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arXiv:cond-mat/0512529 · Journal of Statistical Mechanics Theory and Experiment

Loop expansion around the Bethe–Peierls approximation for lattice models

Giorgio Parisi A, František Slanina

Published Feb 8, 2006• 43 citations
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Abstract

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe–Peierls approximation appears naturally as the saddle-point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetization and correlation function.

DOI
10.1088/1742-5468/2006/02/l02003

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We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe–Peierls approximation appears naturally as the saddle-point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetization and correlation function.
Abstract For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M = 1 . The 1 / M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the 1 / M expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.
The Bethe-Peierls approximation is applied to the classical Ising model in DDD dimensions and in a variational treatment of the Hamiltonian Z(2)Z(2)Z(2) gauge model in 2+1 dimensions. These analytical and simple calculations are in good agreement with exact results.
Abstract A expanded Bethe–Peierls approximation has been used to study the Ising model on honeycomb lattice, square lattice, kagomé lattice, triangular lattice and simple cubic lattice with spin S = 1/2 and 1. The spatial dimensionality of system is distinguished for different lattices. It is found that the transition temperatures on different lattices are much closer to the real values than those obtained by the effective field theory and the Bethe lattice approximation, and the transition temperatures of systems on kagomé lattice (z = 4) and trianguilar lattice (z = 6) are lower than those on square lattice (z = 4) and simple cubic lattice (z = 6), respectively. The static magnetic susceptibilities are also calculated. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We propose a new functional integral representation of the Hubbarda and Anderson models of lattice fermions. The simplest saddle-point approximation leads, at zero temperature, to the results derived from the Gutzwiller variational wave function. This approach uncovers the limitations of the Gutzwiller approximation and clarifies its connection to the "auxiliary-boson" mean-field theory of the Anderson model. This formulation leads to a novel strong-coupling mean-field theory which allows for a unified treatment of antiferromagnetism and ferromagnetism, metal-to-insulator transition, and Kondo compensation effects.
We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d -dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.
A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions d>2d>2d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions d>1d>1d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.