Bootstrapping ground state properties of classical frustrated magnets

TL;DR

This paper presents a semidefinite programming approach to rigorously bound ground state properties of classical frustrated magnets, overcoming non-convex optimization challenges on infinite lattices.

Contribution

It adapts the Lasserre hierarchy to frustrated magnetism, providing convergent, finite-size convex bounds applicable to complex models beyond prior analytical methods.

Findings

Accurately brackets energy densities and observables in 2D frustrated spin models.

Method converges in the thermodynamic limit and applies to non-quadratic Hamiltonians.

Runs efficiently, typically seconds per parameter point.

Abstract

We introduce a method based on semidefinite programming that produces rigorous two-sided bounds on ground state energy densities and correlation functions of translation-invariant classical spin models on infinite lattices. In this method, the challenge of non-convex optimization on an infinite lattice is replaced with a hierarchy of finite-size convex optimizations arising from positivity conditions that any probability distribution over spin configurations must satisfy. This adapts the Lasserre hierarchy in the theory of polynomial optimization to the context of frustrated magnetism, and we prove convergence of this hierarchy in the thermodynamic limit. Our method subsumes the Luttinger--Tisza method and further applies to non-quadratic Hamiltonians and non-Bravais lattices, thus addressing limitations of prior analytical methods. We apply the method to various two-dimensional frustrated spin models, where it brackets the energy densities and observables accurately across large parameter ranges with typical run times of seconds per parameter point.