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.
Highlights
We showed that the consistency conditions of having non-local matrix product operator (MPO) symmetries encoded by a fusion category C in a projected entangled pair state (PEPS) representation of a string-net based on a spherical fusion category D are equivalent to the pentagon equations of a (C, D)-bimodule category M, thereby classifying explicit representations of the PEPS and MPO tensors
These bimodule categories allowed us to construct MPO intertwiners between different PEPS representations of the same string-net providing a generalisation of virtual gauge transformations between PEPS that describe the same state
An important conclusion to be drawn from these MPO intertwiners is that they relate equivalent PEPS tensors with possibly distinct virtual bond dimensions
Summary
A large class of (2+1)D spin systems exhibiting topological order can be constructed using the string-net condensation mechanism introduced by Levin and Wen [1], leading to the string-net models which are classified by the data of a unitary fusion category (UFC) D. The set of MPO symmetries is finite and closed under multiplication, and their properties can be described by the data of a UFC C [4,6] Such MPO symmetries can be used to construct the different ground states of the string-net model on a torus, as well as its anyonic excitations through an explicit construction of the Ocneanu tube algebra elements which form a basis for Z(C), the monoidal center of C. Using the more general PEPS representations for string-nets mentioned above, we define explicit tensor network representations for these boundaries and domain walls While these features have been fully understood abstractly in the setting of category theory [9,10,11,12,13], tensor networks allow to devise actual tensors with all required properties and put those to work on the computer. On the other hand we hope that this formulation will allow readers who are familiar with Turaev-Viro constructions to better understand these particular tensor networks and enrich the computational power for Turaev-Viro models with tensor network methods
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