On Ising and dimer models in two and three dimensions

TL;DR

This paper explores advanced mappings between Ising and dimer models in two and three dimensions, revealing new insights into phase transitions, frustration, and topological order, and introduces an exactly soluble quantum eight vertex model.

Contribution

It introduces a modified Fisher's mapping that enables frustrated Ising models to be represented as positive-weight dimer models and establishes an exact correspondence between 3D Ising models and quantum dimer models.

Findings

Deconfined transition in dimer language for Ising ferromagnets.

New solution for the fully frustrated Ising model on the square lattice.

Exact map between 3D Ising models, gauge theories, and quantum dimer models.

Abstract

Motivated by recent interest in 2+1 dimensional quantum dimer models, we revisit Fisher's mapping of two dimensional Ising models to hardcore dimer models. First, we note that the symmetry breaking transition of the ferromagetic Ising model maps onto a non-symmetry breaking transition in dimer language -- instead it becomes a deconfinement transition for test monomers. Next, we introduce a modification of Fisher's mapping in which a second dimer model, also equivalent to the Ising model, is defined on a generically different lattice derived from the dual. In contrast to Fisher's original mapping, this enables us to reformulate frustrated Ising models as dimer models with positive weights and we illustrate this by providing a new solution of the fully frustrated Ising model on the square lattice. Finally, by means of the modified mapping we show that a large class of three-dimensional Ising models are precisely equivalent, in the time continuum limit, to particular quantum dimer models. As Ising models in three dimensions are dual to Ising gauge theories, this further yields an exact map between the latter and the quantum dimer models. The paramagnetic phase in Ising language maps onto a deconfined, topologically ordered phase in the dimer models. Using this set of ideas, we also construct an exactly soluble quantum eight vertex model.