Regular Article

Adaptive cluster expansion for Ising spin models^★

¹ Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France
² Sorbonne Université, CNRS, Institut de Biologie Paris Seine, Biologie computationnelle et quantitative – LCQB, 75005 Paris, France

^a e-mail: f.zamponi@gmail.com

Received: 13 June 2019
Received in final form: 2 October 2019
Published online: 20 November 2019

Abstract

We propose an algorithm to obtain numerically approximate solutions of the direct Ising problem, that is, to compute the free energy and the equilibrium observables of spin systems with arbitrary two-spin interactions. To this purpose we use the Adaptive Cluster Expansion method [S. Cocco, R. Monasson, Phys. Rev. Lett. 106, 090601 (2011)], originally developed to solve the inverse Ising problem, that is, to infer the interactions from the equilibrium correlations. The method consists in iteratively constructing and selecting clusters of spins, computing their contributions to the free energy and discarding clusters whose contribution is lower than a fixed threshold. The properties of the cluster expansion and its performance are studied in detail on one dimensional, two dimensional, random and fully connected graphs with homogeneous or heterogeneous fields and couplings. We discuss the differences between different representations (Boolean and Ising) of the spin variables.