Finite temperature topological order in 2D topological color codes

TL;DR

This paper investigates how finite temperature affects topological order in 2D color codes, revealing the fragility of topological entropy and the role of gauge symmetry through exact solutions and entropy analysis.

Contribution

It provides an exact solution for entanglement entropy at finite temperature in 2D color codes and analyzes the impact of temperature and topology on topological order.

Findings

Topological entropy vanishes as temperature increases in finite systems.

In the thermodynamic limit, topological order is fragile even at low temperatures.

The order of limits affects the topological entropy, linked to the topology of regions.

Abstract

In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.