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Adiabatic Deformations of Quantum Hall Droplets
by Blagoje Oblak, Benoit Estienne
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Submission summary
| Authors (as registered SciPost users): | Blagoje Oblak |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2212.12935v2 (pdf) |
| Date submitted: | 2023-04-25 10:45 |
| Submitted by: | Oblak, Blagoje |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider area-preserving deformations of the plane, acting on electronic wave functions through "quantomorphisms" that change both the underlying metric and the confining potential. We show that adiabatic sequences of such transformations produce Berry phases that can be written in closed form in terms of the many-body current and density, even in the presence of interactions. For a large class of deformations that generalize squeezing and shearing, the leading piece of the phase is a super-extensive Aharonov-Bohm term (proportional to N$^2$ for N electrons) in the thermodynamic limit. Its gauge-invariant subleading partner only measures the current, whose dominant contribution to the phase stems from a jump at the edge in the limit of strong magnetic fields. This results in a finite Berry curvature per unit area, reminiscent of the Hall viscosity. We show that the latter is in fact included in our formalism, bypassing its standard derivation on a torus and suggesting realistic experimental setups for its observation in quantum simulators.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2023-6-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.12935v2, delivered 2023-06-03, doi: 10.21468/SciPost.Report.7298
Strengths
Explicit `ab-initio' calculation of the effects of area-preserving diffeomorphisms on the Landau Level quantum states, obtaining the Berry phase and other effects.
Weaknesses
This is the first step in a rather long project of extracting universal features directly from the many-body problem. Results do not yet fully compare with the results of the effective field theory approach, which has received confirmation and cannot be questioned, but rather complemented.
Report
I think this is a very honest approach to explicitly calculating universal features of the Laughlin Hall states (primarily integer), in particular by using their area-preserving symmetry. Most of these features have been already obtained by a variety of methods, sometimes direct, sometimes indirect by guessing the effective field theory. This is nonetheless room for better checking the existing effective theory/conformal theory of edge excitations and for obtaining further geometric features, especially for what concerns the bulk excitations, which are currently being investigated by many authors. I would recommend the acceptance of the paper as it is.
Requested changes
Please add the following papers to the references regarding the W-infinity symmetry: Cappelli and Maffi, arXiv:2103.04163, arXiv:1801.03759. These papers are more recent than those already referred to and discuss explicit properties of Laughlin states in the same spirit as the present work.
Report #1 by Anonymous (Referee 4) on 2023-5-31 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.12935v2, delivered 2023-05-31, doi: 10.21468/SciPost.Report.7285
Strengths
Elegant result for the Berry phase for general geometric deformations.
Clear and eloquent presentation.
Weaknesses
Relation to previous calculations of the Hall viscosity not clear.
Claims on the possible measurement of the Berry phase seem a bit far-fetched.
Report
The manuscript addresses the response of many body quantum systems, specifically quantum Hall fluids, to adiabatic geometry deformations, and the resulting Berry phase. The Authors obtain an elegant expression, composed of two parts, one containing the current, which mainly stems from the edge region, and the second representing a bulk Aharonov-Bohm phase. They discuss the relation of their result to the Hall viscosity in the case of specific deformations (more on this below). This is a very nice result, which goes beyond previous discussions in the field (limited to restricted deformations). The presentation is quite eloquent and clear. I therefore believe this manuscript could be suitable for publication in SciPost Physics. However, several points need to be addressed first:
My main concern is the discrepancy between the Berry phase found in this work and the corresponding Hall viscosity calculations in the literature. Perhaps the most relevant here is Ref. [26], which finds the Berry curvature (from which the Berry phase of course follows) for linear transformations (which are enough for calculating the viscosity) in a plane (rather than torus) geometry, similarly to the current manuscript. Sec. II.A shows how by careful calculation the super-extensive contributions cancel out, and the correct Hall viscosity arises. Why does the result in the manuscript seem to be different?
Let me also note that Ref. [26] considers more general states (beyond integer and Laughlin quantum Hall states), and work finds that in general the Hall viscosity is related to the orbital spin. Let me suggest the Authors apply their formula to cases such as neutral p+ip superfluids or the Moore-Read state, and compare the Berry phase they find with the general result for the Hall viscosity.
Another issue I would like to mention is the Author's suggestion that the Berry phase might offer a better way to extract the Hall viscosity than current measurement schemes. Although electron hydrodynamic experiments are challenging, extracting the many body Berry phase seems much harder. Could the Authors comment on that, or else modify their statement?
Requested changes
See above.