Symmetry and Topological Order

TL;DR

This paper establishes conditions for Topological Quantum Order based on Gauge-Like Symmetries, extending understanding of topological phases at zero and finite temperatures, and identifies new systems exhibiting such order.

Contribution

It introduces a unifying symmetry-based framework for Topological Quantum Order, including new systems with Gauge-Like Symmetries and analyzes their physical implications.

Findings

Gauge-Like Symmetries are key to topological order.

Energy spectrum alone cannot confirm topological order.

Thermal fluctuations can cause 'thermal fragility' in quantum systems.

Abstract

We prove sufficient conditions for Topological Quantum Order at both zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (that notably extend and differ from standard local gauge symmetries) and their associated defects, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges), discuss associated braiding, and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on certain suggested quantum computing schemes and lead to "thermal fragility". Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.