A symmetry principle for Topological Quantum Order

TL;DR

This paper introduces a unifying symmetry-based framework for understanding topological quantum order (TQO), emphasizing gauge-like symmetries and entanglement, and provides conditions and methods for constructing TQO states.

Contribution

It proposes low-dimensional gauge-like symmetries as a unifying principle for TQO and offers a systematic way to construct and analyze TQO states without requiring a finite energy gap.

Findings

All known TQO systems exhibit gauge-like symmetries.

Kitaev's toric code and Wen's plaquette model are dual to an Ising chain.

Thermal fluctuations can cause expectation values of toric operators to vanish.

Abstract

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The guiding principle behind our approach is that of symmetries and entanglement. We introduce the concept of low-dimensional Gauge-Like Symmetries (GLSs), and the physical conservation laws (including topological terms and fractionalization) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The topological defects which are associated with the restoration of GLSs lead to TQO. Selection rules associated with the GLSs enable us to systematically construct states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. All currently known examples of TQO display GLSs. We analyze spectral structures and show that Kitaev's toric code model and Wen's plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain. Despite the spectral gap in these systems, the toric operator expectation values may vanish once thermal fluctuations are present. This mapping illustrates that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string orders in general systems with entangled ground states.