Bootstrapping the Ising Model on the Lattice

TL;DR

This paper introduces a bootstrap approach combining spin identities and positivity conditions to rigorously bound spin correlators in the Ising model, achieving higher precision than Monte Carlo methods in 2D and providing initial results for 3D.

Contribution

It develops a novel bootstrap method using semi-definite programming to derive rigorous bounds on spin correlators in the Ising model, surpassing traditional numerical techniques.

Findings

Bounds on 2D Ising model correlators are highly precise.

Method provides initial bounds for 3D Ising model.

Bootstrap bounds outperform Monte Carlo in certain regimes.

Abstract

We study the statistical Ising model of spins on the infinite lattice using a bootstrap method that combines spin-flip identities with positivity conditions, including reflection positivity and Griffiths inequalities, to derive rigorous two-sided bounds on spin correlators through semi-definite programming. For the 2D Ising model on the square lattice, the bootstrap bounds based on correlators supported in a 13-site diamond-shaped region determine the nearest-spin correlator to within a small window, which for a wide range of coupling and magnetic field is narrower than the precision attainable with Monte Carlo methods. We also report preliminary results of the bootstrap bounds for the 3D Ising model on the cubic lattice.