Topological correlations in three dimensional classical Ising models: an exact solution with a continuous phase transition

TL;DR

This paper presents an exact solution for a 3D classical Ising model with imaginary couplings, revealing topological phases and a third order phase transition, with implications for quantum dynamics.

Contribution

It generalizes the Onsager-Kaufman and Kitaev solutions to a 3D model, uncovering topological features and phase transitions in a novel exactly solvable system.

Findings

Topological phase distinctions based on loop observables

Quantized expectation values differing between phases

Partition function related to quantum transition amplitudes

Abstract

We study a three-dimensional (3D) classical Ising model that is exactly solvable when some coupling constants take certain imaginary values. The solution combines and generalizes the Onsager-Kaufman solution of the 2D Ising model and the solution of Kitaev's honeycomb model, leading to a three-parameter phase diagram with a third order phase transition between two distinct phases. Interestingly, the phases of this model are distinguished by topological features: the expectation value of a certain family of loop observables depend only on the topology of the loop (whether the loop is contractible), and are quantized at rational values that differ in the two phases. We show that a related exactly solvable 3D classical statistical model with real coupling constants also shows the topological features of one of these phases. Furthermore, even in the model with complex parameters, the partition function has some physical relevance, as it can be interpreted as the transition amplitude of a quantum dynamical process and may shed light on dynamical quantum phase transitions.