Matrix product operator symmetries and intertwiners in string-nets with
domain walls
- URL: http://arxiv.org/abs/2008.11187v3
- Date: Tue, 16 Feb 2021 16:01:25 GMT
- Title: Matrix product operator symmetries and intertwiners in string-nets with
domain walls
- Authors: Laurens Lootens, J\"urgen Fuchs, Jutho Haegeman, Christoph Schweigert,
Frank Verstraete
- Abstract summary: We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models.
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.
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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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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