Solving frustrated Ising models using tensor networks

TL;DR

This paper introduces a tensor network framework to efficiently analyze frustrated Ising models, enabling accurate residual entropy calculations and outperforming traditional Monte Carlo methods.

Contribution

Develops a general tensor network approach for frustrated Ising models, including cluster optimization and linear programming for improved contractibility.

Findings

Accurately computes residual entropy of kagome lattice Ising model.

Outperforms Monte Carlo methods in speed and accuracy.

Provides a foundation for finite-temperature extensions.

Abstract

Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks %, i.e. tensor networks that can be contracted using standard algorithms for infinite systems. This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e. that the energy can be minimized independently on each cluster. We show that optimizing the choice of clusters, including the weight on shared bonds, is crucial for the contractibility of the tensor networks, and we derive some basic rules and a linear program to implement them. We illustrate the power of the method by computing the residual entropy of a frustrated Ising spin system on the kagome lattice with next-next-nearest neighbour interactions, vastly outperforming Monte Carlo methods in speed and accuracy. The extension to finite-temperature is briefly discussed.