Steady-state topological order

TL;DR

This paper extends the concept of topological order to open quantum systems by analyzing steady states, their degeneracy, entropy, and phase transitions, providing new models and theoretical tools.

Contribution

It introduces models and methods for studying steady-state topological order in open systems, including characterizations via degeneracy, entropy, and gauge theory, and explores phase transitions.

Findings

Liouvillian gap closes in the thermodynamic limit

Steady-state topological order persists with gapless modes

Topological phase transition involves gapping out gapless modes

Abstract

We investigate a generalization of topological order from closed systems to open systems, for which the steady states take the place of ground states. We construct typical lattice models with steady-state topological order, and characterize them by complementary approaches based on topological degeneracy of steady states, topological entropy, and dissipative gauge theory. Whereas the (Liouvillian) level splitting between topologically degenerate steady states is exponentially small with respect to the system size, the Liouvillian gap between the steady states and the rest of the spectrum decays algebraically as the system size grows, and closes in the thermodynamic limit. It is shown that steady-state topological order remains definable in the presence of (Liouvillian) gapless modes. The topological phase transition to the trivial phase, where the topological degeneracy is lifted, is accompanied by gapping out the gapless modes. Our work offers a toolbox for investigating open-system topology of steady states.