ACADEMY OF SCIENCES OF THE CZECH REPUBLIC
INSTITUTE OF PHYSICS
The Limit of High Spatial Dimensions as a Means
for the Construction of Comprehensive
Mean-Field Theories
Václav Janiš
Praha, October 1995
i
For my parents
Věroslava and Václav
ii
Preface
This is a thesis submitted to attain a higher scientific degree, Doctor of Sciences (DSc),
used in the Czech Republic as qualification degree for senior research positions.
The subject of the thesis is the construction of comprehensive mean-field theories
for lattice models via the limit of high spatial dimensions. It consists of two parts.
The first one is written as an introductory and independent (self-contained) review
summarizing relevant general aspects of the construction of mean-field theories using
the limit of high spatial dimensions. The second part, Appendices A-C, consists of
reprints of original papers on the subject published by the author alone and with collaborators. Both parts predominantly deal with original contributions of the author in
this field, particularly in the construction of a global and thermodynamically consistent mean-field theory for interacting electrons. The first part of the thesis is written on
a formal and methodological level with the emphasis on a unifying view on the limit
of high dimensions for various lattice models as developed within the last few years.
Appendix A documents the historical development leading to the exact free-energy
functional for itinerant electrons and its mean-field interpretation. Appendices B and
C complement the general construction of mean-field approximations with practical
applications, computations, and quantitative results especially for interacting and disordered electrons.
The purpose of the first part of the thesis is twofold. First, it serves as an introduction into the problem of the construction of a comprehensive mean-field theory in
statistical mechanics (Chapter 1). Second, it completes the general construction the
major steps of which were published by the author in the original papers reprinted in
Appendix A. The most important new results are contained in Chapter 2 and concern
the construction of the exact free-energy functional in d = ∞ for interacting and disordered lattice models using renormalized perturbation theory of Baym. A unifying
scheme for the construction of mean-field theories for classical and quantum interacting and disordered models is derived and applied to a class of sample models. Chapter
3 brings some new theoretical results concerning tractable mean-field theories of interacting and disordered electrons that also have not yet been published. Applications
of the resulting mean-field theories to strongly correlated and eventually disordered
electrons are left to Appendices B and C.
Second part of the thesis consists of eleven papers dealing with various aspects
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iv
of the construction of mean-field-type approximations for lattice models. They are divided into three thematic groups. First five papers (Appendix A) are related to the general formalism of the limit of high spatial dimensions expounded in Chapter 2. Other
six papers present constructions and applications of tractable mean-field theories for
correlated and disordered electrons from the limit of high dimensions as discussed in
Chapter 3. Appendix B (three papers) deals with an analytic, variational mean-field
theory for the Hubbard model at strong coupling and Appendix C (three papers) with
a numerical solution for the disordered Hubbard model in d = ∞..
Since there are various approaches and interpretations of the limit of high spatial
dimensions at present, it was impossible to make a complete and exhaustive review
either about the limit of high spatial dimensions or the construction of mean-field approximations. The thesis hence was not aimed to become a comprehensive review of
the present status of the research in the field. Only the problems and methods related
to the research activity of the author in this vividly expanding research field of condensed matter theory were chosen to be treated here.
Acknowledgment
I would like to thank cordially all my colleagues and collaborators that helped improve my understanding of the theory of correlated electrons. Especially I am indebted
to Professor Dieter Vollhardt for his hospitality extended to me during my stay at the
Technical University at Aachen (Germany) 1990-1994 and for very fruitful collaboration that continues to last till now. I also would like to thank Professors H. Keiter and
G. Czycholl for their hospitality at the University of Dortmund during my Humboldt
fellowship in 1990 where my first paper on correlated electrons in d = ∞ (Appendix
A4) came to light. Further my thanks go to my colleagues, students and friends from
the Technical University at Aachen, J. Schlipf, G. Uhrig, M. Ulmke, R. Vlaming, R. Strack,
P. van Dongen, W. Metzner, R. Berger, and G. Kruse for useful, illuminating and inspiring
discussions not only on physical problems.
I also would like to thank my colleagues in Prague especially Jan Mašek for his persistent support and fruitful collaboration. I thank Dr. B. Velický for evoking my interest
in the method of the coherent-potential approximation from which my understanding
and interpretation of the role of infinite dimensions in the theory of correlated electrons
arose.
Last but not least I am grateful to my wife Dana and my son Ondřej for their understanding for my devotion to physics that may have sometimes caused their inconveniences. Their enduring support has ever been for me invaluable recourse.
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Contents
1 Introduction
1.1 Development of the concept of mean-field theory
1.2 Microscopic models for mean-field theories . . . .
1.2.1 Classical spin models . . . . . . . . . . . .
1.2.2 Quantum itinerant models . . . . . . . . .
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2 The limit d → ∞
2.1 General formalism . . . . . . . . . . . . . . . . . . .
2.2 Classical lattice models . . . . . . . . . . . . . . . .
2.2.1 Ising and classical Heisenberg models . . . .
2.2.2 Edwards-Anderson model of a spin-glass . .
2.3 Quantum itinerant models . . . . . . . . . . . . . . .
2.3.1 Anderson model of disordered electrons . .
2.3.2 Tight-binding models of correlated electrons
Spinless fermions . . . . . . . . . . . . . . . .
Falicov-Kimball model . . . . . . . . . . . . .
Hubbard model . . . . . . . . . . . . . . . . .
2.3.3 Interacting electrons in a random potential .
2.4 Concluding remarks on the limit d → ∞ . . . . . . .
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3 Mean-field theories for lattice fermions
3.1 Analytic mean-field theories . . . . . . . . . . . . . . . .
3.1.1 Nonrandom case . . . . . . . . . . . . . . . . . .
3.1.2 Random case . . . . . . . . . . . . . . . . . . . .
3.2 Numerical mean-field theory . . . . . . . . . . . . . . .
3.2.1 Monte-Carlo in the Green’s function formalism
3.3 Prospects and limitations of mean-field theories . . . .
References
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vii
Chapter 1
Introduction
1.1 Development of the concept of mean-field theory in
statistical mechanics
Thermodynamics, developed as a phenomenological theory of thermal motion, preceded statistical mechanics. Analogously the notion of mean-field theory was first
used phenomenologically to capture main features of the thermodynamics of phase
transitions. The history of mean-field theories begins with the PhD thesis of van der
Waals (1873). Van der Waals introduced there his famous equation of state with a couple of phenomenological parameters a, b substituting the effects of long-range tails of
intermolecular attraction and of a hard core repulsion of the gas molecules, respectively. Although surprisingly successful in the description of the liquid-gas transition,
the van der Waals theory does not suit to second-order phase transitions with symmetry breaking as represented by the paramagnet-ferromagnet transition. The first phenomenological molecular-field theory for such a transition was developed by Pierre
Weiss, Weiss (1907). This theory not only introduces an effective spin-spin coupling
alike van der Waals theory, but it also uses a ”molecular” magnetic field the value of
which must be determined from an implicit, self-consistent equation. The ”dynamical” molecular field is zero in the high-temperature (paramagnetic) and nonzero in
the low-temperature (ferromagnetic) phase. Weiss hence succeeded to construct a first
phenomenological theory of symmetry-breaking phenomena.
Phenomenological mean-field theory of general continuous phase transitions with
symmetry breaking was accomplished by L. D. Landau, Landau (1937). He proposed
a very general, effective scheme to describe and classify any continuous phase transition and to obtain critical exponents necessary to characterize the nonanalyticity connected with the studied transition. The Landau theory was so successful that it seemed
for some time that the statistical mechanics has only to supply microscopic models to
the Landau mean-field theory of phase transitions. Otherwise the collective phenomena and thermodynamic singularities were thought fully understood and essentially
1
2
CHAPTER 1. INTRODUCTION
grasped by the Landau theory.
Although the phenomenological description of phase transitions reached its pinnacle in the Landau theory, it appeared not to be the final and ultimate solution to the
problem of a proper and adequate description of many-body phenomena. The further
development, however, was entirely enabled by statistical mechanics of microscopic
models. Major step that opened the way to the modern theory of phase transitions
and critical phenomena was the exact solution of the two-dimensional Ising model by
L. Onsager, Onsager (1944). This solution gave ample evidence how important critical
fluctuations, neglected in the mean-field description, are and how valuable it is to have
an underlying microscopic theory for collective and critical phenomena. The most
astonishing on the Onsager’s solution was that the exact critical exponents differed
substantially from those predicted by the Landau (Weiss) mean-field theory. Two fundamental questions were raised by the Onsager’s solution. First, how far can we rely
on mean-field theories and second, what governs the critical fluctuations beyond mean
field? Both questions were answered simultaneously by the Wilson’s renormalizationgroup theory, Wilson (1971), based on the scaling ideas of Widom (1965) and Kadanoff
(1966).
There was a long way from Onsager’s solution to the renormalization group. Onsager’s solution stimulated in the first place an enormous effort to overcome the restrictions of perturbation theory. Thereby many expansion schemes with various degrees of renormalizations have been proposed. Among them linked-cluster and hightemperature expansions proved most successful, Domb and Green (1973). To make
an expansion scheme applicable also to collective and critical phenomena it is necessary to make it partly or fully self-consistent. Self-consistent diagrammatic approximations turn to be again a kind of mean-field theory. Here a series of new, more sophisticated mean-field-type approximations for fluids, Percus and Yevick (1958), van
Leuwen (1959), Meeron (1960), De Dominicis (1962), Stell (1964), and classical spin
systems, Brout (1959), (1960), (1961), Horwitz and Callen (1961), Englert (1963), Bloch
and Langer (1965) were proposed. These theories substantially improve on the meanfield theories of van der Waals and Weiss, since they already contain part of critical
fluctuations.
With many new self-consistent, renormalized theories we may ask which of the
self-consistent approximations do we want to call mean-field theories and what are the
attributes characterizing mean-field approximations. That is, what are the criteria to be
imposed on a self-consistent approximation to be called a mean-field theory? Important contribution to the classification of self-consistent diagrammatic approximations
was made by Baym (1962). He formulated criteria a self-consistent theory must fulfill
to be thermodynamically consistent and conserving, i.e. fulfilling indispensable Ward
identities. Namely, only an approximation conserving all charges generating symmetry transformations of the problem can offer a global theory free of unphysical and
spurious singularities and transitions. Any approximation of mean-field character has
to offer a global and thermodynamically consistent theory simulating behavior of an
1.1. DEVELOPMENT OF THE CONCEPT OF MEAN-FIELD THEORY
3
exact solution. This feature of mean-field theories was acknowledged even later when
the renormalization-group technique proved to be the most appropriate tool for summing divergent perturbation expansion in critical regions. Hence the global character
and consistency have remained as benchmarks of mean-field theories.
It appeared after many years of research that practically only phenomenological
mean-field theories of van der Waals and Weiss remain as flawless sample mean-fields
for classical liquids and magnets, respectively. It became clear that these successful
theories withstanding all consistency tests must have a deeper founding. The most
elegant way to prove that a self-consistent approximation is free of unphysical behavior is to find a model to which the approximation represents an exact solution. To
that purpose a number of ”less realistic” models had been proposed with the hope
that an exact solution could be found. A special model of hard spheres with an attractive long-range potential of Kac was devised to produce a model an exact solution
of which is the van der Waals theory, Kac (1959), van Kampen (1964), Lebowitz and
Penrose (1966). More fruitful for understanding of mean-field theories as exact solutions of special models in statistical mechanics proved classical spin systems. Berlin
and Kac (1952) invented and solved exactly the so-called spherical model in the effort to illuminate Onsager’s solution. However, only a mean-field critical behavior
was found. Lately Stanley (1968) showed that the spherical model is a special limit of
the classical Heisenberg model, namely when the dimension of the spin space tends
to infinity, D → ∞. A mean-field-like solution was shown to be an exact solution of
an ”unphysical” limit of a realistic microscopic model. Also the Weiss solution was
shown to be an exact solution to the Ising (Heisenberg) model in the limit of infinite
spatial dimension, d → ∞, Brout (1960), Fisher and Gaunt (1964), Thompson (1974).
Phenomenological mean-field theories of van der Waals and Weiss got mathematical
rigor and justification of their thermodynamic consistency.
It is important to note that artificial, mathematical limits of realistic models could
produce nonsensical results unless a scaling of fundamental constants governing nonlocal correlations accompanies the limits D → ∞ and d → ∞. The scaling is, however,
determined essentially uniquely from the demand that the total energy be finite. E.g.
the limit to high spatial dimensions in the Heisenberg model enforces linear scaling
of the nearest-neighbor spin-spin coupling J → J ∗ /Z, where Z = 2d is the number of
nearest neighbors. The new constant J ∗ remains fixed in the limit d → ∞. In this special case with linear scaling only weak-coupling theory remains relevant in the limit
of high dimensions, which is no restriction for the Heisenberg model. A definition
of a mean-field theory via a limit D → ∞ or d → ∞ offers not only guarantee that
such a mean-field is a thermodynamically consistent approximation but moreover it
provides a small parameter governing dynamical fluctuations beyond the mean field.
With mean-field theories defined as exact solutions of realistic models in special limits
a consistent and unambiguous way to establish an appropriate mean-field approximation for any lattice model was achieved. Nowadays the construction of a mean-field
approximation through the limit d → ∞ has become canonical for lattice models and
4
CHAPTER 1. INTRODUCTION
serves as a starting point for renormalization-group calculations and dimensional expansion. Exact solution in d = ∞ is now the modern, microscopic understanding of
the concept of mean field. The limit D → ∞ is nowadays less used for the construction of a mean-field theory, since it changes the underlying symmetry of the original
Hamiltonian and suits only to weakly coupled systems.
It may seem that understanding of the Weiss mean-field as an exact solution of the
Ising model in d = ∞ with linear scaling of the spin-spin coupling in the Ising model
ends up our way to the construction of mean-field theories in statistical mechanics.
It is not the case. Namely, the scaling of the spin-spin coupling is not quite universal and cannot be transferred onto other physical, even spin lattice models. It was
notably proved by the Edwards-Anderson model of spin glasses, Edwards and Anderson (1975). This is an Ising model with statically random spin-spin couplings Jij .
Sherrington and Kirkpatrick (1975) showed that to obtain a sensible mean-field model
√
the spin-spin coupling must scale with square root of the dimension, Jij → Jij∗ / 2d,
since only even powers of Jij contribute to the averaged free energy. This spin-glass
model also demonstrated that the mean-field theory (Sherrington-Kirkparick model)
may lead to a highly complicated solution the exact analytic form of which has not
yet been definitely derived from first principles, Binder and Young (1986). In spite of
the technical problems in solving the mean-field theory for classical spin glasses we
can say that the concept of mean-field is already well established at the level of classical spin models of the Heisenberg type. It is worth noting that only lattice models
with well defined (discrete) nearest neighbors provide a suitable framework to define
mean-field theory from the limit of infinite lattice dimensions. There is no such a possibility for continuous, jellium models, since the resulting theory in d = ∞ is either
trivial (noninteracting) or singular.
Mean-field theories at the classical level have their importance as a first and gross
estimate of the critical behavior of the investigated model that is otherwise not available from standard perturbation expansions. In quantum models the situation is quite
different. Due to noncommutativity of canonically conjugate variables (coordinate,
momentum) quantum statistical models are much richer than their classical counterparts. To assess the effects of quantum fluctuations it is important to solve microscopic
models not only in critical regions at finite temperature. Due to the quantum degrees
of freedom, Matsubara frequencies, even the extreme limit of zero temperature is in
quantum physics nontrivial. That is why a consistent, global mean-field theory there
is of even greater interest than in the classical statistical mechanics.
The history of mean-field theories in quantum physics goes back to the thirties to
Hartree (1928), Fock (1930), and Slater (1930) and their theory of self-consistent field.
A mean-field-like approximation with an effective field was derived first within the
framework of quantum mechanics of atoms. The Coulomb repulsion of electrons was
replaced in the Hartree-Fock approximation by an effective local potential determined
self-consistently from a sum rule. This theory generalized to statistical mechanics of
many-body quantum systems played the central role in effective-field approximations
1.1. DEVELOPMENT OF THE CONCEPT OF MEAN-FIELD THEORY
5
to quantum models up to the middle of the sixties. In that time a new era of ”correlated electrons” in quantum many-body systems was launched by Hubbard (1963),
Gutzwiller (1963), and Kanamori (1963). They introduced a tight-binding model of interacting electrons with maximally screened Coulomb repulsion. Although originally
intended as a model for a microscopic description of the Mott metal-insulator transition and for itinerant ferromagnetism, the importance of the Hubbard model exceeded
the initial goal of applicability. Now the Hubbard model is the simplest tight-binding
model of correlated electrons in the same way as the Ising model is the simplest model
for insulating magnets. The main importance of the Hubbard model lies in its ability
to comprise most important quantum effects of correlated electrons. However, only
the approximations to the Hubbard model containing true many-body correlations
can reflect the desired aspects of interacting quantum systems beyond the reach of
perturbation theory. Unfortunately the Hartree-Fock mean-field theory does not fulfill this requirement and can be applied only to weakly-coupled systems (almost free
electrons). Contrary to the classical Heisenberg model important piece of physics goes
thereby lost. That is why a tremendous effort was exerted to extend the weak-coupling
mean-field theory also to the strong coupling. Gutzwiller (1964) and Hubbard (1964)
were the first who proposed different approximations going beyond the limits of weak
coupling. These approximations have mean-field character, but they lack a systematic derivation. Moreover, they are not thermodynamically consistent and conserving
approximations in Baym’s sense. A plethora of other mean-field-like approximations
for the Hubbard and similar Anderson impurity model had been proposed using selfconsistent expansions, Baym and Kadanoff (1961), decouplings of equations of motion,
Czycholl (1986), or static saddle-point approximations, Newns and Read (1986). Almost all of these theories proved useful in various particular limits of the Hubbard
model. However, they failed (except for the Hartree-Fock theory) to produce a global
and thermodynamically consistent mean-field theory of the Weiss type.
It was a real challenge to find a reliable solution at strong coupling. An exact solution of the Hubbard model is hopeless in realistic dimensions d = 2, 3. It was a
great success when an exact solution of the Hubbard model in d = 1 was found by
Lieb and Wu (1968). This solution is based on the algebraic Bethe ansatz and cannot be extended to higher spatial dimensions. Hence the construction of a reliable
global mean-field theory interpolating between the weak and strong coupling regimes
in the Hubbard-like models is of fundamental interest in the theory of correlated electrons. It was surprising that there was no significant effort to define a comprehensive
mean-field theory from the limit d → ∞ in quantum itinerant models. Mean-field
theories were rather searched as static saddle points of various functional-integral representations using slave-boson techniques, Kotliar and Ruckenstein (1986), or, in the
single-impurity Anderson model, as a limit of high orbital degeneracy, N → ∞, Bickers
(1987). However, such ”mean-field” theories were either uncontrollable, inconsistent,
or trivial.
The idea to use seriously the limit of high dimensions for the construction of a com-
6
CHAPTER 1. INTRODUCTION
prehensive mean-field theory in quantum many-body systems with itinerant degrees
of freedom was proposed by Metzner and Vollhardt (1989). Since that intensive effort has been exerted to reach a thermodynamically consistent mean-field theory. The
construction of a quantum mean-field theory is much more complicated than in the
classical statistical mechanics, since the quantum dynamics of the d = ∞ model is no
less complex than that of the d = 2, 3 models. Namely the limit d → ∞ reduces a
quantum theory in (d + 1)-dimensions (d spatial and 1 time dimension) to a (0 + 1)dimensional pure quantum theory, Janiš (1991). I.e. the d = ∞ model is equivalent
to the single-impurity Anderson model with an additional self-consistency condition,
Georges and Kotliar (1992). In between a number of Hubbard-like models were attempted to be solved in d = ∞ by various methods, Vollhardt (1993). However, only
simplified models with classical (spinless fermions) or semiclassical (Falicov-Kimball)
dynamics appears to be exactly solvable in d = ∞, Brandt and Mielsch (1989), (1990),
(1991), van Dongen and Vollhardt (1990), Janiš (1991).
The reason why it took so long before the construction of mean-field theories via
d → ∞, well established in classical statistical mechanics, was correctly understood in
quantum many-body systems lies in the fact that condensed-matter theorists thought
the bandwidth must be kept finite in the mean-field limit. This restriction corresponds
to linear scaling of the hopping amplitude leading to a rather trivial, rigid-band (Hartree)
approximation.
Semiclassical models of noninteracting electrons in a random potential play a special role in the problem of the construction of a comprehensive mean-field theory. They
lie between static, classical spin models and fully dynamical, quantum models of correlated electrons. They proved inspiring and instructive for the construction of a comprehensive mean-field theory of quantum systems consistent in the entire range of
complex energies, i.e. the self-energy possesses the Herglotz properties. The models of
noninteracting electrons in an external random (quenched) potential demand on the
one hand the use of Matsubara frequencies but on the other hand these quantum degrees of freedom do not couple, at least at the level of one-electron Green’s functions
necessary for the thermodynamics. Such disordered electronic models can hence be
called semiclassical. A typical representative for the electrons in a chemically disordered medium (metallic alloy) is the Anderson localization model, Anderson (1958).
It was devised to demonstrate the metal-insulator transition in the electron conductivity in amorphous alloys. Due to the randomness breaking the translational invariance
of the nonrandom model, the Anderson model of disordered electrons is insoluble in
realistic dimensions. Even apart from the localization problem, that appeared due to
the ergodicity breaking almost as complicated as the spin-glass problem, the construction of a consistent mean-field theory appeared a nontrivial problem resolved only in
the end of the sixties.
The first attempts to treat random potentials in a sophisticated way were single-site
T -matrix approximations developed by Watson (1956), (1957), Korringa (1958), Beeby
and Edwards (1962). Self-consistent single-site scattering was constructed and solved
1.1. DEVELOPMENT OF THE CONCEPT OF MEAN-FIELD THEORY
7
by Klauder (1961), Davies and Langer (1963). However first Soven (1967) and Taylor
(1967) succeeded in the construction of a comprehensive mean-field theory using the
multiple-scattering theory of Lax (1951). This solution was called ”coherent-potential
approximation” (CPA) and has been serving up to now as a sample mean-field theory
for disordered electronic and phononic systems, Elliot, Krumhansl, and Leath (1974).
The CPA was not derived from the limit of high spatial dimensions but rather from
the exact solution to a single-site problem. However, as it is the case of other successful, consistent mean-field theories also CPA appeared to have deeper founding. First
its variational character was revealed and its connection to the Hubbard-III approximation from the theory of interacting electrons was established, Velický, Kirkpatrick,
and Ehrenreich (1968). Further an attempt was made to use the CPA as a starting
point of a systematic 1/Z expansion, Schwartz and Siggia (1972). However, the authors incorrectly anticipated that the energy bandwidth must be kept fixed in the limit
Z → ∞. Only later on it became clear that the CPA is an exact solution to the Anderson localization model in d = ∞, Janiš (1986), (1989), Vlaming and Vollhardt (1992),
Janiš and Vollhardt (1992b), where the energy density of states goes over to a gaussian
function. Again the limit d → ∞ proved to serve as a means for the construction of
a consistent mean-field theory, though there is also an analogy to the D → ∞ limit of
the Heisenberg model, Wegner (1979). This latter approach, contrary to the former, defines a mean-field theory only in the weak-disorder limit described by a self-consistent
Born approximation. The CPA reasoning in which a single ”live” site is embedded in
a homogeneous surrounding medium fully reflects the quantum nature of mean-field
theories of itinerant electrons and essentially helped to understand also the mean-field
character of the solution of the Hubbard model in d = ∞, Janiš and Vollhardt (1992b),
Vollhardt (1993).
Models of noninteracting electrons scattered on (randomly) distributed impurities
are the only quantum models that can be solved exactly in d = ∞, that is at the meanfield level. The trouble we encounter in quantum models is that we need infinite many
Matsubara frequencies to describe the thermodynamics. Only if the Matsubara frequencies are decoupled and the equations of motion can be solved separately for each
frequency we can hope to obtain an exact solution in closed form. This is the case of
random-alloy models as well as of the Falicov-Kimball model where one spin species
of the electrons from the Hubbard model is localized. Mean-field theory of quantum
systems cannot work only with finite-many degrees of freedom and hence solubility of
a mean-field quantum model depends on the success to decouple the quantum degrees
of freedom. This is the main difficulty of dynamical quantum mean-field theories that
has not yet been satisfactorily overcome for the Hubbard model. That is why the limit
of infinite spatial dimensions in models of itinerant electrons is still a vividly expanding research area.
Although an exact solution to the Hubbard model in d = ∞ is an open problem, we
learned from the limit of infinite dimensions a lot of the general context of the concept
of mean-field theory in statistical mechanics and its relevance for the analysis of coop-
8
CHAPTER 1. INTRODUCTION
erative phenomena in classical and quantum systems. We can claim that a comprehensive mean-field theory constructed via the limit of high spatial dimensions has become
by now also for quantum itinerant models a well established concept acclaimed by the
community of condensed-matter theorists.
1.2 Microscopic models for mean-field theories
We showed in the preceding section that the mean-field theory developed into a precisely defined concept based on the mathematical limit of infinite spatial dimensions.
To be able to define an appropriate mean-field theory demands to have microscopic
models for which we can construct the limit d → ∞. We chose the limit of high spatial
dimensions rather than the limit of high spin (orbital) degeneracy for the construction
of a consistent mean-field theory. The former approach is superior to the latter, since it
does not generally reduce the mean field only to weak coupling. The latter approach
reduces the theory in the limit D → ∞ to a self-consistent Born approximation, i.e. to
self-consistent second order perturbation theory.
To perform the limit d → ∞ with nontrivial output model we must have a discrete lattice (solid) with well defined distances between the nearest neighbors. We will
consider only translationally invariant lattices, actually hypercubic, and put the lattice
constant to one. Thereby we define a natural length scale. Our ultimate aim is the
construction of a global, consistent mean-field theory for correlated electrons. To stress
the very general character of the construction of mean-field theories via the limit of
high spatial dimensions and to put the still developing mean-field theory for itinerant
models into the context of the well established classical theories, we resume also the
construction of mean fields for spin models.
1.2.1
Classical spin models
The most elementary and also fundamental model for either ferromagnets or antiferromagnets with well defined local magnetic moments is the Ising model, Ising (1925)
defined by the Hamiltonian
X
X
cI = −J
H
Si Sj − h
Si
(1.1)
<ij>
i
where J is the spin-spin exchange, h an external magnetic field, and Si = ±1 are the
z-projections of the local spin moments to the easy quantization axis. The first sum is
taken over all the nearest-neighbor pairs < ij >.
By enlarging the spin dimension in the Ising model we obtain the Heisenberg
model, Heisenberg (1928)
X
X
cHe = −J
Si
(1.2)
Si · Sj − h·
H
<ij>
i
1.2. MICROSCOPIC MODELS FOR MEAN-FIELD THEORIES
9
where the vector of the spin S is generally from a D-dimensional Euclidean space.
If we neglect noncommutativity of the spin operators we speak about the classical
Heisenberg model. For our purposes we remain within the classical version of (1.2).
Physically a global solution to a statistical model means to find a thermodynamic
potential governing the equilibrium thermodynamics of a model under consideration. In classical statistical mechanics the fundamental thermodynamic potential is
the Helmholtz free energy defined
n
o
c
F = −kB T ln Tr exp −βH(S)
(1.3)
where β = (kB T )−1 and the trace Tr is taken over the degrees of freedom of the dynamical variables, spin in our case. It means that a mean-field solution should produce
an approximate formula for the free energy F. We must be able to deduce a global
phase diagram from it. The phase diagram and stability of various solutions is the
most interesting information we can retrieve from a mean-field theory.
Another classical spin model important for the construction of a mean-field theory
through the limit d → ∞ is the Ising spin glass. It was proposed by Edwards and
Anderson (1975) and differs from the Ising model in that it has the spin-spin coupling
constants random
X
X
cSG = −
H
Jij Si Sj − h
Si
(1.4)
<ij>
i
where Jij is a random variable with a gaussian distribution. The Edwards-Anderson
model is not translationally invariant. The free energy determining completely the
thermodynamics of such a model is a functional on the space of configurations of the
spin-spin coupling Jij , i.e.
n
o
c
Fc Jij = −kB T ln Tr exp −βHc (S, Jij )
(1.5)
where the subscript c denotes configurational dependence. This free energy contains
complete information as the thermodynamics concerns, but it is a functional with
infinite-many degrees of freedom. To make the free energy tractable, we must reduce
infinite-many independent variables to a finite number. This is allowed by Brout’s ergodicity argument, Brout (1959). It states that the sum over the whole lattice equals,
in the thermodynamic limit, the averaged value over all configurations. We thus can
replace the configurationally dependent free energy Fc by an averaged free energy and
do not change the global, thermodynamic properties of the model. In the end we have
to evaluate
D
n
oE
cc (S, Jij )
Fav = −kB T ln Tr exp −βH
.
(1.6)
av
This so-called quenched averaging, i.e. averaging of the logarithm of the partition sum,
causes substantial deviations from the thermodynamics of the nonrandom model. That
is why the quenched models are so interesting even at the mean-field level.
10
1.2.2
CHAPTER 1. INTRODUCTION
Quantum itinerant models
The simplest nontrivial itinerant model building a bridge between classical and quantum interacting models is the Anderson model of noninteracting electrons in an external, static random potential, Anderson (1958). The underlying Hamiltonian can be
written in the tight-binding form
cAD =
H
X
<ij>σ
†
tij ciσ
cjσ +
X
†
εi ciσ
ciσ
(1.7)
i
†
where ciσ
and ciσ are creation, annihilation operators of the electron, respectively. The
model parameter tij is the electron hopping between nearest-neighbor sites and εi is
the atomic energy. These parameters are generally configurationally dependent (random). The randomness is experimentally achieved by alloying e. g. two simple metals.
Then a lattice site is occupied either by an atom of type A or B with different ionization
energies for the valence electrons. The hopping matrix can and need not be configurationally dependent. In analogy with spin glasses, the thermodynamics of the Anderson
disordered model will be described by an averaged free energy
D
n
oE
cAD (tij, i )
Fav = −kB T ln Tr exp −βH
av
.
(1.8)
Although Fav determines the thermodynamics and one-particle properties, the quantum dynamics is more complex and cannot be entirely determined from the equilibrium averaged free energy. For instance electric conductivity, essential for the metalinsulator transition and the Anderson localization problem, depends on an averaged
product of two one-electron Green’s functions. Such a quantity must be newly determined apart from the averaged free energy, Velický (1969), Janiš (1989).
The trace in the Anderson model (1.8) contains the sum over the quantum degrees
of freedom, Matsubara frequencies. Since the model Hamiltonian (1.7) does not contain higher than quadratic terms in the creation and annihilation operators, there are
no quantum fluctuations due to a coupling of Matsubara frequencies in the solution.
The Matsubara frequencies do not couple and the quantum origin (noncommutativity) of the creation and annihilation operators is harmless. Only if we introduce a real
particle-particle interaction, the Matsubara frequencies start to mix.
The sample and the most important model for correlated electrons is the Hubbard
model, Hubbard (1963), Gutzwiller (1963), Kanamori (1963), defined by the Hamiltonian
X †
X †
†
cH = −t
ci↓ .
(1.9)
ci↑ ci↑ ci↓
ciσ cjσ + U
H
<ij>σ
i
The fundamental thermodynamic potential governing quantum dynamics of interacting electrons is the grand potential Ω, since it is more natural and easier to keep
the chemical potential fixed and to let the number of particles fluctuate. The grand
1.2. MICROSCOPIC MODELS FOR MEAN-FIELD THEORIES
11
potential to be determined in the Hubbard model is
n o
cH − µN
c
ΩH = −kB T ln Tr exp −β H
(1.10)
c = Pi,σ c† ciσ and µ is the chemical potential. The noncommutativity of the
where N
iσ
two contributions to the Hubbard Hamiltonian, kinetic energy proportional to t and
potential energy proportional to U, causes quantum fluctuations to be relevant.
The Hubbard model is the simplest possible lattice model of interacting, itinerant
electrons. We have only one energy band and spin one half without orbital degeneracy. There exist other quantum models of interacting electrons sharing with the Hubbard model the locality of the electron-electron repulsion. We now list modifications
of the Hubbard model we often meet in the literature on correlated electrons. These
Hubbard-like models can be ordered into two classes: homogeneous or translationally
invariant and impurity models.
A first natural extension of the Hubbard Hamiltonian is to include also nonlocal
electron-electron coupling. Thereby we obtain the so-called extended Hubbard Hamiltonian
X †
X †
X
†
cext = −t
H
ciσ cjσ + U
ci↑ ci↑ ci↓
ci↓ + V
niσ njσ 0
(1.11)
<ij>σ
<ij>σσ 0
i
†
where niσ = ciσ
ciσ is the particle-density operator. In (1.11) we added a nearest-neighbor
interaction that can be either repulsive or attractive. Such a model is often used to
study competition between the spin-density-wave and charge-density-wave ground
states.
Another modification of the Hubbard Hamiltonian is the so-called Falicov-Kimball
model, Falicov and Kimball (1969). Its easiest spinless form is defined by the Hamiltonian
X †
X †
X † †
cFK = −t
H
ciσ cjσ + bi bi + U
ci ci bi bi .
(1.12)
<ij>σ
i
i
This model, in comparison with the Hubbard, looses some important properties. The
most important one is that the Matsubara frequencies do not couple at equilibrium.
A great deal of quantum dynamics goes lost in Hamiltonian (1.12) and the model is
semiclassical as the Anderson localization model (1.7). The Falicov-Kimball model
proved, however, invaluable in the construction of an analytic mean-field theory at
strong coupling of the Hubbard model.
The sample impurity model is the single-impurity Anderson model Anderson (1961)
defined by the Hamiltonian
X †
X
X †
cSIAM = −t
Vi ciσ fσ + Vi∗ fσ† ciσ . (1.13)
fσ† fσ + Uf↑† f↑ f↓† f↓ +
ciσ cjσ + Ef
H
<ij>σ
σ
i,σ
Here we have two types of fermions (two-band model) as in the Falicov-Kimball Hamiltonian with one flat dispersion relation. The Coulomb repulsion is relevant only between the localized electrons, but they are hybridized with the itinerant ones. The spinflip scattering is a fully dynamical process and the Matsubara frequencies are coupled
12
CHAPTER 1. INTRODUCTION
as in the Hubbard model. Such important phenomena as the Kondo effect are present
and the model is generally not exactly soluble. This model plays also important role in
the construction of the mean-field theory of the translationally invariant models, since
the exact solution in d = ∞ can be mapped onto an impurity model with a condition
determining the hybridization self-consistently, Georges and Kotliar (1992).
The last model Hamiltonian we will use in the construction of a comprehensive
mean-field theory is the disordered Hubbard model being a mutual extension of the
Anderson localization and the Hubbard models. Such a model combines the effects of
quantum fluctuations due to the electron-electron coupling with the effects of a static
disorder. It is also the most ”realistic” of the microscopic quantum model considered
here. Its Hamiltonian can be written as
X
X †
X
†
cAH =
H
tij ciσ
cjσ +
i ciσ ciσ + U
ni↑ ni↓ .
(1.14)
<ij>σ
i,σ
i
The averaged grand potential governing the equilibrium thermodynamics is
D
n oE
cAH (tij, i ) − µN
c
Ωav = −kB T ln Tr exp −β H
.
av
(1.15)
Chapter 2
The limit d → ∞ in statistical mechanics
Although we speak about high-dimensional models in statistical mechanics, we concentrate exclusively on lattice models of classical spins and itinerant fermions. We do
not touch such problems as the limit of high dimensions in the theory of fluids using
hard cores, Frisch, Rivier, and Wyler (1985), Wyler, Rivier, and Frisch (1987), Carmesin,
Frisch, and Percus (1989) or the problem of lattice bosons in high dimensions.1 We
also do not aim to cover the field of the infinite-dimensional limit and the mean-field
technique in full, but we rather want to present a construction that is bound with the
author’s research in this field.
We demonstrate on the models introduced in the preceding chapter how the limit
of infinite dimensions generally works in statistical mechanics. Especially we concentrate on the problem of the appropriate scaling of nonlocal input parameters in
different models to obtain a sensible limit. We show the importance of perturbation
(diagrammatic) expansion for the determination of the scaling. Different classes of
diagrams become differently important and only leading-order diagrams remain relevant in the limit d = ∞. This leads to the ”collapse” of diagrams and to a reduction of
perturbation theory. Although the sensible limit of high spatial dimensions can only be
deduced from perturbation theory we, in the end, use a nonperturbative formulation
of the exact solution in d = ∞. To this purpose we use the very general formalism of
Baym enabling to formulate functionally the results of diagram sums. We will call the
exact solution of a model in d = ∞ an exact form of the free-energy functional with a set
of variational functions to be determined self-consistently from stationarity conditions.
Solubility of the equations of motion then depends on the degree of sophistication of
particular models. In the case when the equations in d = ∞ are not exactly soluble by
analytic means (Hubbard model) we have to resort to further approximations to turn
the mean-field theory quantitative. In this chapter we concentrate only on the general
1
The limit d → ∞ for massive latice bosons does not seem attractive since all the particles will condense at low temperatures at the lower band edge. A hypercubic lattice has in d = ∞ an infinite energy
band. There are, however, efforts to include phonons into the d = ∞ treatment via a local electronphonon interaction, Freericks (1992), Freericks and Jarrell (1992).
13
CHAPTER 2. THE LIMIT D → ∞
14
construction and the problem of solving mean-field equations for interacting electrons
will be discussed in the next chapter.
2.1
General formalism
The only way to assess the action of the scaling of the input parameters on the solution of the model under consideration is to use formally perturbation expansion and
to classify the contributions according to powers of the coordination number. Since
a thermodynamic solution means to find an appropriate free energy, we use linkedcluster expansion for the free energy with its standard diagrammatic representation,
Wortis (1973). We will not go into details of perturbation expansion of particular models. We only use very general features of the sums of diagrams. The main idea is to
express the resulting free energy as a functional of only renormalized quantities. To do
so we represent formally the partition function as a functional integral
n
o Z
n
o
(0)−1
(2.1)
Z G
= DϕDϕ∗ exp −ϕ∗ ηG(0)−1 ϕ + h∗ ϕ + ϕ∗ h + U ϕ, ϕ∗
where ϕ and ϕ∗ are fluctuating commuting or anticommuting gaussian fields, G(0)−1
is essentially the dispersion relation of the model represented as an inverse of the
free propagator of one-body excitations. The variable η = ±1 is a sign depending
on whether we deal with bosonic (commuting) or fermionic (anticommuting) fluctuating fields, respectively. Further, h is an external source and U is an interaction, i.e.
a nonquadratic function of the fluctuating fields. In (2.1) we suppressed all internal
degrees of freedom of the fluctuating fields that depend upon particular models under
consideration. The free energy as a functional of G(0)−1 then is
n
o
n
o
F G(0)−1 = −β−1 ln Z G(0)−1 .
(2.2)
It was the idea of Baym (1962) to change the functional dependence in the free
energy by a new representation with a renormalized propagator G. The full propagator
G may be defined from the free energy itself:
δ 2 βF
= η ϕϕ∗ − ϕ ϕ∗ = η
G=− ∗
δh δh
δβF
δβF δβF
−
δG(0)−1 δh∗ δh
.
(2.3)
We now introduce the variable G into the free energy by a substitution
G(0)−1 = G−1 + Σ
(2.4)
where Σ is the self-energy. It is a new variable which also enters the free-energy functional. We can treat the renormalized quantities G and Σ as independent variables in
2.1. GENERAL FORMALISM
15
the free energy. The new free-energy functional must, however, not depend on variations of the new variables G and Σ in order to keep the thermodynamic relations
fulfilled. To secure vanishing of variations of the free energy with respect to G and Σ
we have to modify the free-energy functional, since the variation with respect to G(0)−1
does not vanish. We must add a contribution being a function of only Σ and a contribution being a function of only G. If we denote them FΣ and FG we must
fulfill the
(0)−1
following equations to keep the total free energy F = FΣ + FG + F G
independent
of Σ and G:
δβF
δβFΣ δβFG
=
−
=
.
δΣ
δG−1
δG(0)−1
Using equations (2.3) and (2.4) we easily obtain
n
h
i
h
i o
βFΣ = η tr ln G(0)−1 − Σ + m∗ G(0)−1 − Σ m ,
(2.5)
βFG = −η tr ln G−1 + m∗ G−1 m .
(2.6)
Here we had to introduce new renormalized variables
m=−
δβF
δβF
, m∗ = −
.
∗
δh
δh
We use the above definitions and obtain the free-energy functional
h
i
−βF m; G−1 , Σ = −ηtr ln G(0)−1 − Σ + ηtr ln G−1 − βF h; G−1 + Σ
h
i
− m∗ η G(0)−1 − Σ m + m∗ ηG−1 m ,
(2.7)
(2.8)
In (2.8) the free energy F is stationary (extremal) with respect to all its renormalized
variables m, Σ, and G. The stationarity with respect to the variables m, m∗ leads to trivial equations. We can, however, turn these variables dynamic if we use a substitution
in the functional integral (2.1)
ϕ=Φ+m
(2.9)
where the new fluctuating field Φ has vanishing first moment hΦi = 0. We then obtain
from (2.1), (2.9) and (2.4)
h
i
−βF [m, H; G, Σ] = −ηtr ln G(0)−1 − Σ + ηtr ln G−1 − βF m, H; G−1 + Σ
−m∗ ηG(0)−1 m + h∗ m + m∗ h .
Now the new free energy as a functional of m, H and G−1 + Σ reads
Z
−1
−βF m, H; G + Σ = ln DΦDΦ∗
exp −Φ∗ η G−1 + Σ Φ + H ∗ Φ + Φ∗ H + U [Φ + m, Φ∗ + m∗ ]
(2.10)
(2.11)
16
CHAPTER 2. THE LIMIT D → ∞
where the new external sources H and H ∗ are new variational variables, Legendre
conjugate to m∗ and m, respectively. Vanishing of variations of the free energy with
∗
∗
respect
−1 to H,
H just leads to equations hΦ i = 0, hΦi = 0. Note that the nonlocal part
of G + Σ in (2.11) contributes to the functional integral only via closed loops. It is
due to the choice of the functional integral first moment of which, hΦi, vanishes. The
variational dynamical functions m, H, Σ, and G are determined from the saddle-point
equations defined from stationarity of βF :
δβF δβF δβF δβF
=
=
=
= 0.
δH
δm
δG
δΣ
(2.12)
Expressions (2.10)-(2.12) are exact in any spatial dimension for any model, classical or
quantum.
The free-energy functional (2.10) is not yet the known Baym representation. Although it is a functional of only renormalized quantities, diagrammatic representation
of βF contains the sum of all connected unrenormalized diagrams with the bare prop
−1
agator G(0) = G−1 + Σ . Free energy (2.10) is suitable for the exact solution and for
the cases where the result cannot be generated by a sum of simple skeleton diagrams.
If we have to rely on sums of classes of particular diagrams it is more practical if we
define a new functional
Ψ [m, H; Σ] = ηtr ln G−1 − βF m, H; G−1 + Σ
(2.13)
that is, due to the stationarity equations (2.12), independent of the one-body propagator G. We have no diagrammatic representation for the functional Ψ. But if we perform
a Legendre transform from Ψ to a functional of the propagator G
Φ [m, H; G] = Ψ [m, H; Σ] + ηtrΣG
(2.14)
it will be a sum of all connected diagrams free of self-insertions, i.e. skeleton diagrams
only. Inserting (2.13) and (2.14) into (2.10) we reveal the Baym free-energy functional,
Baym (1962)
−βFe [m, H; G] =
h
i
−ηtr ln G(0)−1 − Σ − ηtrΣG + Φ [m, H; G] − m∗ ηG(0)−1 m + h∗ m + m∗ h .
(2.15)
Both free-energy functionals (2.10) and (2.15) are exact. They are connected by a Legendre transform (2.13), (2.14). While the representation (2.10) is applicable without
restrictions, the direct application of the Baym functional is restricted to cases where
we are able to find a diagrammatic representation for the functional Φ. We then speak
about Φ-derivable approximations. Not all approximations are Φ-derivable. The simplest example for a non- Φ-derivable solution is a 0-dimensional lattice (single site or
atomic solution).
2.1. GENERAL FORMALISM
17
Baym’s representation
is important
for approximate solutions when we are unable
−1
to find the functional F m, H; G + Σ from (2.11) and we have to resort to approximations. Namely, if we use the renormalized version of perturbation expansion with
the one-body renormalized propagator G we have to compensate oversummation of
diagrams due to self-energy insertions in the full propagator by a new potential −Σ.
It means that the excitations in the renormalized perturbation theory do not scatter
on each other only but also on a dynamical potential −Σ. When we approximate the
particle-particle scattering due to the interaction U we must also approximate the additional scattering on the potential −Σ. Since the character of the particle-particle and potential scattering is quite different, not all approximations to the particle-particle scattering are exactly compensated by an approximation on the potential scattering that
can be summed exactly. It is then easier to keep diagrammatic approximations thermodynamically consistent within the Baym formalism with the functional Φ. Thermodynamic consistency of approximations is a very severe restriction disqualifying many
ad hoc approximations. It is of essential importance that mean-field approximations
determine a free energy functional either in the form (2.10) or (2.15).
Free-energy functional (2.10) will be used to derive exact solution in the limit d →
∞. We see that the free energy as a functional of renormalized quantities G and Σ has
a variational character, i.e. the unknown functions or parameters are determined from
stationarity equations (2.12). Such a form of the free-energy functional is mandatory
in order to determine stability of various thermodynamic phases and to fulfill usual
thermodynamic relations. From the defining equations (2.10) and (2.11) we can also
deduce conditions on stability
ReG−1 > 0 , Re G−1 + Σ > 0
(2.16)
that follow from the convergence of gaussian integrals. This stability is connected with
second-order variation
δ2 F
>0
(2.17)
δmδm∗
telling us that the free energy is a local minimum in the configuration space of the
variables m, m∗ .
Another important feature of the representations (2.10) and (2.15) is that the variational variables m and Σ represent sums of classes of Feynman diagrams from perturbation theory. The variable m, due to its symmetry, is just the sum of all connected tree
diagrams, where different lattice sites are connected by just one link. The self-energy Σ
represents the sum of all 1-particle irreducible diagrams. This co-ordination of classes
of diagrams to mathematical functions appears important for the derivation of an exact
solution in d = ∞. However m and Σ are not independent. They are related by a Ward
identity defined by (2.3) and (2.7).
We derived here exact representations for the free energy of renormalized perturbation expansion in the simplest form. I.e. we included only simplest possible symmetry breaking in the form of a ”spontaneous magnetization” m. There can of course
CHAPTER 2. THE LIMIT D → ∞
18
be other symmetries relevant in the exact solution. These can be introduced into the
present scheme via new external sources analogously to h. Possible symmetry breaking terms must, however, always be inserted into the Hamiltonian at the beginning and
cannot be judged from the solution itself. New external sources may make the freeenergy expressions more intricate, especially in quantum statistical mechanics, but do
not change the main line of derivation of the exact expression. Hence the expression
for the free energy must be understood as exact only within a symmetry allowed by
the chosen configuration space.
2.2
Classical lattice models
2.2.1
Ising and classical Heisenberg models
The first step in the construction of the limit of high spatial dimensions we have to
perform is dimensional analysis of relevant, physical quantities. Particularly, we have
to find the dependence of the free energy on the number of nearest neighbors of the
underlying lattice. The coordination number Z of the lattice is the most important
parameter specifying the limit of high dimensions. Although it is possible to consider
various lattice types in higher dimensions we will use, for the sake of simplicity, only
the hypercubic one, Z = 2d.
The only internal input parameter in the Ising (Heisenberg) model is the spin-spin
coupling J. There is also an external characteristic, magnetic field h. We will assume
only the nearest-neighbor coupling. A generalization to farther neighbors is straightforward. Apart from the input parameters there are also output parameters resulting
from solving the equation of state of the model. These are internal characteristics of the
ground state that need not share the symmetry of the Hamiltonian. In the case P
of the
z
−1
(ferromagnetic) Heisenberg model it is homogeneous magnetization m = N
i Si ,
where N is the number of lattice sites. The three parameters J, h, and m determine the
relevant physical quantities with the help of which we can express the energy of the
Heisenberg model
D
E
X
X
cHE = −J
E= H
Si ·Sj − h·
hSi i .
<ij>
(2.18)
i
Here we denote the thermal average with the brackets h i without subscript. We assume the magnetic field oriented along the z-axis. We then can rewrite the energy
E = −J
"
X
<ij>
X
D E X D z z E z D z E X D ⊥ ⊥ E
Szi
+
Si ·Sj − h
Si Sj − Si Sj
Szi Szj +
<ij>
o
n
= −N JZm2 + J 2 Zχk + J 2 Z(D − 1)χ⊥ + hm .
<ij>
#
i
(2.19)
2.2. CLASSICAL LATTICE MODELS
19
Here Z = 2d is the number of nearest neighbors, D the spin dimension, and χk =
z z z z P hD ⊥ ⊥ E ⊥ ⊥ i
1 P
1
,
S
−
S
S
S
χ
=
S0 ·S1 − S0 · S1 are averaged cor1
1
0
0 1
1
⊥
JZ
JZ(D−1)
relation functions for nearest neighbors. The parameters h, χk , χ⊥ , and m remain finite in the high-dimensional limit, Z → ∞. Hence to keep the energy density of the
Heisenberg model finite for any temperature and magnetic field, we have to scale the
spin-spin coupling with increasing Z as
J = J ∗ /Z
(2.20)
where J ∗ is a new energy scale for the limit of infinite coordination number Z. Only
with this scaling the energy density from (2.19) remains finite during the limiting process. Other scalings lead to either trivial (energy density vanishes) or singular (energy
density diverges) models, at least in some cases.
We can now perform the limit of the Heisenberg model to infinite dimensions. In
each diagram of the linked cluster expansion we have a factor Z for any sum over the
nearest neighbors and a factor Z−1 for each link (spin-spin coupling). Then whenever
arbitrary two nearest neighbors in a diagram are connected by two or more links, the
contribution to the energy density is of power Z−1 or smaller. As a consequence we
obtain that in d = ∞ arbitrary two distinct sites are connected maximally with a single
link and the solution is equivalent to the long-range model, Stanley (1971). The diagrammatic expansion reduces to tree diagrams only. There are no closed loops in the
diagrammatic expansion, since the diagonal element containing loops are proportional
at least to Z−1 and the nondiagonal elements at least to Z−3 .
To obtain the free-energy functional in d = ∞ we rewrite the partition sum of the
Heisenberg model via gaussian fluctuating fields, Brézin (1976), Janiš (1986)
)
(
Z
X
1 X −1 D
−βNµ/2 −1
ϕi Jµ
ϕj +
U (ϕi + βhi )
(2.21)
Z=e
Z0 Dϕ exp −
ij
2β i,j
i
where Jµ,ij = µδij + Jij and the effective interaction U is
U1 (ϕ) = ln 2 cosh(ϕ)
D
D≥2
(D−2)/2
−(D−2)/2
U (ϕ) = ln 2
Γ
I(D−2)/2 (ϕ)ϕ
.
2
Here Γ is the gammafunction and Iν is theBessel function of first kind. Further we de R
1 P
noted Z0 = Dϕ exp − 2β i,j ϕi Jµ−1 ϕj . We introduced an auxiliary positive numij
ber µ in order to make the matrix Jµ,ij positive definite. We can now use the free-energy
−1
representation derived in the preceding section (2.10). We put G(0)−1 ij = βJµ ij and
use real fields ϕ = ϕ∗ . There will be factors 1/2 in front of the logarithms. A first consequence of the scaling (2.20) is the locality of the self-energy Σ. Moreover, there are
CHAPTER 2. THE LIMIT D → ∞
20
no closed loops in the free-energy functional and hence there is no renormalization of
the diagonal element of the full correlation function G (propagator). The inverse full
propagator in leading order is
G−1
ij,α
= βJµ
−1
ij
− δij Σii,α .
(2.22)
We must take into account that the self-energy generally may depend on the spin projection α. Using (2.22) we obtain
−1
−1
G + Σ ij,α = βµ δij
and the self-energy and the renormalized propagator fall completely out from the freeenergy functional. Be aware that it does not mean that the self-energy vanishes. The
most general inhomogeneous free energy in d = ∞ then is
−βF [m;
e Σ, G] = −
+
X
i
X
−1
βNµ 1 X
1X
e j,α
ln βµ −
m
e i,α miα +
m
e i,α βJµ m
+
2
2 i,α
2 ij,α
i,α
(
)
∞
Z
D
D
D
Y
X
−1
dϕα
1X
ln
exp −
ϕα βµ ϕα −
mi,α ϕα + UD (ϕ + m
e + βhi ) ,
√
2
2π
α=1
α=1
α=1
−∞
(2.23)
−1
P
e j,α . The last term with the integral is the atomic solution
where miα = j βJµ ij m
of the Heisenberg model with a local spin-spin coupling βµ. If we now replace the
gaussian fluctuating fields by the real spins, m
e i,α by the magnetization mi,α and utilize
the normalization of the spin length to unity, the dependence of free energy on the
auxiliary parameter µ vanishes too. We are then left, in the homogeneous case, with
the Weiss mean-field free energy
(
)
Z
D
D
X
X
β
m2α + N ln dΓ(S) exp β
−βF [m] = − NJ ∗
(hα + J ∗ mα ) Sα
2
α=1
α=1
(2.24)
SD
The integration extends over the surface of a D-dimensional sphere.
The above reasoning may seem a rather complicated way to derive the Weiss meanfield solution, but it follows from the general scheme where the local element of the
correlation function renormalizes and does not generally disappear from the free energy. It is also important to mention that although the self-energy is not explicitly
contained in the free energy functional F [m], it is not zero. It is connected with the
magnetization by a Ward identity (2.3) and (2.7).
2.2. CLASSICAL LATTICE MODELS
2.2.2
21
Edwards-Anderson model of a spin-glass
The Edwards-Anderson model behaves differently in high dimensions than the homogeneous Ising model. If we exclude ferromagnetic long-range
order, i.e. the distribu
tion of the random spin-spin coupling Jij is unbiased, Jij = 0, then
D atEh = 0 the first
. Analyzing
nonzero contribution to the averaged energy is proportional to J 2 = Jij2
av
the first term of (2.19) we obtain for the gaussian distribution of the random couplings
X
Eav ∝ −
mi Jij mj av
<ij>
=
(
*
D E
Jij av hmi i2 − Jij2
av
δmj
mi
δJij
+ )
∝ NJ 2 Zψsg
(2.25)
av
D δm E
j
1 P
where ψSG = N
remains finite in the limit d → ∞. There are no contriij mi δJij
av
butions from odd powers of the coupling Jij and hence to keep the model nontrivial in
the limit d → ∞ we have to scale nearest neighbor spin-spin coupling as follows
√
(2.26)
Jij = Jij∗ / Z
where Jij∗ is an independent random (gaussian) variable with variance
D E
Jij∗2
av
= J ∗2 .
The above scaling leads to a consistent and nontrivial mean-field model called now
Sherrington-Kirkpatrick model, Sherrington and Kirkpatrick (1975). It is worth noting
that it was not a priori clear that just the scaling (2.26) leads to the appropriate meanfield model reproducing the observed behavior of experimentally realized spin glasses.
There was a stream of Japanese works attempting to construct a mean-field solution
via an inhomogeneous Weiss solution (2.24), Matsubara and Sakata (1976), Ueno and
Oguchi (1977), Takano (1980). An important aspect of disordered interacting systems
were missed in these papers. Namely that it is necessary to perform thermal averaging
for an inhomogeneous system and only then average over the randomness. The result
depends upon at which stage we introduce mean-field variational parameters.
It is worth mentioning that it is not quite crazy to apply the scaling (2.26) also to the
homogeneous Ising model. We only have to suppress magnetization in the variational
description. Hence scaling (2.26) can be applied only to the paramagnetic phase in the
absence of the external magnetic field. An approximation for the Ising model resulting
from (2.26) is then Brout’s spherical approximation, Brout (1960), (1961). It is consistent anywhere in the paramagnetic phase (h = 0). However, if we introduce magnetic
field the solution breaks Ward identity (2.3), (2.7). This leads to unphysical behavior
and to a spurious first-order transition in dimensions d < 4, Englert (1963). We cannot
hence extend the Brout’s approximation to the whole range of the input parameters,
CHAPTER 2. THE LIMIT D → ∞
22
since the magnetization must not be treated within renormalized perturbation theory,
otherwise it would generate infinite energy (when considered as the limit of d → ∞).
Nevertheless the problem of consistency of Brout’s approximation can be solved if we
treat the magnetic field perturbatively (for details see Appendices A1- A3).
The most important thing in the derivation of the solution of the SherringtonKirkpatrick model is not to average over the random configurations simultaneously
with averaging the thermal fluctuations, i.e. not to try to introduce global, averaged
mean-field parameters when summing the diagrams. We hence start with the general
formula for the free-energy functional (2.10) that in this particular case explicitly reads
h
i
−1
1
βNµ
1
1 f
− tr
m
e βJµ m
e + ln 1 − βJµ Σ + ln G
−βF m,
e H; G, Σ = −
2
2
2
2
(
)
Z
X
X
−1
1
fi ϕi +
+ ln Dϕ exp − trϕ G + Σ ϕ −
H
ln 2 cosh(ϕi + m
e i + βhi ) . (2.27)
2
i
i
f
The trace is taken over the lattice space. All the renormalized variables G, Σ, m,
e H,
as well as the coupling Jµ are configurationally dependent. The ”chemical potential”
has to be chosen so that µ > J ∗2 . Formula (2.27) is valid in any dimension. We now
introduce simplifications following from the scaling (2.26).
It is clear that the exact solution of the Sherrington-Kirkpatrick model contains diagrams where distinct sites are connected by maximally two links. Only simple loops
contribute to the free energy and the self-energy becomes local. Odd powers of the
bonds must vanish due to the symmetry of the random distribution and only squares
Jij2 matter in the loop contributions. We then have only tree-like diagrams in such
a theory. First we have genuine trees where distinct sites are connected by simple
links. These√diagrams contribute to the local magnetization and lead to the asymptotics mi ∝ Z. Note that hmia v = 0. Second we have loops, contributing to the
self-energy, that collapse to trees with doubled links (self-retrieving paths). The
√ diagonal element of the self-energy is of order 1 and the off-diagonal of order 1/ Z3 .
Since odd powers can couple only to the magnetization, the final contribution of the
nonlocal self-energy to the free energy is of order Z−1 .
To find the exact free-energy functional in d = ∞ we can neglect nonlocal elements
of the self-energy Σ but we must keep track of the local as well as of the nonlocal
parts of the full propagator G. We denote the nondiagonal part of the full propagator
e The only nonlocal elements of G, i.e. G,
e in (2.27) are contained in products of
as G.
the magnetizations and the logarithms. Anyway, the 1-particle irreducible diagrams
contain only the local elements of G that we denote G. If we further denote Λ(G, Σ) the
sum of all the 1-particle irreducible diagrams, the functional integral in (2.27) can then
be represented as
i
h
1
− tr ln G−1 + Σ − Λ(G, Σ) =
2
2.2. CLASSICAL LATTICE MODELS
23
h
i 1
h −1
i
1
1
e
tr ln G + G − tr ln G − tr ln G + Σ − Λ G, Σ .
(2.28)
2
2
2
The first logarithm on the right-hand side of (2.28) compensates −1/2tr ln G from (2.27)
−1
and the last logarithm can be expressed as a local integral with G + Σ analogously to
(2.27). We removed in this way the nonlocal part of the full propagator, that is not
renormalized in the mean-field theory, from the free-energy functional. This is always
the case in d = ∞, and we have only local variational loop parameters G an Σ in the
free-energy functional.
Since only squares of Jij contribute to the free energy we can transform the trace of
the logarithms into a sum over lattice sites. Using a trick with differentiation enabling
to change the trace of a logarithm to a sum of diagonal elements of a propagator we
generally have for a local variable Xi in this model, Janiš (1989),
#
"
X
X
2
2
1X
βJij χj
ln 1 − Xi
χi βJij χj +
tr ln(1 − βJX) =
2 ij
j
i
(2.29)
where the variable χi minimizes the right-hand side of (2.29). We use (2.29) to evaluate
the
ei =
P only remaining nonlocal logarithm in (2.27). We further use a substitution m
j βJµ ij mj , where mi is the Ising magnetization. We then obtain a new free-energy
functional, Janiš (1989)
h
i
X
2
βNµ
1
1
f χ; G, Σ = −
−βF m, H,
mi βµδij + βJij mj + χi βJij χj
−
2
2 ij
2
( "
)
#
∞
X
X Z dϕ
2
1X
Σi
−
ln 1 −
βJij χj + ln 1 − βµΣi + ln Gii +
ln √
2 i
1 − βµΣi j
2π
i
−∞
(
i
1 h −1
fi ϕ + ln 2 cosh
exp − ϕ Gii + Σi ϕ − H
2
ϕ + βµmi +
X
βJij mj + βhi
!)
.
(2.30)
j
We can now get rid of the µ-dependence in the free-energy functional by a substitution,
Janiš (1989),
Σi
σi =
.
(2.31)
1 − βµΣi
fi , and mi we find out that also σi and Gii
If we use the saddle-point equations for Gii , H
disappear from the free energy. We are then left with a free energy with χi and mi as
variational parameters:
2
2
1
1X
2
mi + 1
mi βJij mj + χi βJij χj − χi βJij
−βF m, χ = −
2 ij
2
CHAPTER 2. THE LIMIT D → ∞
24
+
X
i
ln 2 cosh
βhi − mi
X
βJij
j
2
χj +
X
βJij mj
!
,
(2.32)
j
which is essentially the TAP free energy, Thouless, Anderson, and Palmer (1977). The
exact TAP free energy is derived from (2.32) by excluding χi = 1 − m2i from it.
The TAP free energy (2.32) is an exact expression for the free energy of the SherringtonKirkpatrick model for one configuration of the spin-spin coupling Jij . It contains only
local variational parameters mi and χi but they are site dependent. Hence such a meanfield theory makes no sense unless the number of degrees of freedom is reduced. We
can either try to solve the TAP equations for the local magnetizations numerically or
average the free energy over the random configurations. Neither of the procedures is
simple and obvious. In the former approach, Bray and Moore (1979), we obtain an estimate on the ground state and the lowest eigenvalue of the TAP equations and in the
latter technique, if we assume the Brout’s ergodicity argument to be fulfilled, we derive
the Sherrington-Kirkpatrick solution with a single Edwards-Anderson mean-field parameter, Sommers (1978). With this solution we run into troubles in very low temperatures, since the entropy becomes negative, Sherrington and Kirkpatrick (1975). The
only way to bypass the low-temperature inconsistency in the Sherrington-Kirkpatrick
model is to assume that ergodicity is broken in the spin-glass phase. However, in
such a situation we are unable to reduce infinite-many relevant mean-field degrees
of freedom to a finite number, Palmer (1982). There has not yet been found exact
and ansatz-free averaging of the TAP equations. By now the Parisi replica-symmetry
breaking scheme, Parisi (1980), is considered as an exact averaged free energy of the
Sherrington-Kirkpatrick model, Binder and Young (1986). The ergodicity breaking and
averaging of the TAP free energy go beyond the scope of this review.
2.3
2.3.1
Quantum itinerant models
Anderson model of disordered electrons
The simplest but still nontrivial quantum itinerant or tight-binding model in the construction of mean-field theories is the Anderson model of noninteracting electrons in
an external, static random potential (1.7). At this stage we choose the alternative with
diagonal disorder only. That is the hopping matrix is assumed to be nonrandom. It is
the only nonlocal term to be scaled in the limit d → ∞.
The Anderson and other quantum models differ from the classical models studied
in the preceding section in that that they have a new quantum term of kinetic energy.
This operator contains annihilation and creation operators at different lattice sites and
has no classical analogue (only particle densities have classical counterparts). It is
hence a priori not clear how to scale the hopping matrix in the limit of high spatial
dimensions. To find out the scaling law we look at the averaged kinetic energy. It is
2.3. QUANTUM ITINERANT MODELS
25
easy to show, Janiš and Vollhardt (1992a),
E
X
XD †
Ekin = −t
ciσ cjσ
= −it
Gij,σ (0+ ) ∝ Nt2 Z
<ij>σ
av
(2.33)
<ij>σ
D h
iE
†
where Gij,σ (t) = −i T ciσ (t)cjσ (0) is the time-dependent Green’s function, eventually the averaged Green’s function. The scaling of the hopping amplitude between
nearest neighbors follows from (2.33):
√
t = t∗ / Z .
(2.34)
It was derived for the first time by Metzner and Vollhardt (1989) in the context of the
Hubbard model.2 To be sure a sensible model comes out from the scaling (2.34) we
need to show that physical quantities do not depend on odd powers of the hopping
amplitude. Consistency of (2.34) follows from the invariance of quantum models with
respect to the local gauge transformation changing the phase of the wave function, i.e.
of creation and annihilation operators in second quantization.
We see that the scaling of the quantum kinetic energy is formally identical with
the scaling in the classical Edwards-Anderson model (2.26). However, the reasons for
such a scaling in the quantum case are different and also different types of Feynman
diagrams (loops) contribute. First of all it is not true that only squares of tij would
contribute to the free energy. It means that not only loops with self-retrieving paths
are important. We can have true loops in the geometric sense where no bond in the
loop contributes twice. We hence have quite a different free energy due to the kinetic
energy than in the spin-glass model with random couplings. We must hence newly
prove that the scaling (2.34) leads to a finite free-energy density derived from a linked
cluster expansion for this model.
We first show that closed loops from the hopping elements have asymptotics O(d0 )
in the limit d → ∞. We consider only hypercubic (bipartite) lattice. Then any loop, or
a random walk with the step represented by the hopping amplitude tij , consists of an
even number of steps, say 2l. In any axis of the d-dimensional hypercubic lattice we
define positive and negative direction. This can be done for each coordinate independently. Since we consider a closed loop, there must be the same number of positive
and negative movements in each coordinate. The loop under consideration has 2l vertices (lattice sites) that all should be geometrically different. To construct a loop of the
length 2l we choose l vertices where the next step in the random walk with the step
tij goes in positive direction. We have 2ll possibilities to do so. In any such vertex
we have d possible directions, one for each dimension to choose. However, to obtain
a single closed loop we have to avoid possible returns to the sites already visited by
2
There is also an analogy to the long-range Ising model with linear scaling of the hopping matrix,
van Dongen and Vollhardt (1989). Such a model, however, is fairly trivial, since only the mode with the
wave vector q = 0 is relevant.
CHAPTER 2. THE LIMIT D → ∞
26
the random walk. That is, we have to consider a self-avoiding random walk. In the
limit of high spatial dimensions there is, however, no difference between the random
walk and the self-avoiding random walk in leading order of d−1 (multiple returns are
at least of order d−1 ). We hence can sum without restrictions over all nearest neighbors
in positive directions at the chosen vertices. To each positive direction we must choose
negative direction in the same coordinate. For the fixed order of l positive directions
we have just l! possibilities
to choose negative counterparts. Finally for each bond
√
∗
we pay a factor t / 2d. We hence have a contribution due to the number of possible
realizations of a loop of length l
!
2l
Xl (t) =
l!t∗2l (2d)−l dl = (2l − 1)!!t∗2l = O(d0 )
(2.35)
l
Hence any closed loop where two distinct lattice sites are connected by maximally
two paths is relevant for the free energy. The higher order loops, where arbitrary two
distinct lattice sites are connected by more than two paths, vanish in the limit d → ∞.
Adding a new link to a simple loop means to pay at least with a factor d−1/2 whereby we
do not win any new freedom due to summation over nearest neighbors. It hence means
that higher-order loops contribute to the self-energy nontrivially if only the vertices
collapse to one lattice site. We hence speak of a ”collapse of diagrams” or a mapping
of perturbation theory in d = ∞ onto a single-site problem.
We can use (2.35) to evaluate the contribution of a closed loop to the free energy in
d = ∞. We first introduce the one-particle Green’s function
h
i−1
G(z) = tr z1b − bt
(2.36)
where z is a complex number (energy, frequency). The trace represents a sum of all
simple closed loops so we can use (2.35) for the right-hand side of (2.36)
h
i−1
X
tr z1b − bt = z−1
Xl (t)z−2l .
l
To reduce the summations over the whole d = ∞ lattice we rewrite the sum over the
closed loops as an integral over the density of states (DOS). It is easy to show that
Xl (t) = √
∞
Z
1
2πt∗
d2l exp −2 /2t∗2
−∞
and hence we obtain an integral representation for the Green’s function
G(z) = √
1
2πt∗
∞
Z
−∞
d exp −2 /2t∗2
1
.
z−
(2.37)
2.3. QUANTUM ITINERANT MODELS
27
We used the symmetry of the density of states and vanishing of the integrals with odd
powers of . We use (2.37) later on to evaluate the free energy of quantum models in
d = ∞.
We derived (2.37) entirely from arguments in the direct space. There is another
elegant way to derive the DOS in d = ∞ using Fourier transform and momentum
space, Müller-Hartmann (1989). We start with the quantum operator of kinetic energy
X
†
ckin =
H
(k)ckσ
ckσ
(2.38)
k,σ
P
P
where ckσ = i ci exp {ik · Ri }, (k) = −t η exp ik · aη with aη = Ri+η − Ri is a unit
vector to the nearest neighbor in the η direction. The dispersion relation for the hypercubic lattice reads
d
2t∗ X
cos kη
(k) = − √
2d η=1
where the projections of the wave vector lie in the interval [−π, π]. The density of
states is defined
X
ρd () =
δ ( − (k)) .
(2.39)
k
We rewrite (2.39) using an integral representation of the δ-function
1 X
ρd () =
N k
∞
Z
du iu(−(k))
e
.
2π
(2.40)
−∞
We interchange the order of summation and integration and obtain
ρd () =
∞
Z
−∞
π

∞
Z
d Z
d
∗
du iu  dk
2iut
du iu
u2 t∗2
−2

e
exp √ cos k
=
e
1−
+ O(d )
2π
2π
2π
2d
2d
−π
−→
d→∞
−∞
∞
Z
−∞
du iu −u2 t∗2 /2
1
e e
=√
exp −2 /2t∗2
2π
2πt∗
where we used the central-limit theorem. We reproduced the result for the DOS from
(2.37).
Once we have evaluated the sum over closed loops we can derive the (averaged)
free energy for quantum models. The grand potential due to the kinetic energy of a gas
of noninteracting electrons is
Ωkin = −β
−1
∞
X
n=−∞
eiωn 0
+
X
k,σ
ln iωn + µσ − (k)
CHAPTER 2. THE LIMIT D → ∞
28
∞
∞
XZ
N X
iωn 0+
dρ() ln iωn + µσ − .
e
=−
β n=−∞
σ
(2.41)
−∞
where µσ = µ + σh is the chemical potential and external magnetic field, ωn = (2n +
1)πβ−1 . We took into account fermionic structure of the electrons reflected in the sign
of the logarithm (η = −1).
We have now all prerequisites necessary for derivation of the averaged grand potential of the Anderson model of disordered electrons. Since there is no electronelectron interaction we can write the grand potential for one configuration of the random atomic energies per one spin
∞
h
i
X
+
eiωn 0 tr ln iωn + µ 1b − bt − εb
(2.42)
Ω = −β−1
n=−∞
where hats denote matrices in the site indices. The inhomogeneity of the random energies i in combination with the nonlocal hopping tij makes the Anderson model of
disordered electrons insoluble in realistic dimensions. We now use the general expression for the free-energy functional (2.8) with m = 0 and a coherent potential Σn . It is
our variational parameter together with the local element of the averaged one-electron
Green’s function Gn . They contain the entire effect of averaging over the random configurations of the atomic energies onto the homogeneous thermodynamic quantities.
Using (2.41) and (2.37) to evaluate the trace we have
−1
N Ωav [Gn , Σn ] = −β
−1
∞
X
eiωn 0
+
n=−∞
∞
Z

−∞


.
dρ∞ () ln iωn + µ − Σn − + ln Gn + ln G−1
+
Σ
−
ε
n
i av
n

(2.43)
This is the grand potential in the coherent-potential approximation, Janiš (1989). It
was now derived as the exact grand potential for the Anderson model of disordered
electrons in d = ∞. Apart from the scaling of the nearest-neighbor hopping amplitude
(2.34) we did not use any particular property of perturbation theory.3 Stationarity
equations from (2.43) lead to the CPA-equations
Gn =
∞
Z
dρ∞ ()
−∞
1=
3
1
iωn + µ − Σn − 1
1 + Gn (Σn − εi )
.
(2.44)
(2.45)
av
Perturbation theory here means an expansion of the grand potential in the random energies εi and
averaging each term of the expansion independently. There exist several diagrammatic representation
of the expansion in the random potential, Elliot, Krumhasl, and Leath (1974), Janiš (1986).
2.3. QUANTUM ITINERANT MODELS
29
We can now analytically continue the set of complex numbers Gn and Σn from the
Matsubara frequencies iωn with the accumulation point at infinity to analytic functions
in the upper and lower complex half-planes. This analytic continuation is unique provided the Gn , Σn → 0 if n → ∞. The newly defined complex functions must further
fulfill the so called Herglotz, analyticity properties
Σ(z∗ )∗ = Σ(z) ∧ ImΣ(z) ∝ −sgn(Imz)
G(z∗ )∗ = G(z) ∧ ImG(z) ∝ −sgn(Imz) .
(2.46)
It is essential for quantum mean-field theories that the one-electron propagator G(z)
and the self-energy Σ(z) as functions of complex energy z possess the Herglotz analyticity (2.46) in the whole upper and lower complex half-planes. Otherwise we cannot
exclude unphysical behavior due to singularities in complex energies. One may wonder where do the Herglotz properties (2.46) come from. They are realizations of the
general stability conditions (2.16). Namely the gaussian integration cannot be defined
directly for the Matsubara frequencies. We actually have to use a real-frequency representation of the sum over the Matsubara frequencies
β−1
∞
X
n=−∞
eiωn
0+
−1
1
Gn + Σn = −
π
∞
Z
dωf(ω)Im G−1 (ω + i0+ ) + Σ (ω + i0+ )
−∞
with the Fermi function f and label the fluctuating fields by the real frequencies. We
then obtain the Herglotz properties along the real axis immediately from (2.16)
(2.47)
Im G−1 (ω + i0+ ) + Σ (ω + i0+ ) > 0 .
It is insufficient to prove the analyticity only for the solution along the real axis. The
analyticity in the whole upper (lower) complex half-plane is necessary in order to perform unique analytic continuation of G and Σ from the real energies to the imaginary
Matsubara frequencies.
The CPA equations (2.44) and (2.45) are well-known example of Herglotz analyticity as proven by Müller-Hartmann (1973). In this approach the Herglotz properties are
consequence of consistency of the theory derived as an exact solution in d = ∞. The
main difference between classical and quantum mean-field theories lies in the global
analytic properties. While at the classical level the solution should be consistent along
the positive temperature axis, quantum mean-field solutions must fulfill Herglotz analyticity for all complex energies. Quantum theory has a richer content than a classical
one and not necessarily all physical quantities can be derived directly from the (averaged) grand potential. There are dynamical effects such as electrical conductivity,
nonequilibrium excitations, etc. that go in description beyond the equilibrium thermodynamics. In this model the averaged conductivity and the problem of localization of
electrons cannot be derived from the averaged one-particle Green’s functions and the
30
CHAPTER 2. THE LIMIT D → ∞
grand potential. The averaging of products of Green’s functions is inevitable and such
a problem deserves a separate treatment, Velický (1969), Janiš (1988).
The Anderson model of disordered electrons lies between the classical spin models
and quantum many-body systems. On the one side we must use infinite many Matsubara frequencies to describe the equilibrium thermodynamics and the generating
grand potential. On the other side the quantum degrees of freedom do not mix in the
equations (2.44) and (2.45). We hence can treat the Matsubara frequencies separately.
The equations of state at equilibrium become equivalent separately to a classical thermodynamic problem. This is, however, no longer the case if we go over to two-particle
functions (see Appendix A2).
2.3.2
Tight-binding models of correlated electrons
Comprehensive mean-field theory of tight-binding models of correlated electrons constructed via the limit d → ∞ is our ultimate objective. We have already exposed most of
the aspects of this special limit in preceding sections and can almost directly construct
the grand potential of models with correlated electrons in d = ∞. From the Anderson model we now know how to scale the kinetic, hopping energy and from classical
spin models we know how to scale nonlocal coupling. We are able to reduce the lattice
summations onto integrals with the density of states, i.e. to map the theory in d = ∞
to a single-site theory with local variational parameters Gσn and Σσn . Many-body fluctuations due to interparticle correlations may, however, lead to new low-temperature
phases breaking the symmetry of the underlying Hamiltonian. We hence have to modify the grand potential in such a way to allow for symmetry breakings leading to the
state with the lowest energy at zero temperature. The most important symmetry breaking is that leading to spin-density or charge-density waves. Although the mean-field
theory of the d = ∞ model effectively reduces to a single-site dynamical theory, it is
possible to describe within it also solutions with broken translational invariance. It is
our next step to modify the exact free energy (2.10) to the case with broken translational invariance.
First of all, in fermionic systems the fluctuating fields Φ are Grassmann variables
and hence m = m∗ = 0. Let us assume the simplest breaking of translational symmetry,
where we have to distinguish two equivalent sublattices A, B. Within these sublattices
we suppose the translational symmetry to be restored. We now write the full lattice
as a Cartesian product of the two sublattices L = A × B. We separately label the sites
from the sublattice A and B. Since the subllatices are equivalent, we can find a oneto-one mapping A → B. To each site from the sublattice A we ascribe a site in the
sublattice B. We replace a single site i from the sublattice A with a couple of sites
(i, i0 ) ∈ A × B. The couples (i, i0 ) can be considered as a 2-dimensional vector space.
The lattice summations can be reduced to the sublattice A (B). Consequently all the
operators become 2 × 2 matrices. E.g. the matrix of the inverse Green’s function of
2.3. QUANTUM ITINERANT MODELS
31
noninteracting electrons with nearest-neighbor hopping is
iω
+
µ
−t
n
(0)−1
b
G
=
.
−t
iωn + µ
The full grand potential with the renormalized Green’s functions and self-energies is
h
i
h
i
n
o
b Σ
b (0)−1 − Σ
b − βΩ G
b −1 + Σ
b = tr ln G
b − tr ln G
b .
−βΩ G,
(2.48)
Hats denote 2 × 2 matrices and tr runs only over the sublattice A. This is a relatively
trivial change and does not restrict the validity of (2.48) for the chosen symmetry. Only
if we perform the limit d → ∞ expression (2.48) simplifies, since the variational paramb and the self-energy Σ
b
eters become local. Hence the renormalized Green’s function G
become diagonal matrices. Evaluating the trace in the matrix, sublattice, and frequency
space we finally obtain a general exact grand potential in d = ∞
∞
X X
1 X −1
2
+
eiωn 0
Ω [G, Σ] =
Ω Gα + Σα − β−1
N
N α=A,B
σ n=−∞

∞

Z
X
2
ln Gαn .
dρ∞ () ln iωn + µσ − ΣAσn iωn + µσ − ΣBσn − +


α=A,B
(2.49)
−∞
Specific models of interacting and eventually disordered electrons differ only in the
first term of the representation (2.49). It is exact in d = ∞ provided the translational
symmetry is broken maximally
into
n
o two equivalent sublattices A, B. We now con−1
b
b
cretize the functional Ω Gα + Σα to the models of interest.
Spinless fermions
The simplest model of correlated electrons that show a long-range order are spinless
fermions. They are represented by microscopic models with spin-independent interactions. Only nonlocal correlations can be realized in spinless fermions. Because of
the Pauli principle the on-site (Hubbard) electron-electron interaction must be spin
dependent. The simplest interaction is a density-density correlation between nearest
neighbors the energy of which is
+
*
D E
X
ni i2 .
(2.50)
Vb =
Vij n
bin
b j ∝ V NZ hb
<ij>
Hence the nearest-neighbor interaction scales in the same way as the spin-spin coupling in the Ising model. It must be so, since the scaling must obey the correspondence
principle and must not depend upon whether we treat a classical or a quantum model.
The particle densities here correspond to the spins in the Ising model.
CHAPTER 2. THE LIMIT D → ∞
32
An exact grand potential of spinless fermions in d = ∞ follows from (2.49), (2.50)
and linear scaling of the nearest-neighbor density-density coupling V = V ∗ /Z , i.e.
∞
X
2
+
∗
−1
eiωn 0
Ω [G, Σ] = −V nA nB − β
N
n=−∞
+
∞
Z
−∞
(
X
ln [1 + Gαn (Σαn − V ∗ nα0 )]
α=A,B


dρ∞ () ln iωn + µ − ΣAn iωn + µ − ΣBn − 2

(2.51)
where α0 6= α. The densities nA , nB are variational parameters corresponding to the
magnetization in the Ising model. We immediately see that Gαn and Σαn are constants
and do not depend on the Matsubara frequencies iωn . The grand potential (2.51) is
just the Hartree approximation, another consistent mean-field approximation. It can
be simplified in that we get rid of the ”loop” variables G and Σ using stationarity
equations
2
Ω [n] = −V ∗ nA nB
N
∞
∞
P
+ R
eiωn 0
dρ∞ () ln iωn + µ − V ∗ nA iωn + µ − V ∗ nB − 2 .
−β−1
n=−∞
(2.52)
−∞
Spinless fermions in infinite dimensions represent a classical thermodynamic theory
with only static variational parameters, particle densities nA,B . Such a theory does not
contain real quantum fluctuations and many-body correlations. Although grand potential (2.52) leads to a globally consistent mean-field theory, it is reliable only at weak
coupling V ∗ /t∗ 1. It is clear from the fact that in (2.52) only first order of perturbation theory in V is exactly reproduced. We cannot expect quantum many-body effects
in the mean-field theory of spinless fermions. Nevertheless a rich phase diagram is
obtained in this mean-field model away from half filling, Uhrig and Vlaming (1993).
Falicov-Kimball model
First mean-field theory containing many-body correlations is the Falicov-Kimball model
in d = ∞. This model was the first exact solution of quantum itinerant models in d = ∞
derived by Brandt and Mielsch (1989), (1990), (1991). The full grand potential with
the renormalized variational parameters was constructed by Janiš (1991). The spinless Falicov-Kimball Hamiltonian (1.12) does not contain any nonlocal interaction or
hybridization
and hence no scaling of the coupling term is necessary. The functional
−1
Ω Gα + Σα is an atomic solution the partition sum of which contains a sum over
two possible states of the local electrons. The dynamic electrons have a frequency−1
. The atomic partition sum can explicitly be evaldependent propagator G−1
αn + Σαn
2.3. QUANTUM ITINERANT MODELS
33
uated (Appendix A4). We obtain the grand potential
∞
h
i
X
X
2
+
β(µ−Eα )
−1
Ω [G, Σ] = −β
eiωn 0
ln 1 + e
−β
N
n=−∞
α=A,B
+
∞
Z
dρ∞ () ln
iωn + µ − ΣAn
(
X
iωn + µ − ΣBn
−∞
with
Eα = −β
−1
∞
X
e
iωn 0+
ln 1 −
n=−∞
ln [1 + Gαn Σαn ]
α=A,B
UGαn
1 + Gαn Σαn


2
−

(2.53)
.
(2.54)
Conditions on stationarity of the grand potential (2.53) lead to defining equations for
the variational parameters Gαn and Σαn . We have after a few manipulations (Appendix
A4)
∞
Z
iωn + µσ − Σα0 n
(2.55)
Gαn = dρ∞ ()
iωn + µ − ΣAn iωn + µ − ΣBn − 2
−∞
Unα
1 + Gαn (Σαn − U)
(2.56)
1
1 + exp β Eα − µ
(2.57)
Σαn =
where
nα =
is the averaged number of static particles in the Falicov-Kimball model. We see that the
variational parameters Gαn and Σαn now depend explicitly on the Matsubara frequencies and the thermodynamics of the model contains a portion of quantum many-body
fluctuations. However, equations of motion are algebraic. The variables Gαn and Σαn
depend only on one Matsubara frequency ωn . This situation resembles the mean-field
solution of the Anderson model of disordered electrons (2.44), (2.45). It can namely be
easily shown that if we consider a random-alloy with a diagonal disorder where the
constituent A with the atomic energy U has concentration x and the constituent B with
concentration 1 − x has the atomic energy 0, then equations (2.45) and (2.56) are formally equivalent if we identify nα = x. The Falicov-Kimball model is again a semiclassical model with reduced dynamical quantum fluctuations. The dynamical fluctuations
are restricted, since we have only one species of dynamical electrons. They interact
with static electrons, i.e. they are scattered on static impurity potentials distributed in
the lattice. The semiclassical character of the Falicov-Kimball model becomes evident
from the fact that the partition function of this model can be obtained in any dimension
as a static approximation in a special functional-integral representation, Janiš (1994).
Equations (2.55) and (2.56) coincide with the well-known Hubbard-III approximation, Hubbard (1964), if we neglect the static electrons and replace the density of static
CHAPTER 2. THE LIMIT D → ∞
34
particles by the density of the dynamic ones. Then of course the equation (2.57) must
be forgotten and replaced by a sum rule. Analogy between the model of random alloys and the Hubbard-III approximation, disclosed in Velický, Kirkpatrick, and Ehrenreich (1968), led in the end of the seventies to a numerous attempts to improve on
the weak-coupling Hartree-Fock theory using the ”alloy analogy” reasoning, Moriya
(1985), Czycholl (1986). However, mean-field theories constructed in this way and
based on the Hubbard-III approximation become thermodynamically inconsistent and
lead to unphysical behavior, Kawabata (1972). In the next chapter we show how the
exact solution of the Falicov-Kimball model in d = ∞ can be used to construct a thermodynamically consistent mean-field theory of the Hubbard model at strong coupling.
Hubbard model
The Hubbard model is the most important model of correlated electrons and the sample model for the construction of a mean-field theory with dynamical fluctuations. The
Hubbard on-site interaction, contrary to the nonlocal interaction of spinless fermions,
does not undergo any simplification in the limit of high spatial dimensions. Hence
the Hubbard model in d = ∞ remains a fully dynamical model where two types of
electrons, distinguished by the spin projection, dynamically scatter on each other. The
Hubbard model maintains the essential features of the interacting electrons also at the
mean-field level. However, we pay for this quality of the mean-field theory of the
Hubbard model. The exact mean-field grand potential contains infinite-many variational parameters
−1 that are coupled in mean-field equations of motion. The single-site
functional Ω Gα + Σα can be represented as a functional integral only and cannot be
reduced to an analytic expression in closed form. The mean-field theory of the Hubbard model is insoluble by pure analytic means.
The grand potential of the Hubbard model in d = ∞ can be written as, Janiš (1991),
(
Z
∞
X
X X
−1
2
∗
Gασn + Σασn ψασn
Ω [G, Σ] = β−1
ln DψDψ ∗ exp
ψασn
N
σ n=−∞
α=A,B

(
∞
Z
∞

X X
X
iωn 0+
−1
∗
∗
e
ln Gασn
−U dτψα↑ (τ)ψα↑ (τ)ψα↓ (τ)ψα↓ (τ) − β

σ n=−∞
α=A,B
0
+
∞
Z
−∞
∗
dρ∞ () ln
iωn + µσ − ΣAσn
iωn + µσ − ΣBσn


2
.
−

(2.58)
Here ψ and ψ are Grassmann variables and the integral is a Grassmann functional integral. This integral cannot be evaluated in full analytically, since the two terms in the
exponent do not commute and we cannot factorize functional integration into product
of finite-dimensional integrals. It is the reason for the presence of quantum fluctuations in the mean-field solution of the Hubbard model. Quantum degrees of freedom,
2.3. QUANTUM ITINERANT MODELS
35
Matsubara frequencies, are coupled in the functional integral and the equations of motion for the variational variable Σασn can be expressed again only in terms of functional
integration. The mean-field theory of the Hubbard model thus represents a (0 + 1)dimensional field (quantum) theory. The equations of motion in such a theory are
equivalent to the equations derived from the single-impurity Anderson model (1.13)
if the degrees of freedom of the conduction electrons are integrated out and we replace
the effective dynamical propagator of the f-electrons iωn +µ−Ef +iΓ(iωn ) → G−1
ασn +Σασn .
R
−1
Here Γ(iωn ) = dρ() iωn + µ − is the diagonal element of the Green’s function
of the conduction electrons. We then speak about a single-impurity model with an
additive, self-consistency condition binding the f-electron propagator with the renormalized propagator of the conduction electrons, Georges and Kotliar (1992). Hence
the SIAM can be assumed as a simplified version of the Hubbard model in d = ∞.
Since even the SIAM contains quantum-dynamical effects such as Kondo, we can expect nontrivial quantum effects also in the Hubbard model in d = ∞. Possibilities how
to turn the functional of the grand potential (2.58) to a tractable mean-field theory with
finite-many degrees of freedom will be discussed in the next chapter.
2.3.3
Interacting electrons in a random potential
In quantum models of interacting electrons in a random potential we reached the peak
of complexity of the model description of condensed matter. The simultaneous presence of an electron-electron interaction and disorder is expected to lead to new physical phenomena or at least to modifications of the effects known from either disordered,
noninteracting or nonrandom, interacting quantum systems. Although some of interesting effects such as Anderson localization go lost in the limit d → ∞, we nevertheless
expect nontrivial physics in the mean-field theory of interacting electrons subjected to
a static random potential. Mean-field theory as the limit d → ∞ for the interacting,
disordered electrons was formulated for the first time in Janiš and Vollhardt (1992b)
for the diagonal (on-site) random potential and in Dobrosavljević and Kotliar (1993)
for off-diagonal randomness. Since the averaged grand potential for a general random
potential has not yet been published, we derive it here.
There are no problems to treat the on-site disorder. Nondiagonal disorder in the
hopping amplitude is very analogous to the randomness of the Edwards-Anderson
model of a spin glass. In dimension d = ∞ only gaussian fluctuations can contribute,
since distinct lattice sites can be connected maximally by two links. We now distinguish the random and nonrandom parts of the hopping amplitude
(0)
tij = tij + etij
(0)
(2.59)
where tij is the nonrandom part. If εi are configurationally-dependent atomic energies,
CHAPTER 2. THE LIMIT D → ∞
36
we simplify the random hopping as
etij =
s
2
te∗
α(εi )α(εj )
2d
(2.60)
where α(εi ) is a function of the on-site random potential. The random hopping has
2
the variance te∗ /2d. This corresponds to the square-root scaling of the hopping matrix
and the random spin-spin coupling in the Edwards-Anderson model. Contrary to spin
glasses, we have here no ”magnetization”. The independent randomness in the hopping does not change the physics of the nonrandom Hubbard model. Only the dependence of the hopping amplitudes on the chemical occupation of lattice sites α(εi )α(εj )
influences the thermodynamics of the disordered Hubbard model. The randomness in
the hopping leads to an effective random potential
2
εi → εi + te∗ α(εi ) hα(εi )Gii,σn iav .
(2.61)
The averaged quantity contains the local element of the full, configurationally-dependent
propagator. It is a quantum analogue of the Onsager cavity field known from the theory of spin glasses. Since we want to get rid of all renormalized but configurationallydependent variables, we have to introduce a new homogeneous function Γασn which
will be treated variationally in the functional of the averaged grand potential.
There is no ergodicity breaking to expect in the random Hubbard model and hence
we can directly come over to homogeneous variational parameters (for details see Appendix A5). If we again distinguish the sublattices A and B and introduce the variational parameter Γ (corresponds to χ in the spin-glass model (2.29)) we obtain an exact
averaged grand potential
(
∞
X X
X 1 e∗ 2 2
2
iωn 0+
−1
e
Ω [G, Σ, Γ] = −β
ln Gασn + t Γασn
N
2
σ n=−∞
α=A,B
+
∞
Z
dρ∞ () ln
iωn + µσ − ΣAσn
iωn + µσ − ΣBσn
−∞
*
ln
Z
∗
DψDψ exp
(
∞
X X
σ n=−∞
−U
∞
Z
0
∗
ψασn


X
2
− β−1
−

α=A,B
h
i
2
−1
∗
e
Gασn + Σασn − εi − t α(εi )Γασn ψασn
∗
∗
(τ)ψα↓ (τ)av
(τ)ψα↑ (τ)ψα↓
dτψα↑
+


.
(2.62)
av
This is the most general expression for the averaged grand potential of the disordered
Hubbard model in d = ∞ or at the mean-field level. It leads, as in the nonrandom
2.4. CONCLUDING REMARKS ON THE LIMIT D → ∞
37
case, to insoluble dynamical equations of motion for the variational functions G, Σ,
and Γ. We hence have to resort to further approximations to reduce the equations to
a theory with only finite-many coupled degrees of freedom. This will be discussed in
the forthcoming chapter.
The general expression for the averaged grand potential of the disordered Hubbard
model (2.62) can be simplified to other Hubbard-like models. The spinless fermions
as well as the Falicov-Kimball disordered models are exactly soluble by means of elementary analytic functions. We find especially interesting the Falicov-Kimball model
where the one-particle functions remain determined from algebraic equations as in the
nonrandom case, but the two-particle functions such as susceptibilities already mix
different Matsubara frequencies. New thermodynamic phenomena can be expected,
Janiš and Vollhardt (1992b).
2.4
Concluding remarks on the limit d → ∞
In this chapter we presented a very general scheme of the construction of comprehensive and thermodynamically consistent mean-field theories for lattice models via the
limit of high spatial dimensions. Exact solution, i.e. exact free energy (grand potential) can be obtained from a representation of the thermodynamic potential in terms of
renormalized one-particle functions. A sensible model with finite energy density is obtained via an appropriate scaling of nonlocal contributions to the model Hamiltonian.
The scaling is the only step depending on the type of the model under consideration. If
we want to exclude the trivial limit and infinite energy densities, the scaling is practically unambiguously determined by the Hamiltonian. Thereby the limit of high spatial
dimensions offers an unambiguous and systematic way to derive a global mean-field
theory free of unphysical behavior in the entire range of the input parameters.
The derivation of the free-energy functional exact in d = ∞ was based on an analysis of perturbation theory and Feynman diagrams. However, the diagrams were
necessary only to do the first step, i.e. to find the appropriate scaling of nonlocal
quantities. The rest of the derivation was completely nonperturbative where no approximation was ever used. The derivation was performed at the level of generating functionals, thermodynamic potentials, and hence the mean-field theory thereby
derived is automatically thermodynamically consistent and conserving in the Baym
sense. We showed how the free energy of lattice models becomes a variational functional of only local variables, either magnetization or/and local elements of the selfenergy and Green’s function and hence how the mapping onto a single-site problem is
realized. We had not, at any place, to neglect or suppress nondiagonal elements of the
propagator or any other quantity. We even did not need a ”mean-field” argumentation
or interpretation to derive the exact form of the generating thermodynamic potential
in d = ∞. Its form appeared to be consequence of the general representation of perturbation theory with renormalized quantities. The locality of the relevant variational,
38
CHAPTER 2. THE LIMIT D → ∞
renormalized quantities was a direct consequence of the scaling of nonlocal terms of
the underlying Hamiltonians.
The approach presented here seems to be the most general and comprehensive way
to derive the mean-field thermodynamics. It is derived from a generating functional
with a set of variational parameters (functions) clearly distinguishing which quantities
are purely local and which are necessary to be considered extended. Note that other alternative approaches use mostly the local self-energy as the generating functional and
do not work directly with the grand potential. It is sometimes difficult to determine
whether theory generated from the self-energy presents a thermodynamically consistent mean-field theory or not.
In the course of the derivation of mean-field theories for various lattice models we
showed that all the known consistent mean-field approximations, i.e. Weiss, SherringtonKirkpatrick, Hartree-Fock, CPA are really exact solutions of models in the limit of high
spatial dimensions, d = ∞. All these approximations share many common features
that were picked up as the hallmarks of a comprehensive and thermodynamically consistent mean-field theory.
We can conclude what we have learned from the general features of the limit d → ∞
and formulate the basic characteristics of mean-field theories:
MF1. The d = ∞ theory reduces to a variational problem with local variational parameters (functions) encompassing the atomic solution. There is no renormalization
of nonlocal quantities in the d = ∞ theory.
MF2. The explicit free-energy functional has the structure of the exact free-energy functional from renormalized perturbation theory and it generates thermodynamically consistent and conserving theory in Baym’s sense.
MF3. The free energy is fully controllable in the sense that it represents a sum of particular classes of (Feynman) diagrams dominant in the limit d = ∞.
These features can be taken as benchmarks of a comprehensive and thermodynamically consistent mean-field theory of lattice models in statistical mechanics. Unfortunately the mean-field theory for the Hubbard model of interacting electrons is not
exactly soluble in its unrestricted form and further approximations are to be employed
to obtain quantitative results in the theory of strongly correlated electrons. It proves
how complex and difficult the problem of strongly correlated electrons is.
Chapter 3
Mean-field theories for lattice fermions
In the preceding chapter we demonstrated how the limit of high spatial dimensions
leads to simplifications in various lattice models. It became clear that exact solutions
in d = ∞ have mean-field character, i.e. they contain reduced spatial fluctuations and
depend on the lattice structure only via the density of states (number of nearest neighbors). Moreover, we showed that the known, globally consistent mean-field theories
are exact solutions to microscopic models in the limit d → ∞ or D → ∞. Since only the
limit of high spatial dimensions, d → ∞, does not generally suppress the interaction
strength, we may conclude that a thermodynamically consistent mean-field theory is
equivalent to the exact solution of the lattice model in d = ∞. This could end up our
search for the dynamical mean-field theory. However, a mean-field theory to be of
practical use must be able to produce quantitative results, e.g. phase diagrams describing various types of phase transitions and thermodynamic phases. Hence from
this point of view only the models with classical or semiclassical solutions would meet
the criterion of solubility. Mean-field theories of interacting electrons with local (Hubbard) interaction cannot be solved exactly in d = ∞, at least globally. We can ask what
practical value does the Hubbard model in d = ∞ have when it is not soluble. On
the one hand we are sure that an exact solution fulfills all demands and restrictions
on a thermodynamically consistent theory, on the other hand we do not know how to
extract exact quantitative results from it. To make the limit d → ∞ of practical value
for applications, we have to find out a way how to reduce the infinite-many degrees
of freedom to only a finite number of relevant mean-field parameters. Since we are
unable to do it in an exact way we have to resort to further approximations. Not to
loose the profit of the d = ∞ solution, we have to proceed as much as systematically as
possible.
The aim of this chapter is to complete the program of the construction of a tractable
and consistent mean-field theory for interacting lattice fermions. We propose two possible ways how to complete this assignment. The first one is to use further analytic
approximations based on exact solutions of simpler models and extend them, in a selfconsistent manner, onto the Hubbard model. The other possibility is to try to solve the
39
40
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
equations of motion by ”brute force”, i.e. to employ extensively numerical procedures
to evaluate approximately functional integrals. Both the ways have their advantages
and limitations in applicability, which will be discussed in the following.
3.1 Analytic mean-field theories for interacting electrons
The easiest way to attain quantitative results in a controlled way is to construct a thermodynamic potential in closed way. That is, we have free energy represented by elementary analytic functions. Actually, it was a primary goal of the mean-field reasoning
to replace an insoluble model with a simpler effective theory soluble by analytic means.
The limit d → ∞ itself offers no guarantee that the simplification of perturbation theory
due to the collapse of diagrams leads to a soluble theory with only finite-many mutually coupled degrees of freedom. We know that due to noncommutativity of operators,
we are left in interacting quantum systems with infinite-many Matsubara frequencies
coupled in the equations of motion. Exact solution of the Hubbard model in d = ∞ is
hence generally characterized by infinite many parameters. The aim of the mean-field
theory or any other quantitative approach is to reduce the number of relevant quantities to a finite number of ”mean-field” parameters. We hence have to complement the
limit of high spatial dimensions by a reduction scheme leading to finite-many meanfield parameters. Only then we speak about a workable quantitative mean-field theory
of interacting electrons.
It is important to stress that there is no unique way to reduce the exact solution in
d = ∞ to a thermodynamically consistent mean-field theory with a few relevant parameters. More importantly, there is no small parameter that could eventually control
the reduction. There do exist various useful approximation schemes how to obtain
a mean-field-like theory. E. g. the Hartree-Fock decoupling or saddle-point approximations to functional integration, but they do not provide a systematic and general
scheme leading to a globally reliable approximation. It seems indispensable to use
simplified exact solutions to construct reliable and consistent approximations for interacting electrons. An analytically, globally soluble model can consist of noninteracting
electrons that move in a static potential caused either by an external (homogeneous)
force or by other dynamical particles ”frozen” to an averaged or steady state.
Since the fully quantum dynamical models are not soluble in d = ∞, we can use two
simplified models of interacting electrons. It is either the model of spinless fermions
leading to a static mean-field theory or the Falicov-Kimball model which results in a
semiclassical solution with decoupled Matsubara frequencies. However, these simpler models do not possess all the properties and symmetries of the original Hubbard
model. Hence to generate a mean-field theory of the Hubbard model from them, it is
necessary to introduce a set of new variational parameters to recover the full symmetry
of the full interacting model.
If we take the solution of spinless fermions in d = ∞ and extend the exact reduc-
3.1. ANALYTIC MEAN-FIELD THEORIES
41
tion of the nonlocal and spin-independent coupling also to the spin-dependent Hubbard on-site interaction we recover the Hartree approximation. It is a conserving theory that proved useful in the construction of the phase diagram in various situations,
Penn (1966). Although this mean-field theory is fully consistent, it is an analogy of
the Weiss solution, it is reliable only at very low temperatures in almost completely
ordered ground states at half filling or at weak coupling eventually at weak concentration. It is because the Hartree approximation does not contain real quantum manybody correlations. The Hartree approximation describes essentially a Fermi gas with
a renormalized mass of particles. The self-energy due to the electron-electron interaction is a constant depending only on a sum over the Matsubara frequencies, i.e. on a
classical parameter. It is definitely insufficient if we want to describe phenomena connected with strong coupling. The Hartree, or Hartree-Fock, approximation is hence
restricted in applicability only to a weak-coupling regime. It cannot be taken too seriously as concerns the global phase diagram, especially at strong coupling. E.g. it is
well known that the half-filled Hubbard model goes over to the Heisenberg model in
the limit U → ∞ with an effective coupling J ∼ t2 /U, Anderson (1963). This exact
asymtotics cannot be reproduced within the Hartree theory.
There have been various attempts to extend the Hartree-Fock to improve its reliability at strong coupling. Among them very popular was the so-called Gutzwiller
approximation, Gutzwiller (1964), increasing the number of ”classical” mean-field parameters by a factor renormalizing the energy bandwidth. Although such a theory has
a number of attractive features it lacks, contrary to the Hartree theory, a systematic and
controllable derivation. It is then hard to estimate the degree of its reliability in regions
where the Hartree approximation fails. Since the systematics of the derivation is an essential property we demand from a comprehensive mean-field theory, the Gutzwillertype reasoning is not the desired way to reach a mean-field theory at strong coupling.
Moreover, the Gutzwiller approximation is no longer a conserving approximation.
Another soluble model of interacting fermions in d = ∞ is the Falicov-Kimball
model. Its spinless version breaks the symmetry of the two fermionic species mutually
interacting via the Hubbard interaction. The dynamics of the conduction electrons of
this model in d = ∞ is described by the Hubbard-III solution. We can imagine that the
static electrons just only approximate weak kinetic energy of narrow-band electrons.
We then extend the dynamics of the conduction electrons from the Falicov-Kimball
model to the other dynamical electrons of the Hubbard model and thereby restore the
symmetry between the up and down spins. We recover thus the Hubbard-III approximation to the Hubbard model, Hubbard (1964). The reasoning where heavy particles
(forming a narrow band) are replaced by immobile ones suits evidently to situations
with extremely strong Coulomb repulsion close to half filling when the double occupancy hinders the electrons to move. We then speak about strong-coupling mean-field
theory.
We could anticipate that the Hubbard-III is the desired mean-field theory of the
Hubbard model starting with the exact solution of the Falicov-Kimball model. In the
42
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
seventies there was a tremendous effort to use ”alloy-analogy” approximations based
on the Hubbard-III solution to provide a strong-coupling mean-field theory, Cyrot
(1977), Moriya (1985), Czycholl (1986). However, the alloy-analogy theories suffer
from serious deficiencies. There is no explicit grand potential at hand, the approximations are not diagrammatically controlled and the resulting theories are not conserving. The Curie law at strong coupling is also not reproduced. Hence the phase diagram
from the alloy analogy solutions is unreliable as well.
Only the recent construction of Janiš and Vollhardt (1992a) of a mean-field approximation at strong coupling, starting from the exact free energy of the Falicov-Kimball
model in d = ∞ and the atomic solution, extended the Hubbard-III solution from the
paramagnetic phase to a thermodynamically consistent mean-field theory with variational free energy in closed form. This theory is the first that reproduces the Curie law
in the Heisenberg limit of the Hubbard model. Although this approximation does well
at strong coupling, it fails to recover the Fermi-liquid behavior of the Hubbard model
at weak coupling. It is due to the fact that this Hubbard-III-like theory reproduces the
weak-coupling limit only linearly in the interaction strength U. It is second order of
the interaction that is important for the Fermi-liquid theory to hold.
The above two examples of attempts to extend mean-field theories to strong coupling in an uncontrolled way show how easy it is to produce inconsistent approximations in some important limits. Our ultimate aim is to construct a mean-field theory
reliable in the entire range of input parameters. We hence use another, controlled approach to employ simplified exact solutions in d = ∞ for the construction of a comprehensive mean-field theory for the Hubbard model. The best way to obtain a reliable
approximation is to construct a free energy with variational character having a precise
relation to the exact solution, i.e. it is either a lower or an upper bound (at least in
d = ∞). Such a construction was proposed in Janiš and Vollhardt (1993) for general
insoluble Hamiltonians. We use it for the Hubbard Hamiltonian and show how to get
the best out of exact solutions of the simplified models in the construction of global,
conserving approximations of mean-field type being either an upper (Hartree-Fock)
or lower (Janiš-Vollhardt) bounds on the exact free energy in d = ∞. The variational
character of mean-field approximations is essential for the introduction and understanding of effective global parameters reflecting many-body correlations in an averaged way. The variational approach to the mean-field theory is also interesting from
the formal point of view, since it demonstrates that the Hartree-Fock theory represents
the best upper bound and the strong-coupling theory from Janiš and Vollhardt (1992a)
the best lower bound on the free energy of the Hubbard model in d = ∞. Both the
consistent mean-field theories are in some sense complementary. The former, an upper
bound, is reliable at weak coupling, the latter, a lower bound, is reliable at strong coupling or for narrow-band electrons. What we are unable to do at present is to produce
an analytic mean-field theory consistently interpolating between the weak and strong
couplings. It is caused by inability of the present mean-field theories to reproduce correctly two-particle vertex functions as argued in Freericks and Jarrell (1995). However,
3.1. ANALYTIC MEAN-FIELD THEORIES
43
recently the author and J. Schlipf proposed a linked cluster expansion around the existing mean-field theories of the Hubbard model and obtained improved, dynamical
approximations of mean-field character that essentially improve on the two-particle
correlations, Janiš and Schlipf (1995).
Important quantitative results concerning the thermodynamics of the Hubbard model
derived from the strong-coupling mean-field theory of Janiš and Vollhardt (1992a) are
presented in Appendix B3 of this thesis. Here we only give a new systematic derivation of both the lower and the upper bounds on the free energy of the Hubbard model
in d = ∞ suitable for further expansions around mean-field theories. We present this
derivation for nonrandom and random cases separately.
3.1.1
Nonrandom case
The basics of the variational derivation of mean-field theories are two inequalities from
statistical mechanics. If we have a decomposition
c=
H
l
X
α=1
cα
λα H
(3.1)
where λα are real, positive numbers fulfilling a normalization
l
X
λα = 1
(3.2)
α=1
then the free energy due to its convexity Girardeau and Mazo (1973) fulfills an inequality
l
n o
n o
X
cα ≤ F H
c .
λα F H
(3.3)
α=1
n o
cα are known, the left-hand side can serve as a starting point
If the free energies F H
for a mean-field approximation to the free energy on the right-hand side. To turn the
l.h.s. of (3.3) a mean-field theory, it is necessary to introduce a set of variational paramcα that are determined from optimizing the
eters, external sources in the Hamiltonians H
inequality (3.3) and that do not destroy solubility of the model with the Hamiltonian
cα .
H
An upper bound on the free energy is obtained from the Gibbs-Bogoljubov inequality
o
n o
n
c0 + h∆Hi
c 0.
c0 + ∆H
c ≤F H
(3.4)
F H
n
o
c0 .
Here the brackets h i0 denote the average with respect to the weight exp −βH
c0 +∆H
c if new auxiliary
Again the r.h.s. may serve as a mean-field to the Hamiltonian H
44
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
fields are introduced to minimize the r.h.s. of (3.4). It is reached if the average of the
c 0 vanishes.
perturbation h∆Hi
To be able to derive a MFT being either a lower or an upper bound in a unique way
we must decompose the total Hamiltonian into its unperturbed part (leading to the
desired MFT) and a perturbation
c=H
c0 + ∆H.
c
H
(3.5)
The decomposition (3.5) is important not only for the derivation of general mean-field
theories, it is mandatory for us to be able to expand systematically around the meanfield approximation. Only if we are able to expand and estimate fluctuations beyond
the mean-field solution, we can call the mean-field approximation comprehensive,
Janiš and Vollhardt (1993), Janiš and Schlipf (1995). Our aim now is to find suitable
constituents of the decomposition (3.5) for the weak and strong coupling mean-field
theories, or equivalently for an upper or a lower bound.
We start with the simpler case of the Hartree-Fock approximation. For the sake of
simplicity we restrict ourselves only to spatially homogeneous solutions. The variational parameters in the Hartree-Fock theory are the particle densities nσ and we can
write
c=H
cHF + ∆H
cHF
H
(3.6)
with
cHF = −NUn↑ n↓ +
H
cHF = U
∆H
X
X
†
(kσ + Un−σ ) ckσ ckσ
(3.7)
k,σ
ni↑ − n↑ ) (b
ni↓ − n↓ ) .
(b
(3.8)
i
Here we introduced two new variational parameters n↑ and n↓ that have to minimize
the upper bound (3.4). Since the average of the perturbation vanishes, they minimize
the free energy of noninteracting electrons in an external, static and spin-dependent
potential Un−σ .
The situation in the case of the strong-coupling MFT is more complicated. The
decomposition used in Janiš and Vollhardt (1993) leading to a lower bound does not
have a structure of (3.5). To obtain a similar structure to (3.5) we have to distinguish
the static and dynamic degrees of freedom used in the strong-coupling MFT, Janiš and
Schlipf (1995). We decorate the static degrees of freedom by a tilde. For concreteness
we use a decomposition of the Hubbard Hamiltonian into only two Falicov-Kimball
subhamiltonians with equal weights λ = 1/2. We hence define
X
X
UX
†
cσ =
n
b iσ n
e i−σ ,
(3.9)
H
n
e i,−σ +
(k + Eσ ) ckσ ckσ − Eσ
2 i
i
k
†
†
where n
b iσ = ciσ
ciσ , n
e iσ = ceiσ
ceiσ and
X
UX
c=
n
b iσ (b
ni−σ − n
e i−σ )
∆H
Eσ (b
niσ − n
e iσ ) +
2 i,σ
i,σ
(3.10)
3.1. ANALYTIC MEAN-FIELD THEORIES
45
from which the desired decomposition of the Hubbard directly follows Hamiltonian
X
c=
cσ + ∆H
c.
H
H
(3.11)
σ
P c
It is easy to show that F = −β−1 ln Tr exp{−β σ H
σ } leads in d = ∞ to the meanfield free energy of Janiš and Vollhardt (1993) with λat = 0, λσ = 1/2. The variational
parameters here are not n
e iσ (they are operators), but energies Eσ . It is also clear from
c↑ and H
c↓ decouple and FMFT = F↑ + F↓ . The stationarity
(3.9) that the dynamics of H
conditions for such a MFT lead to
niσ iMF = he
niσ iMF .
hb
(3.12)
c in (3.9), one quadratic
We have two different contributions to the perturbation ∆H
and one quartic in creation and annihilation operators. This makes perturbation expansion around the mean field cumbersome. We can, however, drop the quadratic
term without changing the physics of the full Hubbard Hamiltonian. The quadratic
term only shifts the spin-dependent chemical potential of the mobile electrons µ with
the energy Eσ playing the role of the chemical potential for the local (tilde) electrons.
These auxiliary local electrons completely decouple from the physical itinerant electrons in the full solution. The energies Eσ in the physical sector then redefine the origins for the chemical potential µ and the magnetic field h. The physical results do not
differ from those obtained from the original Hubbard Hamiltonian without energies
Eσ . We can hence choose
X
c=H
cMF + ∆H
c MF =
cσ + ∆H
c MF
H
H
(3.13)
σ
and
cMF =
∆H
UX
n
b iσ (b
ni−σ − n
e i−σ ) .
2 i
(3.14)
It is worth noting that decomposition (3.9)-(3.14) enabling a systematic perturbation
expansion is not identical with the strong-coupling MFT from Janiš and Vollhardt
cσ also the atomic Hamil(1992a) where apart from the Falicov-Kimball Hamiltonians H
cat with weights λσ = 1 and λat = −1 were used. To use a decomposition
tonian H
(3.5), no negative terms in the decomposition are allowed. We showed earlier, Janiš
and Vollhardt (1993), that the choice λσ = 1 and λat = −1 is equivalent at T = 0 to
λσ = 1/2, λat = 0 with doubled interaction strength U. There is, however, a difference
in the mean-field theories with λσ = 1 and λat = −1 and λσ = 1/2, λat = 0. The former
keeps the electron-electron interaction U unchanged, while the latter reduces it to U/2.
It means that the latter does not reproduce the atomic limit of the Hubbard model exactly contrary to the former. It may be seen as a deficiency of the variational approach
with a lower bound. But only the variational derivation gives the mean-field theory
46
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
using the Falicov-Kimball models its proper meaning and put it on the same footing
as the mean-field theory based on the Hartree-Fock solution.
Both the mean-field theories for weak and strong couplings have now formally
identical structure. They are based on simplified models the dynamics of which is
modified by auxiliary parameters. They are determined variationally from extremizing
bounds on the exact free energy. As we already mentioned the two mean-field theories
are complementary. The weak-coupling mean-field theory of Hartree and Fock is a
generalization of the exact solution of spinless fermions in d = ∞ and forms an upper
bound on the free energy of the Hubbard model. It has the averaged particle densities
as variational parameters. The strong-coupling mean-field theory based on the exact
solution of the Falicov-Kimball model is a lower bound on the Hubbard free energy
with doubled interaction and has energies (internal fields) as variational parameters,
i.e. Legendre conjugate variables to the particle densities of the Hartree-Fock theory.
This result reflects a general feature of thermodynamic potentials. They are minimized
by extensive variables (magnetization) and maximized by intensive variables (magnetic
field). It is also easy to show that a mean-field theory is nontrivial either as a lower or
as an upper bound. If we construct a lower bound e.g. from the Hartree-Fock theory
we obtain a trivial solution and the same holds for the strong-coupling theory. The
lower and upper bounds can be connected by a Legendre transformation if we replace
the particle densities by their conjugate variables, energies, and vice versa, Janiš and
Vollhardt (1993).
3.1.2
Random case
The general scheme how to derive mean-field theories for disordered interacting models does not differ from the nonrandom case. Only the variational parameters get
site-dependent, i.e. they are decorated with a site index. The first step, the configurationally dependent free energy is just an inhomogeneous nonrandom free energy
with locally fluctuating fields. We can write down the grand potentials for the two
above discussed mean-field theories. The Hartree-Fock grand potential reads
ΩHF {ni↑ , ni↓ } = −U
X
ni↑ ni↓ −β−1
∞
X X
+
eiωn 0 tr ln
σ n=−∞
i
h
i
n−σ − bt (3.15)
iωn + µσ 1b − εb − Ub
where now n
b σ does not mean a quantum operator but rather a site dependent number matrix of local particle densities. Correspondingly the strong-coupling mean-field
grand potential with random atomic energies reads
ΩFK {Gii,σ , Σi,σ ; Eiσ } = −β
−1
∞
X X
eiωn 0
σ n=−∞
+
X
i
)
ln [1 + Gii,σn Σi,σn ] + β−1
X
iσ
+
i
n
h
b σ − bt
tr ln iωn + µσ 1b − εb − Ebσ − Σ
ln 1 + exp β µ−σ + Ei,−σ − Ei,σ
(3.16)
3.1. ANALYTIC MEAN-FIELD THEORIES
47
where
Ei,σ = −β
−1
∞
X
e
iωn 0+
n=−∞
Gii,σn
U
ln 1 −
.
2 1 + Gii,σn Σi,σn
To reach the mean-field theory with finite number of relevant degrees of freedom,
we have to average the expressions (3.15) and (3.16). However as we already know
from the theory of spin glasses, it is not quite possible to reduce an interacting, disordered theory with local, self-consistent parameters as magnetization to a theory with
only finite-many averaged (homogeneous) parameters. It is not a priori clear how
the self-consistent, configurationally dependent variables ni,σ and Ei,σ are distributed.
Their distribution depends on the self-consistency equations. Only discrete random
distributions, such as random alloys, lead to a soluble mean-field theory of disordered,
interacting systems.
Averaging of the expressions (3.15) and (3.16) enables to replace the configurationallydependent functions of Matsubara frequencies (quantum variables) by homogeneous
functions, but we cannot to get rid completely of the configurationally dependent classical variables. This fact corresponds to the situation in the spin-glass problem, where
only loop diagrams can be averaged but not the tree ones. We can use the argumentation used in Section 2.2.1 by averaging the Anderson model of disordered electrons
and in Appendix A5. We then obtain
Z
∞
X X
1
iωn 0+
−1
e
dρ() ln iωn + µσ − − Σσn
ΩHF {Gσn , Σσn ; niσ } = −β
N
σ n=−∞
+ hln [1 + Gσn (Σσn − εi − Uni,−σ )]iav − U hni↑ ni↓ iav ,
(3.17)
Z
∞
h
i
X X
1
iωn 0+
−1
bσ
e
dρ() ln iωn + µσ − − Σ
ΩFK {Gσ , Σσ ; Eiσ } = −β
N
σ n=−∞
X
ln 1 + exp β µ−σ + Ei,−σ − Eσ
+ hln [1 + Gσn (Σσn − εi − Ei,σ )]iav − β−1
av
σ
(3.18)
with
Eσ = −β
−1
∞
X
n=−∞
e
iωn 0+
U
Gσn
ln 1 −
.
2 1 + Gσn Σσn
These are analogues of the TAP mean-field free energy from the Edwards-Anderson
model (2.32). Here we have an advantage that the averaging is completely local and
hence we do not expect ergodicity breaking and the spin-glass phase. Nevertheless the
simultaneous presence of disorder and (strong) interaction may lead to new non-trivial
effects (see Ulmke, Janiš, and Vollhardt (1995), Appendix C3).
48
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
3.2 Numerical mean-field theory for interacting electrons
As we already mentioned, it is the aim of a quantitative mean-field theory to reduce
a problem with infinite-many coupled degrees of freedom to a tractable problem with
only a finite number of relevant variables. The limit of high spatial dimensions reduces the spatially extended degrees of freedom by mapping the lattice summations
to local integrals with the density of states. However, the quantum, interacting models still need infinite-many Matsubara frequencies to allow for quantum dynamical
fluctuations. Analytic mean-field theories, described in the preceding section, use an
”arbitrary” choice of relevant mean-field parameters being introduced in simplified
models of noninteracting electrons (in a static field) as external, static sources. The
quantum dynamical fluctuations are thereby severely reduced and they influence the
mean-field dynamics only in an averaged manner via a self-consistency determining
the mean-field parameters. Although the self-consistency may lead to a substantial
change in the thermodynamics with respect to the simplified models, Janiš, Mašek,
and Vollhardt (1993), we can never describe the details of the electron-electron interaction. We hence cannot make definite conclusion about the actual behavior of quantum
interacting models. It is sometimes useful to have upper and lower bounds, e.g. to
estimate the ground-state energy, Janiš, Mašek, and Vollhardt (1993), but we cannot
decide whether the metal-insulator transition in the paramagnetic phase of the Hubbard model or the Kondo regime in the periodic Anderson model are realized or not.
These questions lie definitely beyond the analytic mean-field theories derived above.
When we cannot solve the Hubbard model in d = ∞ analytically we can attempt to
solve the equations of motion numerically. It is clear that even numerics cannot circumvent the problem of coupling of infinite-many Matsubara frequencies in the Hubbard
model in d = ∞. However, numerical solutions do not discriminate à priori any Matsubara frequencies and hence, essentially contain the full information about the quantum dynamics. To obtain quantitative results we nevertheless have to suppress very
large frequencies. Since all the physical quantities fall off sufficiently rapidly with increasing frequency, this restriction is a matter of precision of the result. However, there
is a severe limitation on the applicability of the numerical solution of the Hubbard and
other interacting quantum models in d = ∞. Namely, we must use discrete Matsubara frequencies, i.e. we are bound to calculate only at finite temperatures. Contrary
to the large frequencies we have no guarantee that performing the continuous limit
(limit to zero temperature) we do not encounter singularities or unexpected behavior. In other words, there is no simple way how to extrapolate the finite-temperature
numerical results to the physically most important and interesting situation of zero
temperature. We cannot numerically reach smaller energy scale as that defined by the
smallest Matsubara frequency ω1 = πT . With the numerical solution we can hence
reproduce quantum dynamics of the Hubbard model only on a coarse-grained energy
scale. The fine or hyperfine structure of quantum dynamics remains beyond the reach
of even the best numerical techniques and most powerful computers. Especially the
3.2. NUMERICAL MEAN-FIELD THEORY
49
spectral properties seem to be hardly derivable from the numerical treatments on the
imaginary axis of Matsubara frequencies. Although methods like maximum-entropy
principle, Gubenatis et al (1991), have been employed to continue the results from the
discrete set of Matsubara frequencies to the axis of real frequencies, the fine structure
of the energy spectrum remains smeared out.
The most powerful numerical method to assess the dynamics of the Hubbard model
in d = ∞ seems to be Monte-Carlo simulations. Since the Hubbard model in d = ∞ is
not described by operators but rather by Green’s functions, we cannot use the standard
equal-time, operator Monte-Carlo method. The Green’s function Monte-Carlo was for
the first time used in the context of the Anderson model by Hirsch and Fye (1986).
In the context of the infinite-dimensional models it was introduced by Jarrell (1992),
Rozenberg, Zhang, and Kotliar (1992), and Georges and Krauth (1992) for the nonrandom Hubbard model and by Janiš, Ulmke, and Vollhardt (1993) and Dobrosavljević
and Kotliar (1993) for the Anderson-Hubbard model with diagonal and nondiagonal
disorder, respectively. Since that a relatively complete picture of the finite-temperature
dynamics of the Hubbard model has been gained. The quantitative results from the
extensive study of the disordered Hubbard model are presented in Appendices C1-C3.
Here we only present the basic ideas of the numerical evaluation of the equations of
state for the Hubbard model with the Monte-Carlo technique.
3.2.1
Monte-Carlo in the Green’s function formalism
We are facing a problem of evaluating moments of a Grassmann functional integral


β


Zβ
Z

X Z
0
∗
∗
0
0 ∗
∗
dτdτ cσ (τ)γσ τ − τ cσ (τ ) − U dτc↑ (τ)c↑ (τ)c↓ (τ)c↓ (τ)
Z =
DcDc exp



 σ
0
0
(3.19)
where cσ∗ (τ) and cσ (τ) are Grassmann variables. The renormalized inverse local propagator reads γσ = G−1
σ + Σσ . The first step in the Monte-Carlo procedure is to discretize
the imaginary-time variable τ. We choose a small scale ∆τ = β/L and write
Zβ
0
#
L−1
X
1
f0 + fL +
fl .
dτf(τ) = ∆τ
2
l=1
"
The discretized fermionic propagator or its inverse fulfills symmetry relations
γσ,−l = −γσ,L−l .
(3.20)
The functional integral (3.19) then in the discrete representation becomes
(
)
#
" L−1
L−1 X
L−1
L−1
XX
X
YY
∗
∗
dcσl dcσl
exp ∆τ 2
cσl
γσ,l−l0 cσl0 − U∆τ
ZL =
n↑l n↓l
l=1
σ
σ
l=0 l0 =0
l=0
(3.21)
50
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
∗
where we denoted nσl = cσl
cσl .
In the next step we use a Hubbard-Stratonovich transformation to get rid of quartic
products of Grassmann variables. We introduce Ising spins with values S = ±1 and
write
(
)
(
)
L−1
L−1
X
XX
exp −U∆τ
(3.22)
n↑l n↓l = TrS exp −∆λ
Sl nσl
σ
l=0
l=0
with a new effective interaction λ defined as
cosh (∆τλ) = exp {∆τU/2} .
Inserting this representation into (3.21) we can perform the grassmannian integration
explicitly and are left with a partition function with spin variables only
n
o
b
b
ZL = TrS det O↑ (S) det O↓ (S)
(3.23)
where we used fundamental spin-dependent matrices
h
i
b
Oσ (S) 0 = ∆τ ∆τγσ,l−l0 − δl,l0 σλSl .
(3.24)
l,l
The Monte-Carlo simulations consist of an approximate evaluation of the trace over
the Ising spins. The configurations are selected with a random-number generator.
However, in contrast to the classical Ising model, the determinants can acquire negative values. It is hence much more complicated and demanding to keep the selection
balanced, i.e. to avoid the ”sign” problem. There must not be too many spin configurations contributing with negative sign to the partition sum. The sign problem interferes
the numerical simulations especially at strong coupling and away from half filling.
The most important function for the numerical Monte-Carlo procedure is the oneelectron Green’s function defined in the discrete representation as
h
i
1
−1
b
b
b
TrS Oσ (S) 0 det O↑ (S) det O↓ (S) .
(3.25)
Gσ,l−l0 =
l,l
ZL
The right-hand side of this definition is used for the Monte-Carlo simulations or, if
the number of spin configurations is small (L ≤ 22), for the exact summation over all
spin realizations. For the self-consistency condition connecting the left-hand side of
(3.25) with the inverse of the effective local propagator γσ we need to employ Fourier
transform having in the discrete-time representation the form
(
" L−1
#)
L−1
X
X
1
e σ (ωn ) = ∆τ Gσ,0 +
Gσ,l eiωn l∆τ −
Gσ,L−l eiωn (L−l)∆τ
(3.26)
G
2 l=1
l=1
with the inverse transform
Gσ,l
X
1 L/2−1
e σ (ωn )e−iωn l∆τ .
G
=
β n=−L/2
(3.27)
3.3. PROSPECTS AND LIMITATIONS OF MEAN-FIELD THEORIES
51
The self-consistency condition matching γσ and Gσ reads
e σ (ωn ) =
G
∞
Z
−∞
dρ∞ ()
1
e −1
iωn + µσ − γeσ (ωn ) + G
σ (ωn ) − .
(3.28)
Equations (3.25) and (3.28) define a couple of equations to be solved simultaneously
in an iterative manner. The most difficult part is of course the Monte-Carlo evaluation
of the temperature Green’s function from (3.25). In the case of a disordered system,
we have to distinguish these equations for different values of possible random atomic
energies with corresponding weights. Although the numerical evaluation of the above
equations is straightforward a number of encumbrances are encountered during the
calculation (cf. Appendix C3).
3.3
Prospects and limitations of mean-field theories for
lattice electrons
Models of interacting electrons are distinguished among the lattice models in that they
do not simplify to exactly solvable problems in the mean-field limit of infinite spatial dimensions. That is why the limit d → ∞ is continuously attracting interest of
condensed-matter theorists. After a long-lasting period of incoherent efforts to reach
simplified solutions of microscopic models of correlated electrons, the limit of high dimensions equipped theorists with a powerful tool to construct local approximations in
a controlled way. The limit of high spatial dimensions is especially useful for investigations of quantum fluctuations in models with strong electron-electron interaction. As
we know quantum character of the dynamics has its origin in noncommutativity of
operators (e.g. of kinetic and potential energy). The limit of high spatial dimensions
does not touch the commutator structure of electronic systems with the local (Hubbard) interaction. Hence the quantum character of the Hubbard-like models is fully
maintained in the limit d → ∞. The simplification brought into interacting models by
a large number of nearest neighbors is a suppression of spatial fluctuations. There is no
conservation of momentum at vertices of perturbation theory except for conservation
of the total momentum and the lattice summations are replaced by energy integrals
with density of states. A mapping of the infinite-dimensional problem onto a singlesite theory is thereby achieved. The quantum solution in d = ∞ retains some spatial
coherence and information about the short-range order (analogue of Onsager’s cavity
field). Reduction of spatial correlations in the critical region does not precludes the existence of long-range order and enables to investigate more profoundly the influence of
quantum fluctuations on phase transitions. Separation of spatial and time fluctuations
present in the infinite-dimensional quantum models offers a perfect setting to study
quantum phase transitions in correlated systems.
52
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
Since the infinite-dimensional models of interacting electrons mimic the full quantum dynamics from finite dimensions, a mean-field theory, understood as an exact
solution in d = ∞, is not attainable. This is the price we pay for keeping the complete
structure of frequency fluctuations. The thermodynamics of quantum models must
be described by infinite number of Matsubara frequencies. The local interaction cause
that they are dynamically coupled. Correlated electrons in d = ∞ become an insoluble
problem with infinite-many relevant degrees of freedom mutually coupled. To make
the mean-field theory of practical use we have to find out a few relevant degrees of
freedom controlling the solution in specific conditions. It is actually the most difficult
part of the construction of a tractable mean-field theory of interacting fermions. We
have two possibilities to do so. We can either try to reduce the quantum degrees of
freedom in such a way that we obtain the thermodynamic potential in closed form or
we use numerical methods. We can follow several guiding lines.
If we want to construct a global thermodynamic theory in closed form we employ
a variational formulation of the thermodynamics. To keep the simplified mean-field
theory consistent we have to utilize exactly soluble models of interacting electrons in
d = ∞. We have only two possibilities, since up to now only two models have been
solved, that of spinless fermions and the Falicov-Kimball model. We introduce appropriate external fields into these simplified models and determine their equilibrium
values from minimizing (maximizing) the simplified free energies. We obtain variational mean-field theories for the Hubbard model. We gain two mean-field theories
with complementary properties. If we start with spinless fermions in d = ∞ we obtain
the mean-field theory of Hartree and Fock. It is an upper bound on the free energy of
the Hubbard model, has particle densities as variational parameters and is reliable at
weak coupling. If we start with the solution of the Falicov-Kimball model in d = ∞ we
end up with a mean-field theory of Janiš and Vollhardt generalizing the Hubbard-III
approximation. It is a lower bound on the free energy of the Hubbard model (in d = ∞),
has energies as variational parameters and is reliable at strong coupling (narrow-band
systems). These two mean-field theories seem to be at present the only global, thermodynamically consistent approximations to the Hubbard model. The limit of high
spatial dimensions proved invaluable in generalizing solutions of simplified models
to consistent mean-field theories for the Hubbard model.
Although we have constructed global, thermodynamically consistent mean-field
theories in the entire range of the input parameters, the detailed structure of the manybody spectrum goes lost in this construction. Particularly there is no reliable theory
properly interpolating between weak and strong couplings. That is due to improper
selection of relevant mean-field parameters in analytical mean-field theories based
mostly on averaged one-particle quantities. Dynamical two-particle correlations are
decisive for the transition from weak to strong couplings. In the weak-coupling as well
as in the strong-coupling mean-field theories we obtain a static solution. It is, however,
inevitable to take into account dynamical fluctuations and to leave static theories if we
want to describe the transition region between the weak and strong couplings. There
3.3. PROSPECTS AND LIMITATIONS OF MEAN-FIELD THEORIES
53
the quantum effects become most pronounced. They may cause a metal-insulator transition, the Kondo effect, or a breakdown of Fermi-liquid theory.
Up to now the only unbiased and systematic reduction of quantum degrees of freedom in the mean-field theory of interacting electrons is the numerical solution of the
dynamical equations for the relevant local quantities. Such a solution, mostly accomplished by Monte-Carlo simulations, is essentially exact at relatively high temperatures. I.e. it predicts pretty well finite-temperature thermodynamic phenomena. A
reliable phase diagram of the (disordered) Hubbard model was constructed. Also
in the numerical solution we pay a price for the unbiased reduction of the infinitedimensional solution to a finite problem. The very low-temperature region and spectral properties of the solutions cannot be reached without further assumptions. The
numerical solution produces reliable results only on a gross energy scale and the genuine critical asymptotics remains inaccessible for the numeric method.
Presently it seems that the here mentioned constructions of tractable mean-field
theories, i.e. analytical and numerical, are the only possibilities to generate a global
mean-field theory. While the analytical way offers a full solution producing all thermodynamic and spectral properties within a simplified dynamics, the numerical one
does not restrict the thermodynamics but fails to reach physically the most interesting
zero-temperature limit. To be able to investigate the behavior of interacting electrons
at very low temperatures and in the transition region between the weak and strong
couplings we have to reduce the quantum degrees of freedom in a specific way using
thereby renormalization-group ideas or other nonperturbative analytical-numerical
techniques. This remains still a challenge in the description of strongly correlated electrons.
54
CHAPTER 3. MEAN-FIELD THEORIES FOR LATTICE FERMIONS
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