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Lieb-Schultz-Mattis theorem in long-range interacting systems and generalizations
by Ruizhi Liu, Jinmin Yi, Shiyu Zhou, Liujun Zou
Submission summary
| Authors (as registered SciPost users): | Liujun Zou |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00041v1 (pdf) |
| Date submitted: | 2024-06-19 05:40 |
| Submitted by: | Zou, Liujun |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In a unified fashion, we establish Lieb-Schultz-Mattis theorem in long-range interacting systems and its generalizations. We show that, for a quantum spin chain, if the multi-spin interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, this theorem holds when the interactions decay faster than $1/r^2$, with $r$ the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.
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
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Report
The authors have generalized the LSM Theorem to long-range interacting systems. Overall the result seems solid and - with a perhaps minor issue that we discuss below in point 3 - correct. Therefore it seems suitable for SciPost Physics eventually.
However we felt that the authors could improve a number of points in the paper at present, which we list below.
1) It seems to us like the new content in the paper is essentially what is in Appendices D and E, and then summarized in the main text -- do the authors agree with this assertion? In some sense this is rather awkward, because the vast majority of the appendices are in A, B and C. Our impression is that this paper essentially just points out that certain results -- maybe Lemma D.1 in particular -- can just be used to generalize LSM to long-range interacting systems with very little work. But in this respect the length of the paper seems a bit confusing, as the actual contribution of the authors was not so lengthy. If Appendices A, B and C are really just intended as a review for the reader, we felt that should be a lot more explicit, and in some sense perhaps the paper could be better organized to highlight the aspects of the work that are new. For example, the proof of Theorem A.5 is not rigorous and the authors refer to Ref. [61] for a rigorous treatment, but this seems like a crucial step in the proofs in Appendices D/E. We assume then that this result is not new. Similarly, it's not clear based on the presentation whether it was known in prior literature how to use C* algebraic results for power-law interacting systems.
2) Relatedly, we are not sure why the authors use the C*-algebraic formalism which seems to take up the vast majority of the paper, given that they state in Theorem III.2 that real systems are finite-size after all. It might be worth commenting on how much simpler some of the key proofs could be if the authors restricted to a genuinely finite system.
3) As far as we can tell, Ref. [37] does not use a C*-algebraic formalism. Is it clear that their results can be ported to C* algebra formalism? Also, Ref. [37] only deals with unique ground states, but in this paper they wish to use it for only locally-unique ones. Why is that justified?
4) Below Definition E.1, depending on the definition of "local Hamiltonian", it is not true that exponential tails can be achieved in finite time evolution, see e.g. the strongest Lieb-Robinson bounds for nearest-neighbor systems in Ref. [51]. If local allows for exponential tails already then the result is true but seems rather semantic as one might as well have just included exponential tails in h_j in Definition E.1 to begin with. The wording can be clarified here.
5) The authors mention locality preserving automorphisms in the main text but do not explain its meaning. From the appendix, this notion is like QCA with the strict locality condition relaxed; some heuristic understanding could be pointed out in the main text.
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