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Fingerprints of quantum criticality in locally resolved transport
by Xiaoyang Huang, Andrew Lucas
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Submission summary
Authors (as registered SciPost users):  Xiaoyang Huang · Andrew Lucas 
Submission information  

Preprint Link:  https://arxiv.org/abs/2105.01075v3 (pdf) 
Date submitted:  20220322 03:01 
Submitted by:  Lucas, Andrew 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Understanding electrical transport in strange metals, including the seeming universality of Planckian $T$linear resistivity, remains a longstanding challenge in condensed matter physics. We propose that local imaging techniques, such as nitrogen vacancy center magnetometry, can locally identify signatures of quantum critical response which are invisible in measurements of a bulk electrical resistivity. As an illustrative example, we use a minimal holographic model for a strange metal in two spatial dimensions to predict how electrical current will flow in regimes dominated by quantum critical dynamics on the Planckian length scale. We describe the crossover between quantum critical transport and hydrodynamic transport (including Ohmic regimes), both in charge neutral and finite density systems. We compare our holographic predictions to experiments on charge neutral graphene, finding quantitative agreement with available data; we suggest further experiments which may determine the relevance of our framework to transport on Planckian scales in this material. More broadly, we propose that locally imaged transport be used to test the universality (or lack thereof) of microscopic dynamics in the diverse set of quantum materials exhibiting $T$linear resistivity.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022420 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2105.01075v3, delivered 20220420, doi: 10.21468/SciPost.Report.4949
Strengths
1  Extends the gamut of experimental observables sensitive to quantum critical (QC) behaviour by considering the local (generalized) conductivities of 2d QC systems.
2  Setup with minimal assumptions; Allows for great theoretical as well as experimental control through only a couple of parameters.
3  The predicted features of the local conductivity in the QC regime are heuristically a consequence of conformal invariance alone. Hence they can be expected to be universal.
4 Results match with existing experiments in graphene after fixing a single input parameter.
Weaknesses
1  Parameter regime necessary to measure QC transport in this setup unlikely to be reached in experiment (in graphene) due to the presence of charge puddles and the difficulty of manufacturing submicron constrictions.
2  Analysis restricted to 2d quantum critical systems.
3  List of transport regimes considered is not exhaustive. "Superdiffusive" conductivity found in [Phys. Rev. B 102 245434 (2020)] not considered.
Report
The main message of the present paper is that locally resolved transport holds information that is distinctly quantum critical (QC). In particular the authors used the local conductivity and the corresponding current flowing in a 2d channel with a constriction to make their case.
I find the results as well as the general message presented in the paper well substantiated and of great importance for future transport experiments in the QC regime. I believe, however, there are a few points that need to be addressed before I can recommend the paper for publication in this journal:
1) In recent years, the superdiffusive (SD) transport regime has been identified in 2d Dirac materials at charge neutrality [Phys. Rev. Lett. 123, 195302 (2019)]. It has been further shown that within the SD regime, the conductivity predicted by hydrodynamics becomes wavenumber dependent [Phys. Rev. B 102, 245434 (2020)].
I think the authors should comment on whether the SD regime is a relevant transport regime to consider in local imaging experiments. If yes, how does the current flow in that regime compare to the flow in the QC one?
2) In order to reach the QC regime in a graphene channel with a constriction of order $1 {\mathrm{\mu m}}$ , the temperature of the graphene sample must be $T = {\cal O}(1 {\rm K})$ . At such low temperatures, however, the effects of charge puddles becomes important [ J. Phys.: Condens. Matter 30 053001 (2018)].
It would be interesting to see a discussion/comment on whether the charge puddles affect the analysis found in the paper, especially around the charge neutrality point. In particular, is there a lower temperature limit set by charge puddles and if yes, does it forbid us from exploring the QC regime?
3) One way to sidestep the charge puddle issue would be to use 3d QC metals. Because of this, I think it would be useful to comment on how the results in the paper depend on the dimensionality of the system, if at all.
Some further, minor points I would like to bring to the attention of the authors are the following:
i) Some of the crossreferences between objects in the paper (tables, equations, sections) are wrong or not link to the proper place.
ii) It seems the authors decided to present their main point in a short letter form and relegate everything else in appendices. I believe the paper would be clearer if some of the appendices were pulled up to the main text e.g. appendix A.
Requested changes
1  Explain why the superdiffusive regime of Phys. Rev. Lett. 123 (2019) is/isn't relevant for the purposes of the paper.
2  Explain how charge puddles may affect the conclusions of the paper and the ability to measure charge transport in QC graphene.
3  Discuss how the results are expected to depend on the dimensionality of the material.
3  Fix crossreferencing issues (e.g. table 1 is referred to as table 6 near the end of the main text).
4  (Optional) Incorporate parts of the appendices into the main text to increase the paper's clarity.
Report #1 by Anonymous (Referee 1) on 202247 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2105.01075v3, delivered 20220407, doi: 10.21468/SciPost.Report.4881
Report
The manuscript argued that novel experimental technique which enables measurement of local current density can detect signatures of quantum critical transport. As a demonstration, the authors calculated the current profile in a constriction geometry under various transport mechanisms: ohmic, ballistic, viscous and quantum critical. The new contribution of the manuscript is to demonstrate that within the quantum critical regime, $\sigma(k)\sim\exp(\alpha k/T)$ and the current profile inside the constriction is sinusoidal. The authors also compared their holographic theory of chargeneutral transport with experiments in graphene and found reasonable agreement. I think the content of the manuscript is interesting and recommend it for publication after the authors addresses the comments below.
1. In Eqs.(2) and (3), the authors separated the electric field into a homogeneous external field and a response field $\tilde{E}_i$ due to boundary, and they proposed an ansatz such that $\tilde{E}_i=0$ outside of the constriction wall. This ansatz is mathematically sufficient for calculating the current profile but physically misleading because it have omitted the field due to bulk charge distribution $\partial_i\mu$. Because the conductivity tensor is transverse, the bulk charge field is a zero mode of Eq.(1) and therefore not effective in current profile computation, but it is crucial for computation of total conductance. The authors should clarify their derivation of Eqs. (2) and (3).
2. In the comparison to graphene experiment, it is unclear to me how the planckian length scale in Eq.(10) is related to the paramters in the theory. For example, how is $\alpha$ in Eq.(8) related to Eq.(10)? Please clarify.
The following two questions are for my personal curiosity:
3. The author's argument for sinusoidal current profile is based on heuristics and numerical data. The current profiles for the ohmic and the viscous regime are obtained by projecting Eq.(1) onto the constriction line and solving the resulting 1D integral equation for the current density. Given the analytic simplicity of the quantum critical conductivity in Eq.(8), I am wondering whether similar procedures can be done in this case and obtain an analytic solution?
4. In studies of ohmicballisticviscous cross over, it was found that the resistance of the system can be approximated as $R=R_{ohmic}+1/(G_{ballistic}+G_{viscous})$ where $G$ denotes the conductances. Do you expect similar epxressions to hold in quantum critical transport?