# Adaptive Cluster Expansion for Ising spin models

## TL;DR

This paper introduces an adaptive cluster expansion algorithm to efficiently approximate solutions of the direct Ising problem, enabling computation of free energy and observables for complex spin systems.

## Contribution

It adapts the cluster expansion method, originally for inverse problems, to directly solve the Ising model, improving computational efficiency and applicability.

## Key findings

- Effective in one and two-dimensional systems
- Handles random and fully connected graphs
- Works with homogeneous and heterogeneous parameters

## 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, 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.