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High-frequency transport and zero-sound in an array of SYK quantum dots
by Aleksey. V. Lunkin, Mikhail V. Feigel'man
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Submission summary
| Authors (as registered SciPost users): | Lunkin Aleksey · Mikhail Feigel'man |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2112.11500v1 (pdf) |
| Date submitted: | 2021-12-23 09:29 |
| Submitted by: | Aleksey, Lunkin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study an array of strongly correlated quantum dots of complex SYK type and account for the effects of quadratic terms added to the SYK Hamiltonian; both local terms and inter-dot tunneling are considered in the non-Fermi-liquid temperature range $T \gg T_{FL}$. Electric $\sigma(\omega,p)$ and thermal $\kappa(\omega,p)$ conductivities are calculated as functions of frequency and momentum, for arbitrary values of the particle-hole asymmetry parameter $\mathcal{E}$. At low-frequencies $\omega \ll T$ we find the Lorentz ratio $L = \kappa(0,0)/\sigma(0,0)$ to be non-universal and temperature-dependent. At $\omega \gg T$ the conductivity $\sigma(\omega,p)$ contains a pole with nearly linear dispersion $\omega \approx sp\ln\frac{\omega}{T}$ reminiscent of the "zero-sound", known for Fermi-liquids.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-2-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.11500v1, delivered 2022-02-04, doi: 10.21468/SciPost.Report.4321
Strengths
The results obtained in this work are original and improve our understanding of the large spectrum of one-dimensional systems that can be constructed from SYK-like quantum dots.
The technical presentation is detailed, allowing the reader to follow and reproduce most of the computations, if desired.
The scope of the paper is clearly defined, and the authors do not digress or speculate beyond it.
Weaknesses
The discussion of the physical interpretation and relevance of the results is mostly neglected in favor of the technical details.
The notation used in this work can feel unfamiliar to readers not sharing a common background with the authors, and may be excessive given the simplicity of the underlying system under consideration.
Given that one of the strong points of SYK-like models is our ability to solve them exactly at large $N$, it is a pity that nowhere in this work numerical solutions are used to complement or at least support the analytical arguments. This would greatly improve the presentation, allowing the reader to more easily visualize the results obtained in the various regimes under consideration.
Report
Building on and extending previous work by the same authors, this paper aims to compute the conductivity properties of a one-dimensional array of $q=4$ random-mass complex SYK systems, coupled by quadratic terms of random tunneling between neighboring sites. Expressions for the electric and thermal conductivities are obtained in the large-$N$ limit as functions of the frequency and momentum, to be later expanded in the high- and low-frequency regimes. A non-universal Lorentz ratio is found in the latter regime, as well as a zero-sound mode in the electric conductivity at high frequencies.
Given that zero-sound is traditionally associated to Fermi-liquids, it is certainly puzzling to observe this phenomenon precisely in an SYK-like system, i.e. a model of non-Fermi liquids. The authors, however, decide not to dwell on this matter and instead concentrate on the technical aspects. This referee feels that the presentation would greatly benefit from moving some of the computational details to an appendix, while devoting more effort to discuss the physics involved, as has already been suggested by another report.
Requested changes
1- A considerably improved introduction should be provided discussing the larger context of the model, the physical interpretation and consequences of zero-sound, etc. The conclusions should also be extended, since at this time they amount to a summary of results presenting no insight into their relevance, connections to other known systems, and so on.
2- At the end of section 3.2 the authors discuss the asymmetry parameter $\mathcal{E}$, mentioning numerical evidence that its absolute value is found to be bounded. The provided citations, however, deal with a different variant of the complex SYK model, namely the mass-deformed case where the coefficient in front of the quadratic term in each quantum dot is not random but simply a fixed constant. This is different from the ${\rm SYK}_{q=4} + {\rm SYK}_{q=2}$ considered here, and can lead to confusion. To avoid this, the authors should clarify the distinctions between ${\rm SYK}_{q=4} + {\rm SYK}_{q=2}$ and the mass-deformed SYK model, and maybe comment on whether their results would also apply to a linear array of mass-deformed complex SYK quantum dots.
3- In equation (12) the authors introduce the parameter $q$ because it "will be useful below for dimensional regularization of some singular expressions." Details of this regularization are scarce to non-existent, and the reader is left wondering whether the results obtained could depend on this regularization scheme. Some discussion of this procedure is therefore in order.
4- While the manuscript is readable as is, the number of typographical errors can be distracting, and there is ample room for improving the English.
Report #1 by Anonymous (Referee 1) on 2022-1-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.11500v1, delivered 2022-01-10, doi: 10.21468/SciPost.Report.4164
Report
The manuscript is devoted to finite frequency and momentum response function of a model based on SYK array. The authors derived dynamic response functions and found an analog of zero-sound mode. This is probably an interesting contribution, which might deserve publication. Unfortunately, it is really hard to judge more definitively due to poor quality of presentation.
Indeed the manuscript is much preoccupied with the technicalities (which are yet
notationally so obscure that they hardly help to understand anything) in expense of discussing the physics of the model. I would urge the authors to significantly expand introduction and/or conclusion sections to explain the assumptions, mechanics and consequences of their results. Some of the items to cover include:
(i) Phase diagram (e.g. energy vs momentum) which demonstrates hierarchy of
energy/momentum scales and expected window of existence of the zero sound.
(ii) Discussion of a usual hydrodynamic sound, its range of existence and overlap (if any) with the zero-sound.
(iii) Since the model is intrinsically disordered, one could expect a range where response functions exhibit diffusive behavior. Is it indeed the case and how does it conform to the presented expressions?
(iv) What are the relative contributions of reparametrization and phase fluctuations in the obtained response functions. E.g. how different are they in
Majorana SYK (no phase fluctuations), vs. Fermi liquid (no reparametrizations)?
(v) What is the role of finite N (if any) for the presented results?
The list can go on and the authors are surely aware of other potential questions, which should be addressed. I want to encourage them not to be shy to provide much more context to their findings.