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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  

Preprint Link:  https://arxiv.org/abs/2212.12935v3 (pdf) 
Date accepted:  20230913 
Date submitted:  20230728 18:11 
Submitted by:  Oblak, Blagoje 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider areapreserving 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 manybody 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 superextensive AharonovBohm term (proportional to N$^2$ for N electrons) in the thermodynamic limit. Its gaugeinvariant 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.
Author comments upon resubmission
List of changes
1. The main concern of referee #1 was "the discrepancy between the Berry phase found in [our] work and the corresponding Hall viscosity calculations in the literature." As explained in the manuscript, there is no discrepancy since what we compute is *not* Hall viscosity, despite striking similarities. This was already stated in the original paper, but we have now emphasized the point even more by adding several sentences throughout the text:

A. In the introduction (bottom of page 4), we've added the sentence "In this sense, there was no reason for the Berry phase (1) to be related to viscosity at all; it just so happens that its extensive piece is proportional to the Hall viscosity," along with the footnote "The proportionality could have been guessed on geometric grounds: the parameter space for linear maps (2) is a hyperbolic plane, which has a unique SL(2, R)invariant Berry curvature up to normalization."

B. Below eq. (75), we've split an earlier sentence in two in order to give more details. The sentence now reads: "The factorization between a and b pieces is manifest, respectively corresponding to deformations of the metric and the (slowly varying) confining potential in the Hamiltonian (36). The resulting Berry phase (49) consists of two parts, respectively involving expectation values of (a†a + aa†) and (b†b + bb†)."

C. Below eq. (78), we've stressed the distinction between our Berry phases and the Hall viscosity by adding the following sentences: "One should not be concerned about this discrepancy: there is no inherent reason for the current portion of the Berry phase (50) to be related to viscosity. The fact that the corresponding Berry curvatures match up to normalization is simply a result of the unique invariant area form on a hyperbolic plane (recall footnote 1). Moreover, it is important to note that the distinction between a and b contributions in the operator (75) is specific to linear deformations (2), and does not align with the separation of (50) into current and AB contributions. Identifying the part of the Berry phase (50) arising solely from metric deformations would pose a significant challenge, extending beyond the scope of the quantomorphisms studied here."

D. Finally in the conclusion: "As explained in the manuscript, such mismatches should not be seen as paradoxes compared with ear lier works on the Hall viscosity, since the splitting between current terms and AB terms in the Berry phase (50) generally has nothing to do with the splitting of quantomorphisms in metric and potential deformations. What is remarkable instead is that the Berry phase (50) contains an extensive gaugeinvariant piece that just happens to be proportional to the Hall viscosity."

2. Again following comments by referee #1, the discussion of fractional quantum Hall states has been extended beyond Laughlin, and now includes any isotropic fractional wave function in the LLL (including MooreRead at filling 1/2). Here we refer to the text surrounding eqs. (79)(84) for details; it is essentially identical to the text of the previous version, save for a slight reordering and an emphasis on the generality of the result. This includes again a sentences that stresses the distinction between our Berry phases and the Hall viscosity, namely: "A similar mismatch occurs, for instance, in the MooreRead state at filling $\nu=1/2$, where [27] predicts a Hall viscosity $\eta_H=\frac{3}{2}\hbar/(8\pi\ell_B^2)$ while our universal current Berry curvature (83) predicts a response coefficient $\frac{1}{2}\hbar/(8\pi\ell_B^2)$. As previously mentioned, this mismatch is not contradictory; instead, it is a distinguishing characteristic. Hall viscosity is a specific response to pure metric deformations, whereas our current Berry curvature (83) encompasses both metric and potential perturbations, and fundamentally does not have any direct connection to viscosity."

3. The last issue raised by referee #1 was the "suggestion that the Berry phase might offer a better way to extract the Hall viscosity than current measurement schemes. (...) Could the Authors comment on that, or else modify their statement?" This is a fair point, so we've now clarified our proposal to measure Berry phases produced by quantomorphisms by adding a sentence in the introduction (below fig. 2): When writing about quantum geometry, "The latter will be crucial in practice, as the likeliest avenue to observe the effects of the phase (1) is to study adiabatic linear response in quantum simulators, where the high degree of control over microscopic details may help overcome issues of decoherence and disorder." The key point here is the emphasis on quantum simulators (as opposed to actual condensed matter samples), where one may hope to control the microscopic Hamiltonian enough to actually implement timedependent quantomorphisms of the kind described in our paper; we're currently discussing this with researchers working on topological photonics, but this is very much in the earliest stages so it's difficult to say anything more definite.

4. As required by referee #2, we've added citations to arXiv:2103.04163 and arXiv:1801.03759.
Published as SciPost Phys. 15, 159 (2023)
Reports on this Submission
Strengths
As in my first report: Explicit `abinitio' calculation of the effects of areapreserving diffeomorphisms on the Landau Level quantum states.
Weaknesses
As in my first report: the issue is not to criticize/correct established effective field theory results, but to complement them by explicit direct calculations.
Report
In my first report, I said that the paper could be published, only some references were missing. They have been added, and the requests by the other referee have been satisfied. So I support the publication.
Strengths
As before:
Elegant result for the Berry phase for general geometric deformations.
Clear and eloquent presentation.
Weaknesses
None (previous ones addressed).
Report
In the response letter and resubmitted manuscript the Authors have, in my opinion, properly addressed previous comments by both Referees. Hence, based on my previous evaluation, I now recommend the publication of the manuscript in SciPost Phys.