Gauge-free cluster variational method by maximal messages and moment matching

Abstract

We present a new implementation of the Cluster Variational Method (CVM) as a message passing algorithm. The kind of message passing algorithms used for CVM, usually named Generalized Belief Propagation, are a generalization of the Belief Propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free-energy is usually not a one-to-one correspondence, because of the existence of a gauge transformation involving the GBP messages.Our contribution is twofold. Firstly we propose a new way of defining messages (fields) in a generic CVM approximation, such that messages arrive on a given region from all its ancestors, and not only from its direct parents, as in the standard Parent-to-Child GBP. We call this approach maximal messages. Secondly we focus on the case of binary variables, re-interpreting the messages as fields enforcing the consistency between the moments of the local (marginal) probability distributions. We provide a precise rule to enforce all consistencies, avoiding any redundancy, that would otherwise lead to a gauge transformation on the messages. This moment matching method is gauge free, i.e. it guarantees that the resulting GBP is not gauge invariant.We apply our maximal messages and moment matching GBP to obtain an analytical expression for the critical temperature of the Ising model in general dimensions at the level of plaquette-CVM. The values obtained outperform Bethe estimates, and are comparable with loop corrected Belief Propagation equations. The method allows for a straightforward generalization to disordered systems.

Loop Corrections in Spin Models through Density Consistency

“…For d ≥ 3 [21] the maximum is realized at r = −1, obtaining: [22]. For comparison, nMF is exact up to the d 0 order, BP is exact up to the d −1 order, and Loop Corrected Bethe (LCB, [13]) and Plaquette CVM (PCVM, [23]) are exact up to the d −2 order.…”

“…Critical values obtained with different approximation schemes of the inverse temperature β marking the onset of spontaneous magnetization in the homogeneous Ising model on infinite d-dimensional hypercubic lattices. The values of βBP , βP CV M and βLCB respectively refer to the Bethe-Peierls, Plaquette Cluster Variational Method [23] and Loop Corrected Bethe [13] approximations, while βp and βm respectively correspond to the maximum β of the paramagnetic DC solution and the minimum β of the magnetized DC solution, βc indicates the currently best known approximation up to numerical accuracy ( [23] for d ≤ 5, [22] for d = 6). Results in bold indicate the closest value to the last column.…”

The Quantum Cluster Variational Method and the Phase Diagram of the quantum ferromagnetic $J_1$-$J_2$ model

“…Since the preferred direction of the system is labeled x, quantum effects appear when the transverse field h z is different from zero. The classical model (h z = 0) has been largely studied [38][39][40][41][42][43][44][45] . However, also there some questions remain open.…”

“…Recently, researchers paid attention also to the effect of external magnetic fields and demonstrated the presence of a nematic phase 40,41 , with slight differences, at low temperatures, between the two studies. This nematic phase is characterized by the presence of orientational order and the lack of positional order and has been reported experimentally 56 .…”

“…The J 1 -J 2 model has captured special attention both because of its apparent simplicity, and because of its resemblance with real materials. The classical version has been studied extensively [38][39][40][41][42][43][44][45] but the quantum version has turned harder to crack. Mainly, because quantum fluctuations add complexity to the inherent frustration of the model inducing a zoology of exotic phases: Stripes like phases, columnar antiferromagnetic (CAF), Néel Anti-ferromagnetic (NAF) phases, Spin-liquid phases and Spin Nematic Phases have been theoretically predicted 40,[46][47][48][49][50][51][52] .…”