Gauge symmetry and non-abelian topological sectors in a geometrically constrained model on the honeycomb lattice

TL;DR

This paper investigates a constrained 2D statistical-mechanical model on the honeycomb lattice, revealing gauge symmetry, non-abelian topological sectors, and unique transfer matrix properties, with implications for topological phases.

Contribution

It introduces a new constrained model with multiple descriptions, analyzes its gauge symmetry and topological sectors, and diagonalizes its transfer matrix revealing Fibonacci eigenvalues.

Findings

Transfer matrix eigenvalues relate to Fibonacci numbers.

Model exhibits gauge symmetry and non-abelian topological sectors.

Evidence suggests the model is not critical at infinite temperature.

Abstract

We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions. They are 1) the Ising model on the honeycomb lattice, constrained to have three up spins and three down spins on every hexagon, 2) the three-color/fully-packed-loop model on the links of the honeycomb lattice, with loops around a single hexagon forbidden, and 3) three Ising models on interleaved triangular lattices, with domain walls of the different Ising models not allowed to cross. Unlike the three-color model, the configuration space on the sphere or plane is connected under local moves. On higher-genus surfaces there are infinitely many dynamical sectors, labeled by a noncontractible set of nonintersecting loops. We demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice. This enables us to diagonalize the original transfer matrix for up to 36 sites, finding an entropy per plaquette ~ 0.3661 and substantial evidence that the model is not critical. We also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of Fibonacci numbers. We comment on the possibility of a topological phase, with infinite topological degeneracy, in an associated two-dimensional quantum model.