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A unified diagrammatic approach to topological fixed point models
by A. Bauer, J. Eisert, C. Wille
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|Authors (as registered SciPost users):||Andreas Bauer · Carolin Wille|
|Preprint Link:||https://arxiv.org/abs/2011.12064v2 (pdf)|
|Date submitted:||2020-12-23 14:17|
|Submitted by:||Bauer, Andreas|
|Submitted to:||SciPost Physics|
We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism directly represent discrete path integrals in Euclidean space-time. This language is more immediately related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We illustrate our formalism at hand of simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.
Published as SciPost Phys. Core 5, 038 (2022)
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:2011.12064v2, delivered 2022-01-30, doi: 10.21468/SciPost.Report.4277
This paper looks at constructing a unified diagrammatic framework to understand topological phases of matter. This is done by considering tensor networks representing Euclidean path integrals, up to certain equivalences. As examples it is shown how this framework can be applied to certain spin systems in 2+1-dimensions, and Fermionic systems in 3+1-dimensions.
The framework considered in this paper is relatively high level, which gives both its major strengths and weaknesses. As the authors concede, they do not consider how such fixed-point models may be constructed, nor does this work seek to classify such systems. Given this, it is difficult to determine what value this framework adds relative to existing approaches. While there are some aspects of this approach that markedly differ from previous work (most notably the focus on path integrals instead of ground states), it is far from clear whether these differences actually add any explanatory or predictive power to the framework.
The structure and style of the paper strike a rather nice balance between the motivation and ideas, and the nitty-gritty details. While it is impressive that their unified framework can be applied to all of the exemplary systems they considered, I don't think that alone is enough for me to recommend this work for publication in its current state. If, however, the authors could focus a little more on the unique aspects of their framework and make a convincing case that it might possess some explanatory/predictive power beyond existing approaches, then I think this work would certainly be worth of publication.
Abstract: "is reminiscent to that of" should be "is reminiscent of".
Abstract: "We illustrate our formalism at hand of simple examples" doesn't scan. Perhaps something like "We illustrate our formalism with simple examples" or something similar.
Page 21: "1+-dimensional case", I think this should read "1+1-dimensional case"?
Page 39: "dimer cristal" should be "dimer crystal"
Page 40: "remain elusive as of to date" should be "remain elusive to date"
Page 63: "coloring" should be "colouring"
- Cite as: Anonymous, Report on arXiv:2011.12064v2, delivered 2021-01-20, doi: 10.21468/SciPost.Report.2434
1. Very clear exposition
2. integrates lots of results scattered in the literature
1. The authors promise to be more general than known approaches, but at the end they only discuss fixed point models
2. Intermingle mathematically rigorous statements with sloppy ones
3. No clear message or results
This paper is rather a review paper than a research paper, and explains how topological field theory and their Euclidean path integral representations in terms of state-sums can be interpreted as tensor networks. The goal is "to introduce a comprehensive systematic, unified, easily accessible, and generalized language for understanding and exploring fixed-point models for quantum phases of matter"; the authors seem to have succeeded partly in their endeavor, but their constructions seem to be effectively equivalent to the ones already present in the literature and it is not clear at all that they are any simpler.
The authors set out to define tensor networks for ground states of quantum spin Hamiltonians of interest by Trotterizing Euclidean path integrals, and claim that this is "the key difference" with known constructions. Such constructions are certainly not new, and seem to have been used to construct the first PEPS representations of topological states such as the toric code.
The main body of the paper discusses fixed point quantum spin models (zero-correlation length) in one, two and three dimensions. The fact that those models can be represented as tensor networks might sound nontrivial, but the authors show convincingly that this indeed trivially follows from the path integral construction (this seems to have been known since the early days of tensor networks). There might however be a fundamental flaw in such constructions for systems that are away from the zero correlation length idealizations such as the AKLT model: in the latter case the Trotterization will lead to an integer spin bond dimension, while it should of course be half-integer. It is not clear how this topological feature emerges from the construction described in the paper.
In summary, this paper is certainly a beautiful summary and exposition of the tensor network way of looking at topological field theories. However, it is difficult to identify any new ideas which would allow this formalism to go beyond what is already known in the field.
Some more remarks:
1. In equation 16 of their paper, the authors formulate a conjecture about Trotterization. The exponential decay seems to be in violation with the bounds derived in e.g [M.B. Hastings, Phys. Rev. B 73, 085115 (2006)]. In the same section, the definitions used for notions such as quasi-adiabatic evolution are very confusing to the reader not yet familiar with them.
2. In the outlook, the authors stress that a big advantage of their method is given by the fact that "the moves turn into polynomial equations for the tensor entries; we can find roots to those equations by applying a Gauss-Newton method". The equations that they have to solve are most probably equivalent to the pentagon equations in the field of fusion category, and those are notoriously hard to solve and definitely not simply solvable by Gauss-Newton methods.