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Matrix product operator symmetries and intertwiners in string-nets with domain walls

by Laurens Lootens, Jürgen Fuchs, Jutho Haegeman, Christoph Schweigert, Frank Verstraete

Submission summary

As Contributors: Laurens Lootens
Arxiv Link: https://arxiv.org/abs/2008.11187v2 (pdf)
Date submitted: 2020-10-09 10:30
Submitted by: Lootens, Laurens
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Mathematical Physics
  • Quantum Algebra
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical

Abstract

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.

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Submission 2008.11187v2 on 9 October 2020

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