Finite--Size Scaling Analysis of Generalized Mean--Field Theories

TL;DR

This paper explores finite-size scaling in generalized mean-field theories using the Peierls--Bogoliubov inequality, linking it to the coherent anomaly method and renormalization group ideas, with applications to the 2D Ising model.

Contribution

It provides a transparent interpretation of the coherent anomaly method within renormalization group frameworks and identifies optimal approximant families for mean-field theories.

Findings

Finite-size scaling yields critical exponents of the full system.

The coherent anomaly method is interpreted via renormalization group ideas.

Optimal approximant families can be identified for mean-field theories.

Abstract

We investigate families of generalized mean--field theories that can be formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians describing mutually non--interacting subsystems of increasing size, the thermodynamics of these mean--field type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean--field critical points. Finite--size scaling analysis of this mean--field critical behaviour allows to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki, which is thus given a transparent interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls--Bogoliubov inequality. In the case of the 2--$d$ Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity of thermodynamic functions.