generated concerning the choice of the appropriate microscopic hamiltonian. In
the Hubbard model the energy levels of each ion are described via a single s band in
a tight-binding basis; competition arises, due to the localizing effect of the repulsive
on-site Coulomb potential and the delocalizing effect of the hopping term between
neighboring sites of the lattice. In the strong coupling limit (U/t > 1), doubly
occupied states become energetically unfavorable and an effective antiferromagnetic
interaction is generated [24]; this corresponds to the tJ model, where the hopping
strength value is simply determined by the canonical relation t/J = 2U/t > 1 .
According to the theory of magnetic insulators [25], Zhang and Rice [23] recently derived a single-band effective hamiltonian from a two-band hamiltonian that is shown
to correspond, approximately, to the strongly interacting two-dimensional Hubbard
model; values of t/J somewhat larger than the canonical value 2U/t are found, suggesting that one should consider the two-dimensional tJ model in the entire range
of the hopping strength values. Similarly, an effective Hubbard hamiltonian, with
a specific coupling value U/t has been shown to reproduce the spectrum of a more
involved CuO hamiltonian [26], so that both the two-dimensional Hubbard model
and the tJ model have recently attracted attention as simple phenomenological models for the description of CuO 2 planes of high-temperature superconductors, e.g., in
La 2 Cu0 4±3 [19].
The question of phase separation is also matter of recent discus-
sion [27] [28]. It is important to establish whether or not simple single band models,
such as the two-dimensional Hubbard model and the tJ model present such behavior,
i.e., whether or not "holes" have a uniform density or separate into two phases of
differing densities. Evidence for phase separation has been established in the range
62
of low t/J values at nonzero doping, for the two-dimensional tJ model, but discussion continues as to whether phase separation occurs at any value of the hopping
strength [20] [29] or vanishes instead above a critical hopping strength [14, 30]; far
from the strong coupling limit, in the two-dimensional Hubbard model, the situation
is also unclear; it has been recently claimed that phase separation is not present in
the two-dimensional Hubbard model on a square lattice [31] and similar conclusions
were recently obtained by Monte Carlo results [27].
In this note the problem of
phase separation for the two-dimensional Hubbard model is analyzed, using a quantum position-space renormalization group calculation. The two-dimensional Hubbard
model and the tJ model, at any value of t/J are considered in the context of a closed
renormalization-group Hubbard hamiltonian. The renormalization group theory of
the tJ model has also been formulated by Falicov and Berker [14] in two and three
dimensions, using the Migdal-Kadanoff technique; it has been shown that phase separation occurs only for low values of the hopping strength t/J and disappears at a
critical hopping strength value (t/J)c ~ 0.23. At the same time, numerical results,
obtained by quantum Monte Carlo methods [34] , high-temperature expansion [35],
and exact-diagonalization technique for small 4x4 small size clusters [30] suggest the
presence of phase separation below a critical value of the hopping strength t/J
~ 1. A
recent calculation for the two-dimensional Hubbard model, limited to the half-filling
case, has also been proposed, using a position-space renormalization-group technique
[8]; it was argued that, in the half-filling case, phase separation would not be present.
63
C=;u
C)
CO
00
CDQ
66
j/j,;a~ijnja
Figre
-1
Firs-
ad
repetiey.(Fg
abot
hss
In~~~~I
T
the
phse
31,rgh
lU
h
6oxsec
oundrie
a]
6ein
64
wth
hae
dsodeed[D,
rt
teght
shon
ul
te
pnl)shw
-_0..
ntferoagetc
opn
ar
eomlzst
orUl-20
moel
3Hubar
hed
iaramof
phse
Caculte
seondordr
uiaV
dtte
an
ful
i
igrmwic
cuves
ym
niiy
eti
sen
haesar
ndT
o
oeo
"IL
CD
.
0.81
a
0.3 SU/t =4.44
U/t =4.44
U/ 4.4
D
D
zt 0.21
J
'-1
P
CD
0.1
\
0. 60.2
4
D0. 4-
Incomm
spin mod
0o.1
.. aa--
a
a
a
S0. 2!
-D
n
0.55
D
I
I
0.6
0.9
Electron density, <ni>
1
-0.1
0
Chemical potential, pU
0.5
1
0
Chemical potential, pAU
xl 0~r
x10-2
x1 G-xl, 0
I
U/t = 0.8
U/t = 0.8
a
0.81-
a-
0.8F
D
D
-0.6
0.6
-0.4
0.4
-C
0.2
0.2
6
0.65
0.8
0.9
1-
1.2
-1.1
Chemical potential, p/U
Electron density, <ni>
Figure 3-3: Calculated phase diagram of the d = 3 Hubbard model for U/t = 0.8.
66
References
[1] G.M. Luke, Phys. Rev. B 42, 7981 (1990).
[2] Y.J. Uemura, J. Appl. Phys. 64, 6087 (1988).
[3] G. Migliorini and A.N. Berker, M.I.T. preprint (1999).
[4] P.W. Anderson, Science 235, 1196 (1987).
[5] G. Baskaran, Z. Zhou, and P.W. Anderson, Solid State Commun. 63, 93 (1987).
[6] J.K. Freericks and L.M. Falicov, Phys. Rev. B 42, 4960 (1990).
[7] P.A. Bares, G.Blatter, and M. Ogata, Phys. Rev. B 44, 130 (1991).
[8] S.A. Cannas, F. Tamarit, and C. Tsallis, Phys. Rev. B 45, 10496 (1992).
[9] J. Hubbard, Proc. Royl Soc. A 276, 238 (1963); 277, 237 (1964); 281, 401
(1964).
[10] C. Castellani, C. Di Castro, and J. Ranningen, Phys. Rev. Lett. 43, 1957 (1979).
[11] M. Blume, V.J. Emery, and R.B. Griffiths, Phys. Rev. A 4, 1071 (1971).
[12] A.N. Berker and M. Wortis, Phys. Rev. B 14, 4946 (1976).
[13] S.A. Kivelson, V.J. Emery, and H.Q. Lin, Phys. Rev. B 42, 6523 (1990).
[14] A. Falicov and A.N. Berker, Phys. Rev. B 51, 12458 (1995).
[15] A. Falicov and A.N. Berker, Turk. J. Phys. 19, 127 (1995).
67
[16] M. Suzuki and H. Takano, Phys. Lett. A 69, 426 (1979).
[17] H. Takano and M. Suzuki, J. Stat. Phys. 26, 635 (1981).
[18] G.S. Rushbrooke and P.J. Wood, Mol. Phys. 6, 409 (1963), calculate this effect
as 14% from series expansion.
[19] F.C. Chou and D.C. Johnston, Phys. Rev. B 54, 572 (1996).
[20] V.J. Emery, S.A. Kivelson, and H.Q. Lin, Phys. Rev. Lett. 64, 475 (1990).
[21] Electron Correlation and Magnetism in Narrow-Band Systems, T. Moriya, ed.
(Springer-Verlag, 1981).
[22] N.F. Mott, Rev. Mod. Phys. 40, 677 (1968).
[23] F.C. Zhang and T.M. Rice Phys. Rev. B 37, 3759 (1988).
[24] K.A. Chao, J.S. Spalek, and A. M. 016s, J. Phys. C 10, L271 (1977).
[25] P.W. Anderson, Phys. Rev. 115, 2 (1959).
[26] M. Schluter and M. Hybersten, Physica C 162-164, 583 (1989).
[27] A.C. Cosentini, M. Capone, L. Guidoni, and G. Bachelet , Phys. Rev. B 58,
R18235 (1998).
[28] M. Grilli, R. Raimondi, C. Castellani, C. Di Castro, and G. Kotliar, Phys. Rev.
Lett. 67, 259 (1991); C. Di Castro and M. Grilli, Physica Scripta T 45, 81
(1992), and reference therein.
[29] C.S. Hellberg and E. Manousakis, Phys. Rev. Lett. 78, 4609 (1997).
[30] E. Dagotto, A. Moreo, F. Ortolani, D. Poilblanc, and J. Riera, Phys. Rev. B
45, 10741 (1992).
[31] G. Su, Phys. Rev. B 54, R8281 (1996).
68
[32] A.A. Migdal, Zh. Eksp. Teor. Fiz 69, 1457 (1957) [Sov. Phys. JETP 42, 743
(1976).
[33] L.P. Kadanoff, Ann. Phys. (N.Y) 100, 359 (1976).
[34] C.T. Shi, Y.C. Chen, and T.K. Lee, cond-mat/9705156.
[35] W. Putikka, M. Luchini, and T.M. Rice, Phys. Rev. Lett. 68, 538 (1992).
69
Chapter 4
Polymers in Two Dimensions
4.1
Introduction
We consider a polymer chain in two dimensions.
We develop a grand-canonical
renormalization-group technique that can be applied to many interesting problems
in polymer physics, including polymers in random media and self-interacting heteropolymers. The collapsed/extended phase diagram of a linear polymer is studied
considering renormalization-group theory for a lattice model in two dimensions. In
order to construct the recursion relations, we consider an exhaustive enumeration of
polymeric paths on the square lattice for different values of the rescaling factor b. We
obtain the phase diagram, in the variables of bending energy, monomer-unit chemical
potential, and temperature. At low chemical potential and favorable bending energy,
we observe a collapsed phase, where the end-to-end separation R grows as the power
v = 0.5 of the chain lenght N. The results are rapidly converging when considering
increasing values of the rescaling factor b, suggesting that our approach is convergent.
70
Collapsing Transition of a Polymer
4.2
in Two Dimensions: Grand-Canonical
Renormalization- Group Theory
2 3
Gabriele Migliorini1 and A. Nihat Berker1 , ,
'Department of Physics and Center for Materials Science and Engineering
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
2
Department of Physics, Istanbul Technical University
Maslak, Istanbul 80626, Turkey
3 Feza
Giirsey Research Center for Basic Sciences, Qengelk6y, Istanbul 81220, Turkey
Abstract
The collapsed/extended phase diagram of a linear polymer is obtained by application
of renormalization-group theory to a two-dimensional lattice model under a grandcanonical ensemble. The phase diagram is in the variables of bending energy, monomerunit chemical potential, and temperature.
At low chemical potential and favorable
bending energy, a collapsed phase occurs in which the polymer fills space compactly
and the end-to-end separation scales, with respect to the chain length, with the exponent v = 1/d = 0.5. At high chemical potential and unfavorable bending energy, an
extended phase occurs in which the end-to-end separation scales as the chain length,
*
=
1. The phase boundary in-between is second order, giving, with v = 0.817 ± 0.005,
a new lattice-walk exponent in two dimensions. Within the extended phase, the local
behavior changes in different regions, from thick rod (high bending ratio) to thin rod
(negligible bending ratio). The latter could be a precursor, in the absence of attraction,
to the globule (necklace) phase. Results obtained in successive levels of approximations,
including the enumeration of over 500 polymer configurations for the recursion relations,
suggest that the phase diagram and critical exponent in our results are convergent.
Bridging the microscopic and macroscopic characteristics of macromolecules is an
important current challenge of statistical physics and materials science. Semiflexible
polymeric chains[1] have been studied for arbitrary stiffness and a collapse transition
has been predicted, with the chain stiffness being the major driving force.[2, 3, 4] In
this context, we have studied a polymer on a two-dimensional (d = 2) lattice[5] by the
renormalization-group method in a new, grand-canonical formulation: On a square
lattice, a linear infinite polymer is forced to enter and to exit the system through
opposite sides of a lattice, with a chemical potential p associated with each monomer
unit within the lattice and with an energy B associated with each bent configuration
of a pair of consecutive bonds. The phase diagram is obtained in the variables of
bending energy and monomer-unit chemical potential, divided by temperature. Our
calculation yields both the long-range property of the scaling of end-to-end separation
as a function of chain length and the local property of the density of segmental bends.
We find two phases that are analogous to the phases of real polymers, and possibly the precursor for a third such polymeric phase. In one of the two found phases,
occurring at low monomer-unit chemical potential and low bending energy, the polymer fills the lattice compactly, with the end-to-end separation R scaling as the power
v
=
1/d = 0.5 of the number of monomer units N. This phase is analogous to the
collapsed phase of physical polymers. In the other phase, occurring at high chemical
potential and high bending energy, the polymer extends linearly at large scales, v
=
1.
This phase is analogous to the rod phase of physical polymers. The phase boundary between the collapsed and rod phases is second order, with v = 0.817 ± 0.005,
which thus provides a new lattice-walk exponent in two dimensions. The rod phase
comprises two regimes, thick rod and thin rod, which have the same large-scale property (v = 1), but which are distinguished by a characteristic of the local scale: The
thick-rod phase, occurring close to the phase boundary, has a segmental bending density (93%) even higher than that of the collapsed phase (89%). The thin-rod phase,
occurring away from the phase boundary, has negligible segmental bending density.
With the addition of monomer attraction, the thick-rod phase could yield a globule
(necklace) phase, by a microsegragation of the monomers constrained along the chain.
72
In the lattice, consider the various ways in which the polymer can occur on a
given elementary square: The polymer could cross the square along a single edge
only, with Boltzmann weight u
-
e"/kT;
it could cross along two consecutive edges,
effectively crossing the square diagonally, with weight v =
2 e(2
+B)/kT, where the
factor of 2 reflects the two ways of achieving this diagonal crossing; it could cross
along one edge, wander outside the square, and cross the given square along the
opposite edge, thus double-crossing the square, with weight w = e(2p+W)/kT, where
W is the monomer-monomer interaction energy. Even for W = 0 in the original
(unrenormalized) system, this interaction acquires an entropy-induced non-zero value
under renormalization-group transformations. Finally, the polymer could cross the
square along three consecutive edges, a U-turn, but this is taken to be sterically
forbidden. The renormalization-group transformation with a length-rescaling factor
of b is constructed by considering a bxb block and calculating, by summation of
local configurations, the renormalized Boltzmann weights u' for single-crossing, v' for
diagonal-crossing, and w' for double-crossing this bxb block.
Fig. 4.1 shows the 412 local configurations of the unrenormalized polymer that
contribute to a single-crossing while wandering into one 4x4 cell. The sum of the
Boltzmann weights of these configurations, called ii, is a polynomial in (u, v, w). For
6
example, the first configuration in this figure contributes u8 v w' to this polynomial.
This renormalized single-crossing is also accomplished by the polymer wandering into
the cell on the other side of the direct path. Thus, u' = ub + 2ii, where the first
term comes from the linear single-crossing, is an approximate recursion relation for
the weight of single-crossing in the renormalized system, containing the weights of
825 local configurations of the unrenormalized system. Similarly, Fig. 4.2 shows the
439 local configurations within a 4x4 cell that contribute to the diagonal-crossing in
the renormalized system. Summing the weights of these diagrams gives a polynomial
for v'. Finally, Fig. 4.3 shows the 534 local configurations taking place within a
single 4x4 cell that contribute to the renormalized double-crossing. Calling the sum
of these weights i, and also taking into account each segment wandering into an
adjoining cell, the approximate recursion relation w' = @ + 2ubii + 3
73
2
is obtained,
containing the weights of 510, 590 local configurations of the unrenormalized system.
For b = 2, 3, w does not affect u' and v' and thus need not to be recurred.
The global connectivity of the renormalization-group flows given by these recursion
relations yields the phase diagram of the system.[2] Furthermore, let (n,, no, nw) be
the total numbers single-crossings, diagonal-crossings, and double-crossings divided
by the end-to-end distance R, which is the distance between the entry and exit points.
Then the total chain length is
(4.1)
N = (nu + 2n, + 2nr)R/2,
where the division by 2 comes from the fact that each monomer segment participates
into two elementary squares. The densities of the unrenormalized and renormalized
(primed) systems are related by
(nu , nv, nw) = (I1/b)(n' , n' , n')
T,
(4.2)
where T is the derivative of the renormalized dimensionless interactions (log u',log v',
log(w') with respect to the unrenormalized dimensionless interactions (log u,log v,
log w). These recursion relations for the densities are iterated until R' = 1 in the
multiply renormalized system, where (n' , n' , n',) are dictated by the boundary conditions. We use (n', n'4, n',)
=
(2, 0, 0) for this multiply renormalized system (all
boundary conditions yield the same scaling behavior), corresponding to an Rx2R
rectangular system with the polymer entry and exit points at the centers of the long
sides. Thus, the densities of the original system are calculated.
The calculated phase diagram is shown in Fig. 4.4, in terms of the monomer-unit
chemical potential p and the bending energy B, divided by temperature kT. It is seen
that the phase boundaries obtained with successive calculations using b = 2, 3, and 4
are almost identical, indicating that the calculations are convergent. The numbers of
local configurations entering the successive calculations are given in Table I.
At p < 0 and at lower values of B for / > 0, a phase occurs in which, under
repeated rescalings, all three generalized fugacities (u, v, w) renormalize to +oo, as
74
2log u = 2log v = log w, while Eq.(4.2) reduces to (nu, n, nw) = (A/b)(n' , n', n' ),
where A calculates within our approximations to be only weakly dependent on b.
Thus, at large length scales, the linear polymer coils around to cover the two-dimensional lattice such that R
N"', with v
=
log b/ log A, calculated to be 0.5, 0.519, 0.5 for
b = 2, 3, 4, respectively. We believe that this is an indication, within our approximate
scheme, that v = 1/2 = 1/d, the compact filling of space by the collapsed polymer.
Thus, for a favorable chemical potential (pu < 0), coiling in order to achieve maximal
coverage at large length scales takes place for any non-prohibitive bending energy
(B < +oo).
An unfavorable chemical potential (p > 0), on the other hand, is
overcome by a sufficiently favorable bending energy (B < 0), to achieve the largescale collapse.
The sliver of this collapsed phase reaching to p~>50 and B > 0 is
entropy-driven.
In the phase at higher values of B for p > 0, all three generalized fugacities (U,v, w)
renormalize to zero under repeated rescalings and Eq.(4.2) reduces to (nu, nro, 0) =
(ni, I', 0). At large length scales, the chain length scales as the end-to-end distance,
v = 1. The polymer covers space minimally, i.e., rectilinearly, between the entry and
exit boundary points. This is the extended (rod) phase.
The phase boundary between the two phases is second order. It is governed by
a fixed point, for example for b = 4 at (u*, v*, w*)
=
(0.573, 0.253, 0.300), with one
relevant eigenvalue exponent y = 1.253,1.232, 1.223 for b = 2, 3, 4, respectively, which
reflects v: The end-to-end distance R rescales as b- 1 . From their normalization given
before Eq.(4.1), the densities (nu, no, r)
b-.
conjugate to (logu, logy, logw) rescale as
From Eq.(4.1), the chain length N rescales as b-Y.
identify the critical exponent at the phase boundary, v
=
Thus, R
-
N/Y, and we
1/y = 0.798, 0.812, 0.817 as
calculated for b=2,3,4, respectively, leading us to deduce v = 0.817 ± 0.005.
Figs. 4.5 and 4.6 gives the fraction of bent articulations, nr/(nu/2 + nv + nw),
calculated, via the renormalization-group transformations and Eq.(4.2), for the original (unrenormalized) system. For a scan at fixed pI/kT = 0.1, the fraction of bent
articulations is large and constant (89%) in the collapsed phase, remarkably peaks
(93%) just inside the rod phase, and precipitously disappears further into the rod
75
phase. The latter two regimes correspond to the thick-rod regime (local behavior
of bending ratio even higher than in the collapsed phase, but long-range behavior of
v = 1) and thin-rod regime (negligible bending ratio, v = 1). In Figs. 4.5 and 4.6, the
scan at p/kT = -0.1 occurs entirely within the collapsed phase. This curve for the
bending ratio is monotonic, being large and constant at favorable bending energies
and precipitously dropping at unfavorable bending energies. With the addition of
attraction between consecutive monomers, the thick-rod phase could yield a globule
(necklace) phase, by a microsegragation of the monomers constrained along the chain.
Further interesting results can be expected in the application of this method to
branched and cross-linked polymers, heteropolymers, polymers with quenched randomness [7] in themselves or in their environment [8], and three-dimensional systems.
We thank A.R. Atilgan, I. Bahar, B. Erman, and A. Kabakgioglu for inspiring and
instructive interactions. ANB gratefully acknowledges the hospitality of the Polymer
Research Center of Bosphorous University. This research was supported by the Italian
Istituto Nazionale di Fisica Nucleare (INFN), U.S. Department of Energy under Grant
No. DE-FG02-92ER45473, by the U.S. National Science Foundation under Grant
No. DMR-94-00334, and by the Scientific and Technical Research Council of Turkey
(TUBITAK).
76
Rescaling factor Number of configurations Calculated exponent v
collapsed critical rod
i
'
ii
b =2
2
3
0.500
0.798
1
b =3
20
19
0.519
0.812
1
4
412
439
0.500
0.817
1
b
=
534
Table 4.1: Number of local polymer configurations contributing to the recursion relations and calculated exponents for R ~ N", for different values of the length rescaling
factor b of the renormalization-group transformation.
4.3
Conclusion
It is seen that renormalization-group theory can be used to link the microscopic and
macroscopic properties of polymers. For the case of a lattice polymer in two dimensions, a phase diagram, critical properties, full scaling curves, and densities are calculated. The model and calculation could be further developed for branched and/or
crossed-linked polymers. For heteropolymers, two distinct chemical potentials and
three distinct bending energies have to be used. For the latter system, and for polymers adsorbed on a irregular substrate, renormalization-group for quenched disorder
has to be performed. Generalizations to three-dimensional lattices are also possible.
Finally, with the inclusion of polymer-solvent and polymer-polymer interactions, a
true phase transition between the globule and rod regimes can be expected.
4.4
Appendix: Polynomials in the Recursion Relations
The polynomials for ii, v', iv- are composed of the following terms, where (c, x, y, z)
represents cuxvywz, listed in Table 4.2:
77
Table 4.2: The polynomials for ii, v', iv-.
iiV__
c
x
y
z
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
1
0
0
0
0
0
1
0
1
0
0
0
1
0
24
20
24
12
2
40
64
8
62
4
40
12
22
4
4
12
12
20
4
2
32
20
26
8
2
4
4
1
2
8
8
6
4
c
x
y
z
c x
y
z
2
0
6
0
1
4
6
6 0
8
2 0
3
12 2
4
24 2
5
28 2
6
16 2
7
2 2
8
24 2
9
8 2
16 2 10
2
12 4
3
20 4
4
6 4
6
36 4
7
28 4
8
48 4
8
20 4
8
2 4
9
16 4
9
4 4
2 4 10
1
2 6
4
4 6
5
8 6
6
20 6
6
20 6
7
20 6
7
16 6
8
4 6
4
2 8
5
4 8
4 8 5
6
2 8
4
6
6
7
12
37
40
23
10
4
2
2
1
8
22
50
4
43
4
16
16
14
2
2
1
10
13
2
4
8
20
2
4
8
0
2
2
3
2
4
2
5
2
6
2
7
2
8
2
9
2
2 10
2 11
4
0
2
4
3
4
4
4
5
4
4 5
6
4
6
4
7
4
4 8
4 9
9
4
4 10
2
6
3
6
4
6
4
6
5
6
6
6
7
6
6 7
8
6
_
_
___~v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
8
0
6
2
7
2
8
2
9
2
2 10
4
4
5
4
5
2
6
4
6
2
7
4
7
2
8
4
8
2
9
4
2
6
3
6
4
6
4
4
4
2
5
6
5
4
6
6
6
4
6
2
7
6
9
2
0
8
2
8
3
8
3
6
8 4
4
6
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
0
0
1
2
0
1
0
1
2
0
2
0
0
0
1
0
1
2
1
2
4
1
2
6
8
8
8 1
2 0
5 0
4
4
2
4
6
6
4
7
2
2
8
6
0
4
6
6 2
2
8
6
1
78
a)
2s K3(e
Figure 4-1: Local configurations of the unrenormalized polymer contributing to the
recursion relations for the calculation with length rescaling factor b = 4: 412 local
configurations contributing to ii.
79
Q 6R)z /d(,S
5!
S S9Sk f%
i57,
z rJ (S§V S P2 S vJ '1 -v 5)j'5'
N J2 S;9 fv~~l s z5
5 Z
8
000
0
R-2! 6S\)
r 0F6
5S5 z7
/1
. F0
c
-D
iq0
I
ZY)
S, , Y
I
0
(I
I~jIc~~ZS
'i~8
I
I
4F-
rod
phase
'N
2F
OF
-2F-
-4'
collapsed
phase
Ii
0
II
0.5
I
1
Monomer-Unit Chemical Potential, p/kT
Figure 4-4: Calculated phase diagram for the lattice polymer. The phase boundary
between the collapsed and rod phases is second order. Calculations with b = 2, 3, 4,
shown here, give almost identical phase boundary curves. The scaling exponents are
v = 0.5 = 1/d, 0.817, 1, respectively in the collapsed phase, on the phase boundary,
and in the rod phase.
82
1.5
1.0
0.8-
S0.6 -
0.4-
01
-4
-2
2
0
4
6
Bending Energy, B/kT
Figure 4-5: Calculated density of the number of successive segments that are bent,
with respect to the total number of segments: (a) This scan is for p/kT = 0.1. The
arrow indicates the position of the phase transition between the collapsed and rod
phases. The bending density is large and constant (89%) in the coil phase, actually
increases as this second-order phase transition is approached, and goes through a
maximum (93%) and abruptly drops in the rod phase, the latter two indicating thickrod and thin-rod regimes. (b) This scan, for pu/kT = -0.1, occurs entirely within
the collapsed phase. The bending density is monotonic, being large and constant for
favorable bending energies, and abruptly dropping for unfavorable bending energies.
83
8
20
I
I
I
.I
.I
I
-2
0
2
4
6
,I
S 10-
-4
Bending Energy, B/kT
Figure 4-6: Calculated ratio of bent and unbent successive segments: (a) This scan
is for [/kT = 0.1. The arrow indicates the position of the phase transition between
the collapsed and rod phases.(b) This scan, for p/kT = -0.1, occurs entirely within
the collapsed phase.
84
8
References
[1] P.J. Flory, Proc. Royal Soc. 234, 60 (1956).
[2] C.B. Post and B.H. Zimm, Biopolymers 18, 1487 (1079).
[3] M. Muthukumar, J. Chem. Phys. 81, 6272 (1984).
[4] A.L. Kholodenko and K.F. Freed, J. Phys. A 17, 2703 (1984).
[5] For previous studies of lattice polymers, see J. Bascle, T. Garel, and H. Orland,
J. Phys. A 25, L1323 (1992); S. Doniach, T. Garel, and H. Orland, J. Chem.
Phys. 105, 1601 (1996).
[6] A.N. Berker and M. Wortis, Phys. Rev. B 14, 4946 (1978).
[7] G. Migliorini and A.N. Berker, Phys. Rev. B 57, 426 (1998).
[8] A. Lopatnikova and A.N. Berker, Phys. Rev. B 56, 11865 (1998).
85
Chapter 5
Heteropolymers
5.1
Introduction
Heteropolymeric chains are composed by hundreds of monomers [1]. Each monomer
can be of one of the 20 possibles types due to the different types of residues found in
nature. The chain is in self-interaction, meaning that the different monomers interact
between themselves with an interaction which depends on the chemical nature of
the monomers themselves. The minimization of the free energy of an heteropolymer
chain is a very difficult problem, both in numerical and analytic approaches, and
it is a direct consequence of the metastable nature of the folding process. During
the folding time of a protein, a typical example of a heteropolymer chain, only a
very small fraction of the configuration space can be visited. This phenomenon is
known as Levinthal paradox. Indeed proteins are well-known to fold reversibly into
a unique (or few) structures. This led directly to the hypothesis that proteins are
a very special kind of heteropolymers selected during the natural evolution for their
particular folding properties. On the other hand, if the folding properties were too
atypical amongst the huge set of ~ 20100 possible heteropolymers, one should end
up with a paradoxical situation: according to Levinthal's hypothesis only a few of all
possible polymeric chains have been visited during evolution. To explain this apparent
paradox, the hypothesis is made that the folding properties are shared by real proteins
with a large set of possible heteropolymers, with some kind of randomness set in.
86
To investigate this possibility many models of disordered heteropolymers have been
proposed [2, 3, 4].
A mean-field theory has been formulated by Shaknovich and
Gutin [5]. It has been shown that under a replica symmetric hypothesis, in meanfield theory, the model is equivalent to a directed polymer in a random field. This
suggests the application of the knowledge about this subject [6, 7, 8, 9], to study the
frozen phase of random heteropolymers. It is important to notice, that differently
from spin-glass models, where many metastable states dominate the thermodynamic
behavior, for random heteropolymers a unique folded structure dominates in the
partition function.
In the next section of this chapter I will review some of the mean-field properties of heteropolymer models, showing how to relate the difficult problem of random
heteropolymers in self-interaction with the problem of a directed polymer in a random medium.
The usual approach introduced for a directed polymer in random
media, in the presence of a 6-correlated external potential is then generalized to the
case of a hierarchical 6-correlated potential. The two problems are considered in the
variational approach of Garel and Orland [3], showing how the Hartree-Fock approximation involved produces the same set of equations for the order parameter, for both
problems.
87
5.2
Mean-Field Theory of Heteropolymers
Gabriele Migliorini
Department of Physics and Center for Material Science and Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, U.S.A.
Abstract
We review recent results obtained in the mean-field theory
of random heteropolymers. The mean-field variational approach is considered and is related to the problem of a directed polymer in random media. Indeed, it is shown in details how, in the case of replica symmetric potential, the two
problems are equivalent. The problem of how to properly
break the replica symmetry in the case of the heteropolymer
model of Shaknovich and Gutin is also considered and results
obtained in the past are critically reviewed.
The problem of heteropolymer chains in self-interaction has been recently studied
both in the phenomenological approach [2] and the microscopic field theory approach
[3, 5, 6]. Despite many efforts to study this problem, a full description of the freezing
transition, which is expected to characterize the low-temperature behavior of the
heteropolymer chain, is still missing. At the phenomenological level it was shown
[2, 6] how a "Random Energy Model-like" freezing transition occurs. Assuming that
the homopolymeric interaction term is negative, so that the system is in the collapsed
phase, the freezing behavior can be understood in the following way. In the collapsed
phase the monomer density p is finite and constant in space. For a given number of
monomers N we will also assume that the ratio N/p is fixed. It is then clear that
the self-interaction term in the energy will be the only term in the hamiltonian that
depends explicitly on the conformation of the chain, once we assume to neglect the
elastic energy of the chain. In this case, the correlation E 1 E 2 between the energies
E 1 and E 2 of two different conformations r 1 (t), r 2 (t) of the collapsed chain is given
by
E 1E 2 =
C2
2
(5.1)
p2,
where C is the three-body interaction energy between monomers along the chain and
p is the constant density of the globule. Similarly
C
2
(5.2)
Np,
E2 =12
where N is the number of monomers. These phenomenological expressions are easy
This also imply that the joint
to derive. A clear derivation is found in Ref. [6].
probability of having two energies E1, E 2 is given by
lim P(E1 , E 2 ) ~ exp{-
(E+2
N- oo
E2)
2
NC2 P
,
(5.3)
which is nothing but the Random Energy Model behavior of Ref. [10]. Let us consider
the model for a random heteropolymer of Refs.
[2, 3, 6, 5, 4], which is defined
according to the following rules. The hamiltonian for a polymer chain made of N
89
monomers, labeled by t = 1, ... , N , in position x(t) in a d dimensional space, is given
by
H[x(t)]
Ho[x(t)] + H1[x(t)],
=
(5.4)
where the first term lets all the monomers interact with the same energies B 0 and C.
Ho[x(t)] =
-
2 0
-
Bo
+
C
dtx(t)
dt dt'5(x(t) - x(t'))
dt dt' dt"6(x(t) - x(t'))6(x(t) - x(t")),
(5.5)
and
H1[x(t)]
f dt dtB(t,t')6(x(t) - x(t')).
(5.6)
The hamiltonian Ho corresponds to the homopolymer problem [11]. For B 0 > 0,
C < 0, without the term H 1 , the system is in a dense phase, characterized by a
constant density po
=
Bo/2C. In this phase the polymer is in a liquid-like phase
and all the configurations with equilibrium density are equally likely, and are visited
during temporal evolution. In H1 the interactions B(t, t') are independent gaussian
variables with zero mean and variance given by
B(t, t')B(s, s')
=
6(t - s)6(t' - s')B 2 /2.
(5.7)
The effect of H1 to the original problem Ho can be understood as follows. Let us
express the original problem Ho
=
HoJ + H02 + H3 in discrete spin notation. Let
si
=
27
exp(-ini),
q
(5.8)
where now s'i is a Potts variable ranging along the possible q roots of unity on the
complex plane. One then has
H02 =
-Bo E 6,gi,,
ij
90
(5.9)
where now 6 a,b is a discrete Kronecker 6-function. The effect of the original term H2 is
then a mean-field like interaction between the Potts spins along the one-dimensional
backbone of the polymer chain. The actual value of each Potts variable corresponds
to the actual Cartesian position of each monomer on a lattice, where
i=1...N
=Ln
,
where L is the monomer to monomer distance and ni = 1, .., q is an integer, introduced
in equation (5.8). The three-body term in HO can be obviously expressed in term of
Ising-Potts variables as well.
In a similar fashion one can regularize the elastic energy term HOJ according to
H1
Z(ni
2
2
N
q-1
(5.10)
A,+. 'EE
i=1 nA=1
This is rather interesting because in these terms the original problem of a homopolymer chain is not only in discrete (lattice) notation, but can be analyzed using the tools
of Ising spin models. The partition function is as a convolution of a one-dimensional
"Potts-like" chain and a mean-field Potts model, where all pairs of monomers interact,
along the backbone of the chain, with the same energy BO. A careful investigation of
the simple homopolymer problem, once the model is regularized on the lattice according to (5.8), is under current investigation, using a transfer matrix technique [12]. It
is also important to notice that, in the case of Ising models, where the exchange interaction between spins naturally decay with the distance, the homopolymer problem
is, according to equation (5.9) a mean-field problem in nature. This physically means
that the 6 contact interaction between monomers is the same for any pair of chain
subunits. This is also clear considering that the index i is a lattice index in the case of
Ising spin models, directly connected to the Cartesian position of spins in real space,
while, in the context of homopolymers the index i is the monomer index, while the
information about the Cartesian position of the monomers is coded by the Potts spin
directions in the complex plain. In this notation, the original homopolymer partition
function reads
ZH =
E exp{9i
3
i
B0 E
i,j
nA
91
6
gs
}.
(5.11)
The second term H1 in the hamiltonian (5.5) can eventually lead to a freezing
transition. To understand this, consider again the discrete formulation of the original
hamiltonian, now including the self-interaction term H1 [x(t), x(t')]. In terms of Potts
variables this gives
H1[x(t), x(t')]
=
ZB(x(t),x(t')),s .
(5.12)
i~j
This problem, where one neglects entirely the homopolymer term Ho, has been carefully investigated in the past and correspond to the usual mean-field Potts glass
[13, 14, 15, 16], once one assumes that the exchange interaction between the spins
2
is a quenched variable, of zero mean and given variance B /2, according to equation
(5.7). Indeed, the mean-field Potts glass, in the limit of q becoming large, for a fixed
number of lattice sites N, behaves like the one-step replica-symmetry broken Random
Energy Model of Derrida [10], which is the bare essential model characterized by the
freezing behavior, that we now naturally interpret, in the language of heteropolymers,
in terms of a frozen conformation of monomers in space. Let us assume now that the
system is in a dense phase, and let us consider the continuous notation that naturally
apply in the polymer context. Consider the Hamiltonian
H=
dt[x~t)2 +
1
/H'
02
2
V(z(t))],5.3
with V an external potential, so that the system is at a given density. The hamiltonian
describes a polymer in d + 1 dimensions, directed in the direction labeled by t. Two
monomers interact if they are at the same transverse height x.
Because of the mean-field nature of the heteropolymer self-interacting problem it
is natural to attack the problem in the replica approach [13]. The n-th moment of
the partition function is now, (T = 1),[5]
Zn
=
=
(JDx(t) exp-H(x(t))
f7JjDQab(xy)exp
a,b
-B2Q) Q2
a<b
92
(5.14)
where
[Q]
=
f{~a=1 'D
H,[xa(t)] + B 2
-:
(t)} exp
(a=1
afb
dtQab(Xa(t), Xb(t))
.
(5.15)
In mean-field theory, the functional integral over the field Qab(X, y) is evaluated by
the saddle-point method [3]. The self-consistency equation for Qab(X, y) now reads
Qab(X, y) =
J
dt (6(x - x,(t))6(y - xb(t))),
(5.16)
where the mean is taken integrating over the variables x(t)a with the Boltzmann
measure (5.15). As usual with the replica method the quantities that have a direct
physical meaning are the moments
Mk = Mn
-
1
S Qe(x, y).
(5.17)
a~b
For example the first moment (k = 1) represents
M1 =
-P(y,t),
dt-PB (x , t)?
(5.18)
where PB(x, t)) is the probability, for a given realization of the {B(t, t')}, that x(t) =
x. Similarly the second moment is
M2=
(5.19)
dtds PB(x, t; x, s))PB(y, t; y, s)),
where PB(X, t; Y, s)) is the probability for a given sample that x(t) = x, x(s) = y
Considering for simplicity the polymer with ring boundary conditions, ([Q] is the
partition function at temperature 1/N of n quantum particles with Hamiltonian
=
1
5a
-a
+ E V(Xa) - B 2
a
5 Qab(Xa,
a<b
93
Xb).
(5.20)
For N -+ oc, C is dominated by the ground-state of ft, and is given by
(5.21)
([Q] ~ e-N<Vl/IlV>
where
4
is the ground-state wave function of H. The interaction potential Qab(X, y)
is to be determined self consistently through:
dx...dx|b(x1...x")|26(x-xa)6(x-xb),
Qab(X, Y) = N < 4|6(x-xa)6(x-xb)|1 >= N
(5.22)
and the free energy density of the heteropolymer is given by
F
N =<
B2
I@
+2
+>
f
2.
L Qab(X, y).(5.23)
dxdy a<b
As a first operative approximation let us consider the replica-symmetric case, i.e.,
Qab(X, Y) = Q(x, y),
(5.24)
a f b.
([Q] reduces in this case to the replicated partition function of a directed polymer
with Hamiltonian
h =
where
#t(x(t))
(5.25)
dt [ , + V(x(t)) + #t(x(t))],
is a quenched Gaussian random potential [8], with correlations given
by
#tW(x)O't(y) = 6tt B 2 Q(x, y).
(5.26)
This formula can be easily interpreted in terms of a mean-field theory. A meanfield theory can be built considering a (small) section of the polymer in the selfconsistent field due to the rest of the chain. The self-consistent field
#t(x(t))
itself
will be a random variable, because of the random nature of the monomer to monomer
interaction and depends on the transverse and longitudinal position of the monomer.
Within the replica symmetric hypothesis for Qab(X, y), replica symmetry breaking
effects are not to be expected at the level of the ground-state wave function [8]. This
94
suggests that this hypothesis is self-consistent. The crucial point to understand is if
it leads to a stable solution of the saddle point equations.
An operative possibility is to simulate directed polymers for various forms of the
correlations of the random potential, to solve the saddle point equation (5.16) at the
numerical level. This should is clearly possible in dimension d = 1 where the polymer
can be simulated by transfer-matrix techniques.
This research was supported by the U.S. National Science Foundation under
Grant. No. DMR-94-00334 and by the Scientific and Technical Research Council
of Turkey (TUBITAK). The author wish to acknowledge his gratitude to Prof. A.N.
Berker, A.Y. Grosberg, and M. Kardar for very interesting discussions while this work
was done.
95
5.3
The Bose Formulation of Heteropolymer Chains
One of the most interesting topics is the physics of disordered systems [9] is the
behavior of directed polymers in random media and random heteropolymers in selfinteraction. A relatively recent method for examining properties of quenched random
systems is the replica method [9]. Many observables can be obtained in the n -+ 0
limit of an n times replicated original system, when the proper coupling between
replicas is introduced. In this note the model of a two-dimensional heteropolymer in
a 6-correlated hierarchical potential is considered. After a brief review of the replica
symmetric case, that has been shown to correspond to the well-known problem of a directed polymer in random media, and where a Bethe ansatz solution has been recently
given [9], we start breaking the replica-symmetry within the Bethe ansatz scheme. A
generalized wave function with no symmetry restriction is proposed, and the problem of constructing the proper eigenstates for the system is shown to correspond to
the original Yang matrix problem [17, 18], where a new set of recursive relations for
the coefficients entering the wave function is given. We consider n identical particles
interacting via a 6-function potential that is hierarchical in nature. The information
required for such an operation is to extend the usual non-linear Schroedinger model to
the case of multicomponents. The different components have an extra "color" label,
that would code the hierarchical nature of the potential and of the related hierarchical boundary conditions, so that the terms involving 6's coming from kinetic energy
derivatives cancel against terms from the interaction Hamiltonian V applied to the
state. In the first section connection between this model and the usual heteropolymer models, e.g., the Independent Interaction Model [3, 5], is reviewed in light of this
findings. In the second section a Bethe ansatz solution is proposed. In section three
conclusions are made. Let us consider then the following problem:
H= -1
a
a + AZ 6(Xa
-
xb)qab.
(5.27)
a,b
It has been shown that for the Independent Interaction Model, in mean-field theory, the saddle point analysis leads to consider an "effective" replicated hamiltonian,
96
where a general hierarchical potential Uab has to be considered. The main point of the
previous section was indeed that the very nature of heteropolymers in self-interaction
is the hierarchical nature of the potential. Any replica-symmetric potential would reduce the problem to another problem, mainly the problem of a polymer in a random
medium. A careful analysis shows that the equations obtained by Shaknovich and
Gutin [5] are mutatis mutandis nothing but the equations obtained by Mezard and
Parisi [8, 19, 20] in the general case of an interface in a random medium. Now just
consider the usual Hubbard - Stratonovich transformation, in order to deduce the
microscopic heteropolymer model that correspond to the original hamiltonian. We
simply have
=
exp{ dt 1 (_t))2
+ A
dt E
6(Xa(t) -
Xb(t))qab},
(5.28)
where, using the notation introduced by Shaknovich and Gutin,
qab =
drQab(r,r),
Qab(r, r') =
f
dtdt'6(Xa(t) - Xa)
6
(Xb(t) - Xb),
(5.29)
and where Zeff corresponds to the effective problem introduced in the previous sec-
tion. A few simple algebraic steps show how the original problem corresponds to
the original problem of Ref. [5], where now the self-interaction Btt, (x, y) between the
monomers has naturally a spatial dependence (x, y). Indeed, the previous expression
now becomes
=
exp{J dtdt'
dxdx'Bt,t (x, y)6(x-x(t))6(y-x(t'))} = exp{J dtdt'B t')x (t )] }.
(5.30)
The simple choice of a 6-correlated hierarchical potential has a direct meaning in the
language of heteropolymers and corresponds to a problem that can now be faced in
the two-dimensional case, using the machinery of Bethe ansatz.
For the simple case of a replica-symmetric potential we have a simple non-linear
97
Schroedinger model [21], i.e.,
(5.31)
H = f dx{8O * 00 + cO * 0 * #0}.
It has been extensively shown that this is indeed, in second quantization, the problem
corresponding to a gas of particles interacting with a 6 contact potential. We will
restrict our analysis to the case of c < 0 eventhough the other case c > 0 is similarly
tractable. [22] The eigenfunctions are simply
<d,(ki,7..,
kn) >in= fdx1..dXn($
n
kixi)
sImiH {
ic
ki -
k. E(xi-xj)}#*(x1)..#*(Xn)10 >
3
i<j<N
i=1
(5.32)
where E represents a sum of two step functions according to the two possible replica
coordinates.
Now consider the creation operators
#'
where the extra index a is a
"color" index and runs from a = 1...Nc, the total number of colors. In the replica
symmetry-breaking scheme, we will construct blocks of size m in the nxn interaction
matrix, and consider different couplings between and within blocks. This suggests to
consider
H
= JdZxqx2a
(5.33)
*
* O+ca
a
ab
where, for the very simple case of n = 2 one considers eigenfunctions in the form
Ik1, k2 >= Jdxidx2 #(Xi, x 2 )q*l(xl)q*, (x 2 ) o >
.
(5.34)
The Bethe ansatz for this problem is a generalization of the identical boson wave
function considered above,
4(X 1 , x 2 ) = 1 [P, Q]O(xQ1 < xQ2 ) exp(i[kpixQ1 + kp 2xQ 2 )]).
(5.35)
[P,Q]
The state above will be an eigenstate, provided that the 6-function terms from the
kinetic energy and interaction Hamiltonians are made to cancel. This is equivalent
to the condition of the first derivative of the wave function having a discontinuity
98
at x,
= Xb.
The pure bosonic case, where replica-symmetry breaking is not present,
reads
21 =
Yl 2k
12 ,
(5.36)
where properties of the operator Yj have been summarized in Ref.[21] [22]. The
pure bosonic case with n particles is a simple generalization of the previous matrix
equation and gives
+',
n i=
where
tp
is a column vector of the n! x n! matrix of the coefficients
(5.37)
[Q, P].
A
matrix structure for the boundary conditions of the hierarchical potential case have
been considered and analyzed in detail for the simple case of a four-particle system
where particles are divided in two subblocks, and numerically for the eight-particle
problem, where two subblocks of four particles is under current study. Whether or
not the problem can be formulated in a compact way, as in the case of n identical
particles, according to equation (5.37) is not clear yet, but will be one of my main
priorities in the next future.
99
5.4
Conclusion
The wave function form to possibly solve the two-dimensional heteropolymer model
introduced at the beginning of this section is proposed, considering a hierarchical
boundary matrix set of equations for the generalized Bethe ansatz. The usual "kstring" solution for the bound state in the pure bosonic case, made of equispaced
imaginary values for the momentum, yielding the simple result
EG - -n3
(5.38)
could possibly be generalized to a hierarchical "k-string" in terms of replica-symmetry
breaking parameters n, m. The result, for n/m = 2 could be confirmed by previous
results, obtained with perturbative technique [23], considering one step of replica
symmetry breaking and introducing one coupling between replicas of the same block
and a second one for replicas within blocks.
This is extremely encouraging and
suggests that both the spectra of the replica symmetric and replica symmetry-broken
models should be studied in deep detail. The crucial difference with respect to to
the theory of Shaknovich and Gutin [5] is that the two spectra are well distinguished
and no ambiguous or superuniversal result is found. Differently, as stated above,
the variational method of Ref.[5] reduce the difficult problem of heteropolymers and
proteins to the still interesting but very different in nature problem of a directed
polymer in random medium.
100
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Chapter 6
Conclusions and Future Prospects
In this thesis the properties of physical systems in the presence of quenched disorder
are studied. The phase diagrams and the statistical mechanics of classical Ising systems that go undergo phase transitions are obtained for the case of a random external
field and a random exchange spin-glass interaction between spins. Uniting phase diagrams, for the general case of a system that both include spin-glass in the exchange
interaction and the external fields, are obtained via renormalization-group theory in
spatial dimensions d = 2 and d = 3. The strong violation of critical phenomena universality, previously found at random-bond tricriticality in d = 3 is now seen in the
case of the spin-glass in d = 2. We considered a very detailed renormalization-group
study that both include bond randomness and field randomness. We show how the
close connection between these two problems becomes manifest in the language of
renormalization-group theory. In this language, several conclusions are obtained for
both the random-field and spin-glass Ising model in d = 2 and d = 3. The probability
distribution for the bond variables J and the site variables H is obtained explicitly,
iterating the recursion relations that relate renormalized and unrenormalized quantities. It is crucial to note that the form of the recursions obtained at the level of
the Migdal-Kadanoff approximations do not predict in the phase diagram a spin-glass
phase when a uniform external field is applied to the system. A crossover regime is
observed in the whole p E [0.33, 0.5] region, when a small magnetic field is included
in the initial conditions for the renormalization-group trajectories. The trajectory in103
deed flows towards the spin-glass fixed distribution, where the distributions of the J+
and J_ are centered around ±oo respectively, but eventually flows back to the paramagnetic fixed distribution after a certain amount of renormalization group steps. It is
also important to mention that the techniques developed here for the representation
of the probability distributions can have wide applications. These techniques can
indeed be crucial to study any pure system where a renormalization-group theory is
present to probe and detect the effects of quenched randomness in the phase diagram.
In this direction a renormalization-group study for the Potts model in the presence
of bond and field randomness is going to be attacked in the near future.
A position-space renormalization-group study of both the Hubbard and the tJ
model has been presented, at arbitrary filling. The presence/absence of phase separation, a crucial aspect to understand the conducting properties of strongly correlated
electronic models is also studied for a general model that include both the Hubbard
and tJ models at arbitrary antiferromagnetic coupling, for any value of the on-site
repulsive term U. From the physical point of view, new features and multiple reentrances are observed in the phase diagram, when exploring the new region of small
on-site repulsion to t hopping strength ratio, in three dimensions. The particle-hole
symmetry appears naturally in our generalized phase diagram and two new phases in
the low-temperature region are seen, for large enough electron (hole) density doping
percentage. At half filling, still in three dimensions, antiferromagnetic ordering is seen
below a critical temperature value. The correct behavior of the critical N el point that
one expects when the on-site repulsion to hopping strength ratio U/t changes is seen
to agree with the results that one can obtain from elementary quantum mechanics
considerations and represents one of the many preliminary tests we performed before
exploring the arbitrary filling region in the phase diagram. In two dimensions phase
separation, as mentioned above, is studied and a three-dimensional phase diagram,
where temperature, on-site repulsion to hopping strength and antiferromagnetic to
hopping strength ratios are considered simultaneously. It is crucial to note that, differently from classical spin systems, the method we used is approximate even in one
dimension because of the commutation relations that are neglected beyond site blocks
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of a certain size. It would be essential to check and improve the one-dimensional recursion relations we obtained, considering the results that one can infer about the
one-dimensional Hubbard model by Bethe ansatz and field theory. In this sense, a
simple but operative starting point would be to analyze the properties of the quantum anisotropic Heisenberg model. Simple recursion relations are obtained for this
problem, at different values of the scaling factor b. In particular recursion relations
have been obtained up to b = 3 and b = 4 by the present author. Numerical recursion
relations can be obtained for arbitrary values of the scaling factor b.
It is seen that renormalization-group theory can be used to link the microscopic
and macroscopic properties of polymers. For the case of a lattice polymer in two dimensions, a phase diagram, critical properties, full scaling curves, and densities have
been calculated. The model and calculation could be further developed for branched
and/or crossed-linked polymers. For heteropolymers, two distinct chemical potentials and three distinct bending energies have to be used. For the latter system, and
for polymers adsorbed on a irregular substrate, renormalization-group for quenched
disorder has to be performed. Generalizations to three dimensional lattices are also
possible. Finally, with the inclusion of polymer-solvent and polymer-polymer interactions, a true phase transition between the globule and rod regimes can be expected.
The close connection between self-interacting heteropolymer chains and directed
polymers in random media has been investigated at the mean-field level. In particular
it has been shown that, in the case of a replica symmetric external field, the two
problems are closely related.
Connections and discrepancies with the variational
Hartree-Fock-like method of Shaknovich and Gutin have been carefully analyzed.
This suggests that the exactly solvable model of directed polymers in random media,
where the external random potential is 6 correlated, can be generalized to a simple
heteropolymer folding model, where the potential would couple different replicas of
the system via a 6 potential that is now hierarchical in form. The exact integrability
of a the directed polymer problem in 1 + 1 dimensions via Bethe ansatz is closely
related to the quantum inverse method, and the exact solution is indeed obtained, for
the multicomponent non-linear Schroedinger model, via a transfer matrix approach.
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Future investigations for both the directed polymer problem and the self-interacting
heteropolymer chains are also one of my priorities in my future research.
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