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Higher categorical symmetries and gauging in two-dimensional spin systems

by Clement Delcamp, Apoorv Tiwari

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Clement Delcamp · Apoorv Tiwari
Submission information
Preprint Link: scipost_202303_00011v1  (pdf)
Date submitted: 2023-03-09 17:39
Submitted by: Delcamp, Clement
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries. We then demonstrate that the resulting model has a symmetry structure encoded into the Morita dual of the input fusion 2-category with respect to the corresponding module 2-category. We exemplify this approach by specialising to certain finite group generalisations of the transverse-field Ising model, for which we explicitly define lattice symmetry operators organised into fusion 2-categories of higher representations of higher groups.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 2301.01259v2 on 17 February 2024

Reports on this Submission

Anonymous Report 2 on 2023-12-21 (Invited Report)


1) Provides a concrete realization of the abstract data defining 2-representations of groups and 2-groups as symmetry operators of lattice models.
2) Formulates condensation defects in a Hamiltonian formalism as operators acting on fictitious degrees of freedom instead of a product of projectors, hence connecting to their higher-gauging construction in field theory. Futhermore, investigates condensation defects that could be constructed by 1-gauging non-invertible symmetries (e.g., the surface operators of $2\text{-}\mathrm{Rep}(S_3)$).
3) Provides a framework for gauging symmetries in Hamiltonian lattice models $\textit{with}$ a general choice of twist (i.e., discrete torsion) that has an accompanying "duality operator" implementing the gauging.
4) The paper is well written: it is self-contained, reviews relevant maths, and includes many examples.


1) Many of the dual symmetry and gauging results obtained are known already, albeit from an alternative non-lattice perspective. It would be nice if the authors could go further than this, which is uniquely possible in their lattice models. For example, they could use the dual symmetry to understand how the phases and phase transitions of the original and Morita dual models map onto one another, which can be used to constrain the universality classes of 2nd order phase transitions of either model.
2) The discussion throughout the paper is very formal without a lack of complementary physical descriptions. I suspect this makes the paper challenging to follow for most people who appreciate explicit lattice model constructions.
3) The physical consequences of the symmetries in their model is not discussed.
4) The paper is long and technical but does not have a summary section.


This study thoroughly investigates gauging finite invertible $0$-form symmetries $G$ in two-dimensional Hamiltonian lattice models. Gauging $G$ in $1+1$D models is well understood, and a few places in the literature have investigated higher dimensional generalizations. In this paper, the authors recover these results directly in the context of lattice models and, in doing so, develop a general framework for gauging a $G$ symmetry in Hamiltonian lattice models with a generic choice of twist. It is a nice result and benefits the community by working in a concrete class of models rather than an abstract, model-independent formalism and developing a general framework for gauging ordinary symmetries in such models. The authors apply these results to numerous examples arising as special cases of their general model.

The paper should be published, and I don't request any major changes to be made. That said, there is room for the paper to be improved. Below are some suggestions and comments I have for the authors to consider, as well as some questions the authors may consider answering in this or future work.

- Since the authors study Hamiltonian lattice models, mentioning subsystem symmetries may be worthwhile in the introduction. Otherwise, according to the author's discussion in the introduction, string-net models would have no exact symmetries since the string operators commuting with the Hamiltonian are not topological outside of the ground state subspace.
- A handful of statements made throughout the paper apply only to discrete or finite symmetries but are presented as if generally true. For example, only $\textit{finite}$ non-invertible symmetries are described by fusion $d$-categories, not all non-invertible symmetries. Another example is dual symmetries and gauging, which only applies to gauging discrete sub-symmetries.
-When defining the general Hamiltonian $H \equiv \sum_v h_v$ in section 3.1, would it be possible to write the operator $\mathbb{h}_v$ in a purely two-dimensional way instead of using the pitched cobordism Eq. (3.3)? I found the fictitious interval used to be different from other lattice models I am used to and hard to understand. These fictitious intervals, along with similar graphical representations and string diagrams, are used elsewhere throughout the paper, and I always found them challenging to understand. I understand it may be too inconvenient for the authors to rework this throughout their entire paper, so as an alternative, I suggest they revise the introduction and discussion on these graphical representations to make it more pedagogical.
-It would be interesting to see how the model in section 3.1 can become the Heisenberg model, as stated in the text. This may be demonstrated alongside the transverse field Ising model in that subsection: both models have the same Hilbert spaces, but different choices of coefficients $h_{v,i}$ yield the different models. Also, interestingly, one model has a finite symmetry while the other has a continuous symmetry.
- The starting model before gauging has an ordinary symmetry whose group elements define the Hilbert space. Do the authors expect the generalization to a model with a general fusion 2-category symmetry, which they state they can do in section 6, to be straightforward? I would suspect that the gauging procedure in sec 3.2 would need to be modified since non-invertible symmetries do not have gauge fields. Maybe a more general starting point for gauging is not minimal coupling but instead using symmetry defects. It may be interesting to comment on this in section 6 or elsewhere.
- It is common with Hamiltonian lattice models to assume the Hilbert space has a tensor product decomposition. This isn't the case in this paper since the Morita dual model has a flatness condition imposed at the Hilbert space level. Of course, this flatness condition can be implemented energetically, in which case the low-energy Hilbert space would have the dual symmetry studied here. Do the authors have an idea about what the microscopic symmetry affecting the entire Hilbert space is in such a modified Morita dual model?
- When reading Sec. 3.7, I found myself frequently jumping back to Sec. 3.6 to remind me of the definition of the 2-group $\mathbb{G}$, particularly whether $Q$ or $L$ was the homotopy group of degree $1$ or $2$. I suggest the authors change the notation and denote $Q$ and $L$ as $\pi_1$ and $\pi_2$, so the group $G$ becomes $\pi_1 \ltimes_\phi \pi_2$ in Sec. 3.7. It would make it much smoother to read.
- Could the authors comment on any challenges arising from considering a more general $G$ in section 3.7, such as one that is a nontrivial group extension? The authors do briefly remark in Section 6 that the semidirect product $G$ is not the most general and could be anomalous. But I am curious why they didn't explore such a more general $G$ given how general other parts of the paper are and, therefore, wonder if there are some technical hurdles.
-I found the examples considered in Sec 3.8, 4, and 5 very clear. My only comment is they were lengthy, and there was a lot of overlap behind the general structure of the examples. If the authors could streamline the discussion of the examples, that would benefit the reader.

Requested changes

I only have one requested change. To me, two theories are dual if they are two different-looking theories that describe the same physical system (e.g., particle vortex dualities). The authors use the term "duality" in a much weaker way to mean two theories whose local symmetric operators can be mapped to one another. I am okay with that since it is used this way else way in the literature (e.g., Wen's work since 2018). Still, I think it would be nice for the authors to explicitly remark somewhere that when they say two models are "dual," it does not mean that their partition functions are equivalent nor that they describe the same physical system. Rather, their spectrum and correlation functions match $\textit{only}$ in the symmetric sub-Hilbert space.

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

Report 1 by Matthew Yu on 2023-7-27 (Invited Report)


The main focus is to study 2d lattice models and gauge invertible sub-symmetries. In the Morita dual theory there will be some dual symmetry operators and the authors show how to construct them.

The applications of lattice operators to higher representations is interesting and gives a concrete way of studying these representations.

The authors provide a very clear overview of the transverse-field Ising model and use its gaugings to obtain non-invertible surface operators.

The paper also does a great job at explaining the mathematical background that is needed in a self contained way.


The referee appreciates this lattice point of view of gauging and constructing non-invertible operators. Being able to make abstract categorical statements concrete through realizing them as lattice operators allows for the authors to prove that the resulting model after gauging has the desired symmetry operators.

It would be interesting to see if noninvertible symmetries can really be used to explain or constrain the phases of some lattice system. I also wonder how far these methods generalize to even higher dimensions?

I would recommend this paper for publication

Requested changes

One point that I got confused about: when the authors say two-dimensions, do they really mean (2+1)d in all the cases?

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

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