Variational methods for characterizing matrix product operator symmetries

TL;DR

This paper introduces a variational method to analyze topological order in strongly correlated 2D systems using iPEPS, extracting anyon fusion and braiding properties through symmetry and tensor network techniques.

Contribution

It develops a novel approach combining variational tensor network methods with the fundamental theorem of MPS to characterize topological order and anyon statistics in 2D quantum states.

Findings

Successfully applied to various topological models including toric code and double Fibonacci.

Extracted topological $F$-symbols, $S$, and $T$ matrices from ground states.

Demonstrated the method's ability to identify and classify topological phases.

Abstract

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$-symbols of the underlying fusion category, which unequivocally identify the topological order of the ground state. We bring the $F$-symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual- and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of $Z_3$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of $S_3$ and ${\rm Rep}(S_3)$ string net models.