SciPost Submission Page
Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
by Ken Shiozaki
Submission summary
| Authors (as registered SciPost users): | Ken Shiozaki |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2507.19932v1 (pdf) |
| Date submitted: | Aug. 20, 2025, 3:48 a.m. |
| Submitted by: | Ken Shiozaki |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
Generative AI tools (OpenAI ChatGPT, July 2025) were used for English language editing and translation of parts of the manuscript. No AI tool was used for generating research content or results.
Abstract
We analyze topological phase transitions and higher Berry curvature in one-dimensional quantum spin systems, using a framework that explicitly incorporates the symmetry group action on the parameter space. Based on a $G$-compatible discretization of the parameter space, we incorporate both group cochains and parameter-space differentials, enabling the systematic construction of equivariant topological invariants. We derive a fixed-point formula for the higher Berry invariant in the case where the symmetry action has isolated fixed points. This reveals that the phase transition point between Haldane and trivial phases acts as a monopole-like defect where higher Berry curvature emanates. We further discuss hierarchical structures of topological defects in the parameter space, governed by symmetry reductions and compatibility with subgroup structures.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- Provides a clear introduction and review of prior work on higher Berry phase and higher Berry curvature, and cleanly extends it to the situation of parameter spaces with a group action.
2- Introduces a new kind of topological invariant exclusive to the equivariant setting.
Weaknesses
Report
Requested changes
1- In deriving the formulas for topological invariants and geometric phases defined only in the presence of a group action, i.e. eq. (40-41) and (201), it would be helpful to explain how the formula was derived, if it was obtained through a systematic procedure. (Is it related to the simplicial G complex of appendix A?) If no systematic procedure is known to the author, it would also be helpful to have some indication that this is the case.
2- A natural question about the topological invariant defined only in the presence of a group action is whether the 1d invariant in (201) can be understood as a pump of the 0d invariant in (40-41). A comment on this would be helpful.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
Report
Higher Berry phases in quantum many-body systems have attracted substantial attention in recent years, motivated by the desire to probe the topology of the space of gapped Hamiltonians. In this work, the author offers a fresh perspective by extending the notion of symmetry to parameter space itself. When the parameter space admits fixed points (points invariant under the symmetry action) the associated higher Berry phase localizes and becomes tied to topological invariants defined on these lower-dimensional fixed-point submanifolds.
The concepts and ideas introduced in this manuscript are highly original, particularly the generalization of symmetry actions to parameter space. I believe this framework will inspire many follow-up studies, and I strongly recommend the publication of this work.
I list a few minor comments below; the author may feel free to address them with brief remarks:
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Regarding the mathematical underpinnings, the framework appears closely related to Borel localization in equivariant cohomology. If so, a remark clarifying this connection would be very helpful.
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In Sec. 5, the manuscript discusses certain “constraints’’ on the codimension of defects. I wonder whether the present framework can further elucidate the physical nature of these defects. For example, if a defect is gapless, what kinds of gapless theories are allowed?
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Although this work focuses on invertible gapped systems, I believe the method can be generalized to non-invertible gapped systems, such as the topological orders in (2+1)d. The author may comment on possible interesting applications in this direction.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
1- The paper contains a broad range of new results, which are all interconnected. 2- It is clearly explained how these results connect to previously-known concepts in the theory of invertible phases of matter, and the outlines of an elegant unifying framework are provided. 2- The manuscript is very well-written and clear. The introduction and section 2 are particularly helpful.
Weaknesses
Report
I believe that (1) these results are novel and interesting, and (2) the connections made in this work provide the seeds for extensive follow-up work. Because of these reasons, I think the manuscript meets the acceptance criteria.
Requested changes
I have a few detailed comments on the manuscript that I would like the author to address:
1- Between Eqs (15) and (16): how is \Delta^\circ defined? (in the second bullet point)
2- Eq. (71) contains factors \Lambda^{2/3}. It is not clear to me that this provides a unique definition of the higher Berry connection. For example, what if we define the higher Berry connection with three factors of \Lambda^2 instead, and ensure normalization by dividing by tr(\Lambda(\tau_0)^2 \Lambda(\tau_1)^2\Lambda(\tau_2)^2). It seems that this definition of the higher Berry connection would also meet all the mathematical requirements? Would this affect the topological invariants?
3- On the bottom of page the author mentions that the pair of DDKS defects with values +1 and -1 are prohibited to annihilate because of the Z2 equivariance. This is not immediately clear to me - could some more detailed explanation be provided?
4- Section 5.1 is too telegraphic. A bit more explanation on the bullet points would be helpful.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)