Topological and Error-Correcting Properties for Symmetry-Protected Topological Order

TL;DR

This paper explores the error-correcting and topological properties of symmetry-protected topological (SPT) orders in bosonic systems, revealing their stability and connections to symmetry-breaking orders through information-theoretic measures.

Contribution

It demonstrates that SPT orders form quantum error-correcting codes with macroscopic classical distance and introduces new topological entanglement entropy measures to distinguish different quantum phases.

Findings

SPT ground states form quantum error-correcting codes with macroscopic classical distance.

New topological entanglement entropy types can differentiate topological, SPT, and symmetry-breaking orders.

SPT orders are stable against local errors and linked to symmetry-breaking properties.

Abstract

We discuss the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic distance, providing a new angle for the hidden symmetry-breaking properties in SPT orders. We propose new types of topological entanglement entropy that probe the STP orders hidden in their symmetric ground states, which also signal the topological phase transitions protected by symmetry. Combined with the original definition of topological entanglement entropy that probes the 'intrinsic topological orders', and the recent proposed one that probes the symmetry-breaking orders, the set of different types of topological entanglement entropy may hence distinguish topological orders, SPT orders, and symmetry-breaking orders, which may be mixed up in a single system.