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

### Submission summary

 As Contributors: Jutho Haegeman · Laurens Lootens · Christoph Schweigert · Frank Verstraete Arxiv Link: https://arxiv.org/abs/2008.11187v3 (pdf) Date accepted: 2021-02-24 Date submitted: 2021-02-17 19:59 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.

Published as SciPost Phys. 10, 053 (2021)

We thank the referees for their reading of the manuscript and their positive reports. To address the comments of the first referee:

The first mention of the bimodule associators on page 8 occurs without their definition (apart from a brief reference to the appendix). Since these equations (13) are already rather complicated, this made the section difficult to follow on first reading. At the very least, I'd recommend a more direct reference to the relevant sections of the appendices, including how to read the triple line notation'.

We have added additional information on these associators before they are used in equations (13), and provided a direct reference to the relevant section in the appendix. A more detailed explanation and their precise definition would require the background of fusion, module and biomdule categories, which we feel would obscure the fact that these objects appear naturally in the tensor network description.

I found the discussion of the TFT boundaries rather confusing. This is not my area of expertise, and I struggled to follow why the two boundaries are treated so differently. Should I think of the choice of a physical boundary as using up' that end of the manifold?

We have added an additional paragraph explaining what would happen if we consider both boundary conditions to be either gluing or physical boundaries. The physical boundaries have no degrees of freedom on them and can be thought of as boundary conditions; in this sense, they indeed `use up' a particular part of the manifold.

I feel that the existing literature on topological phase transitions should be cited more prominently. The concept of anyon condensation has for example been pioneered in F. A. Bais and J. K. Slingerland Phys. Rev. B 79, 045316.

We have added references on this particular topic.

Overall, it might make it more clear when emphasizing which parts are reformulations of existing relations and what are new results that emerge from the formulation in terms of tensor networks.

The tensor network perspective, including the various equations that in the end correspond to pentagon equations of a bimodule category, has been previously described and various properties of these systems (i.e. topological entanglement entropy) have been studied in the tensor network language before. We think the main contribution of this work is the realization that the MPO symmetries and the string-net model do not have to be described by a single fusion category, but rather that there is additional flexibility in this description and that a complete description of MPO symmetries requires the consideration of invertible bimodule categories, which is mentioned in the introduction. This observation provides insight into the nature of PEPS itself, but we also expect it to provide a useful framework for considering e.g. phase transitions. Other new results include a PEPS description of boundaries and domain walls in string-net models, which requires the flexibility of these more general representations, as well as a generalized interpretation in terms of Turaev-Viro state sum models with boundary.

Given that the “pull through condition” is used extensively, it might be helpful to introduce it in some more detail right in the beginning.

We have added extra motivation for this pulling through condition, and why it is a natural feature of models with topological order.

### List of changes

- Various spelling/grammar mistakes corrected
- Additional explanation on the pulling-through conditions in Section 2
- Additional reference to the appendix regarding the bimodule associators used in Eqs (13)
- Corrected a mistake in Eq (27)
- Additional explanation on the difference between gluing and physical boundary in Section 5, after Eq (51)