Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling

TL;DR

This paper provides an exact solution to the classical and quantum dimer models on the recently discovered spectre aperiodic monotile, revealing a deconfined phase with infinitely separable monomers at zero energy cost.

Contribution

It introduces an exact solution for the dimer and quantum dimer models on the spectre aperiodic monotile, a novel aperiodic tiling shape, and uncovers a deconfined phase in the quantum model.

Findings

Exact partition function for the classical dimer model.

Identification of an eigenbasis for the quantum dimer model at all interaction strengths.

Existence of a deconfined phase with infinitely separated monomers at zero energy cost.

Abstract

The decades-long search for a shape that tiles the plane only aperiodically under translations and rotations recently ended with the discovery of the `spectre' aperiodic monotile. In this setting we study the dimer model, in which dimers are placed along tile edges such that each vertex meets precisely one dimer. The complexity of the tiling combines with the dimer constraint to allow an exact solution to the model. The partition function is $\mathcal{Z}=2^{N_{\textrm{Mystic}}+1}$ where $N_{\textrm{Mystic}}$ is the number of `Mystic' tiles. We exactly solve the quantum dimer (Rokhsar Kivelson) model in the same setting by identifying an eigenbasis at all interaction strengths $V/t$. We find that test monomers, once created, can be infinitely separated at zero energy cost for all $V/t$, constituting a deconfined phase in a 2+1D bipartite quantum dimer model.