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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
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Submission summary
| Authors (as registered SciPost users): | Jürgen Fuchs · Jutho Haegeman · Laurens Lootens · Christoph Schweigert · Frank Verstraete |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2008.11187v2 (pdf) |
| Date submitted: | 2020-10-09 10:30 |
| Submitted by: | Lootens, Laurens |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-1-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2008.11187v2, delivered 2021-01-24, doi: 10.21468/SciPost.Report.2460
Report
The authors use bimodule category to formulate the basic concepts of topologically ordered phases using tensor networks. It is shown in detail how the consistency conditions arising from non-local matrix-product operator (MPO) symmetries are equivalent to pentagon equations. These bimodule categories furthermore allow to construct MPO “intertwiners” between different projected entangled pair states (PEPS) representations of the same string-net. This result provides a generalization of virtual gauge transformations between PEPS that describe the same state.
The manuscript is on an exciting topic and overall clearly written. The “dictionary” between the formulation in terms of topological fields theories and tensor networks will be useful for a practitioners in the field. As pointed out by the referees, the framework could be very useful for simulations of error-correcting codes based on string-net models.
I thus find the manuscript interesting and sufficiently relevant for publication in SciPost Physics after addressing the minor comments below.
Requested changes
- 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.
- 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.
- Given that the “pull through condition” is used extensively, it might be helpful to introduce it in some more detail right in the beginning.
Report #1 by Anonymous (Referee 1) on 2020-11-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2008.11187v2, delivered 2020-11-16, doi: 10.21468/SciPost.Report.2197
Report
The authors examine the structure of MPO symmetries in PEPS states. In particular, they identify that the complete set of equations obeyed by the PEPS tensors, virtual MPO symmetry tensors, and the associated fusion tensors, as the set of pentagon equations obeyed by two fusion categories $\mathcal{C},\mathcal{D}$, and a $(\mathcal{C},\mathcal{D})$ bimodule category. The PEPS tensor itself is realized by the `module' F symbols ${}^3\!F$, allowing the generalization of previous string-net PEPS to the full Morita class. Additionally, explicit virtual MPO intertwiners between different string-nets are examined.
Finally, the authors show how these PEPS representations, realized by general invertible bimodules, correspond to TFTs, and therefore place them on a firm mathematical footing.
As the authors state, realizing PEPS via distinct module categories in this way may be useful for studying phase transitions out of the string-net phase. In particular, how the underlying microscopic model impacts this behavior. It is the framework for this work that I see as the main contribution of this work.
I find this paper both interesting, and generally well written, and recommend publication. I have some recommended changes:
- The first mention of the bimodule associators ${}^1\!F-{}^3\!F$ 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'.
- 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 $\Sigma\times\{0\}$ and $\Sigma\times\{1\}$ are treated so differently. Should I think of the choice of a physical boundary as `using up' that end of the manifold?