Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity

TL;DR

This paper redefines topological order using the concept of gapped quantum liquids, introduces new invariants to distinguish order types, and explores how stochastic local transformations reveal the emergence of unitarity in topologically ordered states.

Contribution

It introduces the concept of gapped quantum liquids, links topological order to stable ground-state degeneracy, and demonstrates how stochastic local transformations distinguish topological from symmetry-breaking orders.

Findings

Gapped quantum liquids can dissolve product states and relate to topological order.

Topological entanglement entropy probes symmetry-breaking properties without explicit symmetry knowledge.

Topological orders are stable under stochastic local transformations, indicating emergent unitarity.

Abstract

In this work we present some new understanding of topological order, including three main aspects: (1) It was believed that classifying topological orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of \emph{gapped quantum liquid} as a special kind of gapped quantum states that can "dissolve" any product states on additional sites. Topologically ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both topologically ordered states and symmetry-breaking states. This allows us to use a gLU invariant -- topological entanglement entropy -- to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking orders even without knowing the symmetry and the associated order parameters. (3) The universality classes of topological orders and symmetry-breaking orders can be distinguished by \emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible transformations}): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the topological-order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a new definition of long-range entanglement based on SL transformations, under which only topologically ordered states are long-range entangled.