Anyons in a highly-entangled toric xy model

TL;DR

This paper demonstrates that the classical xy model, when gauge invariance is enforced, exhibits purely topological order similar to Kitaev's quantum double model, revealing a deep connection between classical and quantum topological phases.

Contribution

It shows that the classical xy model can be made topologically ordered by enforcing gauge invariance, linking it to quantum topological models like Kitaev's quantum double.

Findings

Classical xy model with gauge invariance exhibits topological order.

Quantum xy topological order corresponds to the infinite lattice limit of Kitaev's quantum double model.

The obstruction to topological order in the xy model is due to non-topological U(1) gauge action.

Abstract

While ostensibly coined in 1989 by Xiao-Gang Wen, the term "topological order" has been in use since 1972 to describe the behavior of the classical xy model. It has been noted that the xy model does not have Wen's topological order since it is also subject a non-topological U(1) gauge action. We show in a sense this is the only obstruction. That is, if gauge invariance is enforced energetically then the $xy$ model becomes purely topologically ordered. In fact, we show that the quantum $xy$ topological order is an infinite lattice limit of Kitaev's quantum double model applied to the group G=Z.