Galois conjugates of String-net Model

TL;DR

This paper explores non-Hermitian Galois conjugates of Hermitian string-net models, showing they have real spectra and recover topological properties through bi-orthogonal basis analysis, with implications for understanding non-Hermitian topological phases.

Contribution

It introduces a framework for analyzing Galois conjugates of string-net models, demonstrating real spectra and topological invariants in a bi-orthogonal basis, extending topological order concepts to non-Hermitian systems.

Findings

Models have real spectra despite non-Hermiticity

Topological entanglement entropy relates to quantum dimension

Results primarily demonstrated in the Yang-Lee model

Abstract

We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate bi-orthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the bi-orthogonal basis thetopological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates.