Constructing and proving the ground state of a generalized Ising model by the cluster tree optimization algorithm

TL;DR

This paper introduces the cluster tree optimization algorithm, an efficient method for finding and proving the ground states of generalized Ising models, including complex and aperiodic systems, with improved tractability over existing methods.

Contribution

The paper presents a novel cluster tree optimization algorithm that efficiently determines the ground state of generalized Ising models of arbitrary complexity, with provable bounds on error.

Findings

Algorithm converges to true ground state energy for aperiodic systems.

Significantly reduces computational complexity compared to polytope method.

Provides a practical tool for studying low-energy states in frustrated systems.

Abstract

Generalized Ising models, also known as cluster expansions, are an important tool in many areas of condensed-matter physics and materials science, as they are often used in the study of lattice thermodynamics, solid-solid phase transitions, magnetic and thermal properties of solids, and fluid mechanics. However, the problem of finding the global ground state of generalized Ising model has remained unresolved, with only a limited number of results for simple systems known. We propose a method to efficiently find the periodic ground state of a generalized Ising model of arbitrary complexity by a new algorithm which we term cluster tree optimization. Importantly, we are able to show that even in the case of an aperiodic ground state, our algorithm produces a sequence of states with energy converging to the true ground state energy, with a provable bound on error. Compared to the current state-of-the-art polytope method, this algorithm eliminates the necessity of introducing an exponential number of variables to counter frustration, and thus significantly improves tractability. We believe that the cluster tree algorithm offers an intuitive and efficient approach to finding and proving ground states of generalized Ising Hamiltonians of arbitrary complexity, which will help validate assumptions regarding local vs. global optimality in lattice models, as well as offer insights into the low-energy behavior of highly frustrated systems.