Symmetry-deformed toric codes and the quantum dimer model

TL;DR

This paper explores symmetry-based deformations of the toric code, revealing how certain modifications lead to models with subsystem symmetries and rich phase diagrams, but do not support topological or fracton order.

Contribution

It systematically deconstructs the toric code to introduce global symmetries, analyzes the resulting models' properties, and studies the quantum dimer model's phase behavior and emergent symmetries.

Findings

Deformation models exhibit subsystem symmetries and ground-state degeneracies.

Loss of gauge symmetry prevents these models from supporting topological or fracton order.

Quantum dimer model shows an emergent SO(2) symmetry and instability towards valence bond solid formation.

Abstract

Motivated by the recent introduction of a $U(1)$-symmetric toric code model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the toric code (TC) Hamiltonian. This "term-dropping" protocol introduces global symmetries that go beyond the alternative framework of "ungauging" topological order in symmetry-deformed models and gives rise to models such as the $U(1)$TC or $XY$TC. These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do not support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.