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Communications in Theoretical Physics

Effective Mean-Field Theory Based on Cumulant Expansion: A New Approach Method Order by Order

Ying An, Yong-Xin Song

Published Sep 15, 1995• 1 citation
Publisher

Abstract

We propose in this paper an effective mean-field theory, which makes the calculation for any observable being performed in a general form of cumulant expansion. The effective mean-field equation with reasonable physical annotation, which is obtained from the lower bound extreme value condition of free energy, can be applied to general statistic mechanics and Lagrangian quantum theory. By applying it to lattices, our theory, for various models containing nonuniform system and nonequilibrium phenomenon, can give a realistic scheme to study the nonperturbative effect order by order. As a testing example to two-dimensional Ising model, we obtain a result approaching the Onsager solution order by order.

DOI
10.1088/0253-6102/24/2/233

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We propose in this paper an effective mean-field theory, which makes the calculation for any observable being performed in a general form of cumulant expansion. The effective mean-field equation with reasonable physical annotation, which is obtained from the lower bound extreme value condition of free energy, can be applied to general statistic mechanics and Lagrangian quantum theory. By applying it to lattices, our theory, for various models containing nonuniform system and nonequilibrium phenomenon, can give a realistic scheme to study the nonperturbative effect order by order. As a testing example to two-dimensional Ising model, we obtain a result approaching the Onsager solution order by order.
An effective mean-field theory based on cumulant expansion is expanded to deal with the first-order transition of the Potts model on general d-dimensional hypercubic lattice in this paper. It is a generalized mean-field method developed from the cumulant expansion of average free energy. As we shall see, it gives the traditional mean-field result at its first-order approximation and the effective mean field introduced by this method will be amended with the approximation's order increasing. For a full view of this method, we have carried out calculations up to the seventh-order approximation on a square (d=2)(d=2)(d=2) lattice since some known exact quantites of this case are terminals that should be approached. The results show this method has the potential to be a useful tool to calculate thermodynamical observables of spin systems.
A general self-consistency approach allows a thorough treatment of the corrections to the standard mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a new point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a new coarse graining procedure. We outline and justify our method by some preliminary calculations. Concrete results are given for the critical temperature and the Landau parameters of the ϕd4ϕ^4_dϕd4​-theory - the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions - the problem for the calculation of the true critical temperature, is considered within the framework of the present approach. A comprehensive description of the ground state properties of many-body systems is also demonstrated.
A general self-consistency approach allows for a thorough treatment of the corrections to the mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a coarse-graining procedure. We outline and justify our method by some preliminary calculations. Results are given for the critical temperature and the Landau parameters of the phi(4) theory--the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions--the problem for the calculation of the true critical temperature--is considered within the framework of the present approach. A comprehensive description of the ground-state properties of many-body systems is also demonstrated.
Introducing low-energy effective Hamiltonians is usual to grasp most correlations in quantum many-body problems. For instance, such effective Hamiltonians can be treated at the mean-field level to reproduce some physical properties of interest. Employing effective Hamiltonians that contain many-body correlations renders the use of perturbative many-body techniques difficult because of the overcounting of correlations. In this work, we develop a strategy to apply an extension of the many-body perturbation theory starting from an effective interaction that contains correlations beyond the mean-field level. The goal is to re-organize the many-body calculation to avoid the overcounting of correlations originating from the introduction of correlated effective Hamiltonians in the description. For this purpose, we generalize the formulation of the Rayleigh-Schrödinger perturbation theory by including free parameters adjusted to reproduce the appropriate limits. In particular, the expansion in the bare weak-coupling regime and the strong-coupling limit serves as a valuable input to fix the value of the free parameters appearing in the resulting expression. This method avoids double counting of correlations using beyond-mean-field strategies for the description of many-body systems. The ground state energy of various systems relevant for ultracold atomic, nuclear, and condensed matter physics is reproduced qualitatively beyond the domain of validity of the standard many-body perturbation theory. Finally, our method suggests interpreting the formal results obtained as an effective field theory using the proposed reorganization of the many-body calculation. The results, like ground state energies, are improved systematically by considering higher orders in the extended many-body perturbation theory while maintaining a straightforward polynomial expansion.
For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite NNN. Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large-NNN limit depends on how the model parameters scale with NNN, and the convergence of cumulant expansions may be nonuniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with NNN and may exceed that of mean-field theory.