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Fractional domain wall statistics in spin chains with anomalous symmetries
by Jose Garre Rubio, Norbert Schuch
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | José Garre-Rubio |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2405.00439v1 (pdf) |
| Date submitted: | 2024-05-15 19:05 |
| Submitted by: | Garre-Rubio, José |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the statistics of domain wall excitations in quantum spin chains. We focus on systems with finite symmetry groups represented by matrix product unitaries (MPUs), i.e. finite depth quantum circuits. Such symmetries can be anomalous, in which case gapped phases which they support must break the symmetry. The lowest lying excitations of those systems are thus domain wall excitations. We investigate the behavior of these domain walls under exchange, and find that they can exhibit non-trivial exchange statistics. This statistics is completely determined by the anomaly of the symmetry, and we provide a direct relation between the known classification of MPU symmetry actions on ground states and the domain wall statistics. Already for the simplest case of a $\mathbb Z_2$ symmetry, we obtain that the presence of an anomalous MPU symmetry gives rise to domain wall excitations which behave neither as bosons nor as fermions, but rather exhibit fractional statistics. Finally, we show that the exchange statistics of domain walls is a physically accessible quantity, by devising explicit measurement operators through which it can be determined.
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
- Generality of the method
- Clarity of the presentation
Report
This paper clarifies the relationship between the statistics of domain-wall excitations and the anomaly of symmetries in one-dimensional systems.
Requested changes
1. Throughout the manuscript, the notation for the system size alternates between n and N. These should be unified.
2. Regarding Eq. (6), is $m >\geq0$ a typo?
3. For the domain wall tensor $e_{AB}^i$, is there a necessity for the leg $i$ to have the same dimension as the physical leg?
4. Regarding Eq. (8), the MPS tensors from site $1$ to $k$ are missing.
5. Regarding Eq. (10), while this equation seems to be a sufficient condition, is it also a necessary condition? Could you provide the derivation of Eq.(10)?
6. In Phys. Rev. B 107, 155136, they introduce symmetry operators truncated to a segment (referred to as “patch operators”) and specifically calculate the F-symbol for anomalous $Z_2$ symmetry. Since the calculation for $Z_2$ symmetry appears to be essentially the same, wouldn’t it be appropriate to cite this work?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
- The interplay between symmetry anomalies and topological features in low-dimensional quantum systems represent a novel and timely topic in condensed matter physics
- The use of tensor networks (MPS and MPUs) is rigorous and builds on a well-established framework
- The paper both general and specific cases, providing clarity on how anomalies manifest in exchange statistics.
- The paper is well-organized, with detailed mathematical derivations.
Weaknesses
- The paper is largely theoretical and could potentially benefit from accompanying numerical simulations (e.g. by tuning away from fixed points of the model and measuring the finite sie operator proposed in Sec. V).
Report
The paper investigates the exchange statistics of domain wall excitations in one-dimensional (1D) quantum spin chains with anomalous symmetries, utilizing Matrix Product States (MPS) and Matrix Product Unitaries (MPU) as key theoretical frameworks. It provides a connection between the anomaly of the symmetry and the physical properties of domain wall excitations, presenting a method to measure these properties experimentally. The work extends the classification of anomalous gapped phases beyond ground state properties and links them to physically measurable quantities in low-energy excitations.
Requested changes
- It might be helpful to clarify the emergence of anyons in 1D and their stability. Generally, one would expect that anyons can only emerge in presence of topological order, which is absent in 2D.
- Maybe there are some typos following Eq. (1): Should it be Z_i = Z_j = 1 instead of I=j=1? Also, should it read \mu > 1 instead of \mu>0?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)