Probing topological order with R\'enyi entropy

TL;DR

This paper analytically investigates the quantum phase transition in a topologically ordered system, showing that the topological Renyi entropy acts as a discrete invariant distinguishing phases.

Contribution

It demonstrates that the topological Renyi entropy exhibits a discontinuity at the phase transition, establishing it as a reliable topological invariant.

Findings

Discontinuity in topological Renyi entropy at the phase transition

Different constant values of entropy characterize the two phases

Exact and perturbative methods confirm the entropy's role as a topological invariant

Abstract

We present an analytical study of the quantum phase transition between the topologically ordered toric-code-model ground state and the disordered spin-polarized state. The phase transition is induced by applying an external magnetic field, and the variation in topological order is detected via two non-local quantities: the Wilson loop and the topological Renyi entropy of order 2. By exploiting an equivalence with the transverse-field Ising model and considering two different variants of the problem, we investigate the field dependence of these quantities by means of an exact treatment in the exactly solvable variant and complementary perturbation theories around the limits of zero and infinite fields in both variants. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological Renyi entropy and that the two phases around the phase transition point are characterized by its different constant values. Our results therefore indicate that the topological Renyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.