Fingerprints of quantum criticality in locally resolved transport

Apologies that we had prepared a nice LaTeXed PDF with our reply but I cannot figure out how to upload it... so I have cut and paste and some math formulas are a little annoying to read.

Response to Referee 1:

"Explain why the superdiffusive regime of Phys. Rev. Lett. 123 (2019) (also Phys. Rev. B 102, 245434 (2020)) is/isn't relevant for the purposes of the paper."

Note that these papers are Ref.[34,35] in our paper. The main purpose of our paper is to show that the sinusoidal current profile is a fingerprint for quantum critical flow -- at least near certain quantum critical points, including those coming from holographic models -- and is distinguishable from hydrodynamic flows.

In the interesting papers mentioned by the referee, evaluating the explicit formulas for $\sigma(k,\omega)$ given there lead to, in the $\omega \rightarrow 0$ limit, the form $\sigma\sim 1/(1+v^2\tau_{c,1}\tau_{c,2}q^2)$ when $q<(v\tau_{c,2})^{-1}$, so they can interpret between $\sigma\sim 1$ ($q\ll (v\tau_{c,1})^{-1}$) and $\sigma\sim 1/q^2$ ($q\gg (v\tau_{c,1})^{-1}$). This will look quite a lot like viscous flow, and we suppose this comes from the emergence of approximate momentum conservation of electron and hole fluids separately. We have added a few sentences to Section 6 of the main text commenting on this interesting observation. But we also note there that this behavior is quite different from the quantum critical flow, $\mathrm{Re}~\sigma\sim \exp(-\gamma k)$.

One can certainly do an analysis of experimental data using this kinetic theory instead. One merit of our approach is that we only required a one-parameter fitting between our holographic model and the experimental data, which already led a good fit. Moreover, our estimated fit parameter agrees quantitatively with another independent experiment. Moreover, our approach is capable of interpreting between the hydrodynamic (Ohmic, in the case of charge-neutral systems) and quantum critical regimes. If one wants to analyze the quantum critical regime using approaches based on kinetic theories, the breakdown of semiclassical dynamics on length scales smaller than $\hbar v_{\mathrm{F}}/k_{\mathrm{B}} T$ must be accounted for quite carefully. This could be an interesting future project.

"In order to reach the QC regime in a graphene channel with a constriction of order $1\;\mu$m, the temperature of the graphene sample must be $T=\mathcal{O}(1K)$. At such low temperatures, however, the effects of charge puddles becomes important. Explain how charge puddles may affect the conclusions of the paper and the ability to measure charge transport in QC graphene."

First, we want to point out that in order to reach the quantum critical regime with a constriction of width $1\;\mu$m, the temperature could be $T \lesssim 40$ K, instead of $T \lesssim 1$ K. We found this by simply plugging in numbers into our Eq. (11) to find the Planckian length scale.

In experiments on graphene samples using hBN substrates (the default method used today) one typically finds that the amplitude of charge puddles is around 30 K, and their size is around 100 nm. Since both the size of charge puddles is small relative to the device sizes we are advocating using, and it is possible to enter the quantum critical regime at temperatures above 30 K, we believe it is definitely possible to image these phenomena in experiment if our theory is accurate. We emphasize that in Fig. 4 in the main text, we evaluated (albeit in a model neglecting charge puddles) and saw that for 600 nm constrictions and $T\sim 128\;$ K, it should be possible to see quite strong deviations from Ohmic or viscous flow patterns! And given the numbers highlighted above, this seems entirely reasonable to try.

We have added a few comments in the main text discussing these points and explaining the relevant length/energy scales of charge puddles. We thank the referee for the suggestion.

"Discuss how the results are expected to depend on the dimensionality of the material."

From the perspective of holographic duality, we expect the exponential decaying conductivity $\mathrm{Re}~\sigma\sim \exp(-\alpha k)$ to be held in any dimension since the spectral weight will always decay exponentially toward the horizon. A field theory analysis similar to Ref.[29] would also suggest that the quantum critical sinusoidal profile is insensitive to spatial dimensions. Of course, the imaging procedure would be quite difficult in a three dimensional model, so we expect our ideas will be most interesting in two-dimensional systems.

Response to Referee 2:

"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)."

Our Eqs.(1), (2) and (3) aim to compute spatially resolved current profiles through $k$-dependent conductivities. We agree that these formulae do not capture the bulk charge distribution, therefore, we explained in Appendix I that a generalization of Eq.(1) to spatially resolved charge density $n(x)$ is required to compute the conductance $G$. A detailed discussion of this issue is provided in Appendix I. Nevertheless, to help direct the reader's attention to the ease of this generalization, we have moved one equation (now Eq.(4)) and some discussion from Appendix I into the main text in Section 2.

"In the comparison to graphene experiment, it is unclear to me how the planckian length scale in Eq.(11) is related to the paramters in the theory. For example, how is $\alpha$ in Eq.(9) related to Eq.(11)? Please clarify."

At a fixed temperature $T$, our holographic model has no fit parameters. $\alpha$ is an $\mathcal{O}(1)$ number whose exact value needs to be determined numerically as discussed in Appendices E and F. To compare to experiment, we need to fix the number $C$. In the holographic model using natural units, we have $C=1$. We run the holographic simulation for a fixed width device at various $T$ (as its output depends only on the product $Tw$, in dimensionless units), generate the current flow patterns, and find which one is the best fit to the experimental data. That gives us $T_{\mathrm{fit}}$ (which assumes $C=1$). However, we find $T_{\mathrm{fit}}$ is not equal to the experimental temperature $T_{\mathrm{exp}}$ but they satisfy $1/T_{\mathrm{exp}}w_x\propto 1/T_{\mathrm{fit}}w_x$. Hence, according to Eq.(11), we can determine $C$ through $1/T_{\mathrm{exp}}w_x=C\times 1/T_{\mathrm{fit}}w_x$.

We have added a sentence explaining this to the manuscript in the experimental section and hope this clarifies the issue.

"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.(9), I am wondering whether similar procedures can be done in this case and obtain an analytic solution?"

Unfortunately, we were unable to find analytic expressions for the quantum critical current profile when we tried to solve these integral equations. However, we believe that the sinusoidal profile captures the main feature of the quantum critical flow, at least around the center of the constriction. The obstacle to get a closed form analytical expression is in part fixing boundary conditions, but since the experimental data has limited accuracy near the boundary due to finite resolution, the universal feature near the center would likely be more important. But it could be interesting if future authors are capable of solving this problem!

"In studies of ohmic-ballistic-viscous 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?"

Our holographic model is capable of showing the breakdown of Matthiessen's rule in the hydrodynamic regime (see Eqn.(10) and Eq.(F6)). But we doubt that, for example, one could just replace $G_{ballistic}$ or $G_{viscous}$ with $G_{qc}$ in a quantum critical regime; the precise mathematical formula the referee quotes is probably rather bit special to the models studied in earlier papers.

The changes we made to the manuscript are explained in the responses above.