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A unified diagrammatic approach to topological fixed point models

by A. Bauer, J. Eisert, C. Wille

Submission summary

As Contributors: Andreas Bauer
Arxiv Link: (pdf)
Date submitted: 2020-12-23 14:17
Submitted by: Bauer, Andreas
Submitted to: SciPost Physics
Academic field: Physics
  • Quantum Algebra
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


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.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2011.12064v2 on 23 December 2020

Reports on this Submission

Anonymous Report 1 on 2021-1-20 Invited Report


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.

  • validity: high
  • significance: ok
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: excellent

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