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Electron Currents from Gradual Heating in Tilted Dirac Cone Materials

by Ahmadreza Moradpouri, Mahdi Torabian, Seyed Akbar Jafari

This is not the latest submitted version.

Submission summary

As Contributors: Seyed Akbar Jafari
Preprint link: scipost_202110_00027v2
Date submitted: 2022-02-04 06:32
Submitted by: Jafari, Seyed Akbar
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Materials hosting tilted Dirac/Weyl fermions provide an emergent spacetime structure for the solid state physics. They admit a geometric description in terms of an effective spacetime metric. Using this metric that is rooted in the long-distance behavior of the underlying lattice, we formulate the hydrodynamic theory for tilted Dirac/Weyl materials in $2+1$ spacetime dimensions. We find that the mingling of space and time through the off-diagonal components of the metric gives rise to: (i) heat and electric currents in response to the {\em temporal} gradient of temperature, $\partial_t T$ and (ii) a non-zero symmetric Hall-like conductance $\sigma^{ij}\propto \zeta^i\zeta^j$ where $\zeta^j$ parameterize the tilt in $j$'th space direction. The finding (i) above that can be demonstrated in the laboratory in state of the art cooling/heating rate settings, implies that the non-trivial emergent spacetime geometry in these materials empowers them with a fascinating capability to harvest the naturally available sources of $\partial_t T$ of hot deserts to produce electric energy. We further find a tilt-induced contribution to the conductivity which is an offspring of Drude pole and can be experimentally disentangled from the Drude pole itself.

Current status:
Has been resubmitted

Author comments upon resubmission

We are happy that the scrutiny of the referees and editors has led to a reformulation of the most important results of our paper and a better understanding of the novel results of our paper. Here we give a response to essential criticisms of the referees.

  1. One major concern shared by the referees is that effect of Coulomb interactions on the structure of the spacetime. Referees are right. As long as the electron fluid is still described by quasiparticles, the metric structure is preserved, although the tilt parameters can be renormalized with respect to those obtained from band structure. We have clearly stated this point in the introduction of the paper.

  2. Another major concern of the referees is the nature of Hall response. In the revised version we have argued that the rotational invariance leads to the property that the fact that the off-diagonal (Hall) response in normal conductors has only antisymmetric part. Any symmetric part would vanish by a rotation (that leaves the system invariant). However, in the tilted Dirac cone materials, the anisotropy arising from the tilt breaks the rotational invariance and hence the Hall response can not be used to rule out symmetric Hall response. Of course such a Hall-like response is anomalous in the sense that it does not require external B field. Upon application of B field, the antisymmetric part of the response also becomes active.

  3. Regarding the controversial Eq. (38) of the previous version of the paper, we have now added a new section 3.1 that properly takes "external" sources into account. Furthermore, the interpretation as "accumulative" current applies only to finite samples. We have argued that the t-linear pressure gradient will be eventually balanced by a counter reacting electric field. When the system is part of an external circuit, such heat/electric currents flow in the circuit and do not lead to accumulation. We have corrected the earlier misinterpretation and have clearly stated it in the paper. Furthermore, proper handling of the external sources resolves all the issues about the self-consistency of a t-linear term. In the revised version where the sources are properly included, the t-linear pressure that is the root cause of all t-linear terms is actually rooted in the t-linear increase of the temperature (external source) for which we have explicitly solved the equations.

  4. Regarding the concern of referee 2, we have clearly stated and separated the effects that arise from "generic" anisotropy from the particular form of anisotropy that can be encoded into the spacetime metric. So our line of thought is to study only the forms of anisotropy that can be encoded into a spacetime metric. Because then the resulting effect can be attributed to the structure of the underlying spacetime.

  5. We have tried our best to carefully rewrite the paper in order to reduce the language/grammatical errors.

  6. As for numerical estimates, in the conclusion we have given estimates based on the state of the art cooling rates Technologies. In this case, the vector transport coefficients introduced by us, after conversion to spatial gradients via the Fermi velocity, will be comparable to thermoelectric coefficients and hence they can be in principle demonstrated in the laboratory.

List of changes

In the revised version we have highlighted the major changes with red color to assist referees and the editors in clearly seeing the changes that are summarized below:
- Inconsistencies in the notations are resolved.
- The entire abstract, introduction and conclusion was rewritten.
- An entirely new subsection 3.1 has been added that properly deals with "external sources". The lack of such section was source of our own confusion about the self-consistency of t-linear currents.
- A discussion in section 3.3 is added to argue why a symmetric Hall response can arise in our system. To emphasize this, every "Hall" in the paper has been replaced by "Hall-like".
- We have totally rewritten the discussion of t-linear currents in section 3.4 and have corrected our misinterpretation of "accumulative currents" that only applies to finite (closed) system.
- After properly adding the external temperature sources to the theory, the vector transport coefficients in Eqs. (59) and (60) of section 3.5 are also corrected. These are essential result of our paper and quantify the response of the fluid to gradual heating. This section is also entirely rewritten.
- The section 4 on viscous fluids has not changed much except for small rephrasing here and there.
- The section 5 on conclusion has also been rewritten to account for the above list of changes.

Reports on this Submission

Anonymous Report 3 on 2022-3-6 (Invited Report)


The paper is on an interesting topic, as I wrote previously.


I am concerned that the paper still has fundamental mistakes in it. It seems as though the response of the system might be incompatible with basic principles, and I believe the authors still need to address some important points listed in my report.


1) Normally in hydrodynamics if you turn on an "external temperature gradient" as the authors did in Section 3.1.1, you could "cancel" the effect by having the actual temperature change so as to cancel the effect out; similarly, the electric field could in principle be cancelled off by a x-dependent chemical potential. For example taking the latter case (in the absence of tilt) I'd have

d_mu T^mu i = n E^i = d_i P = n d_i mu

so taking mu = E^i x_i I could find consistency. It seems this cannot happen in Eq. (27) because there would be a zeta^j d_j mu term on the left hand side of the equation but there is no similar term on the right hand side. This makes me worry that the calculation in Section 3 might in some way be inconsistent. It seems like a plausible argument on physical grounds that you wouldn't want the external electromagnetic fields to see the metric tilt...but that may have then led to this later issue, and I worry this is a serious enough point that it might mean this plausible assumption was wrong. I am not 100% sure if this is a mistake that needs to be fixed or just a funny feature of this system; however, at a minimum some discussion of this point should be had, and I do honestly lean towards thinking that the calculation is just not correct in some way.

2) In Eq. (33), why is there a tau_imp that appeared seemingly out of nowhere? I.e. it has nothing to do with d_t T, but was not included in Eq. (27).

3) In Section 3.2 I am still concerned about a few points. If I use the memory matrix formalism (see e.g. the discussion in "Holographic Quantum Matter"), I would expect that if I have long-lived momentum that the conductivity (ignoring incoherent part for a moment) could be approximated as

\sigma_{ij} = n^2 \Gamma_{ij}^{-1},

where \Gamma_{ij} relates to momentum relaxation and would encode the anisotropy, while n = \chi_{JP} is a susceptibility which is fixed. Even in an anisotropic system, \chi_{JP} does not become anisotropic in general. In this formalism it seems as if the structure that arises is rather different though. Since the derivation based on memory matrix methods would be rather formal and microscopics-independent, it should be valid for this system too. So the authors either ought to explain why the above expectation is wrong, or correct Section 3.2 (and possibly earlier sections too) in order to resolve the issue.

4) Below Eq. (43), omega_2 can be arbitrarily large compared to omega_1. More importantly, I would expect that you would find in this anisotropic system that (if we align the tilt axis with a particular direction, let's say x) that sigma_{xx} has a Drude peak with a different pole than sigma_{yy} and sigma_{zz}. If there are 2 poles visible in the same 'component' of sigma, it would simply be that the axes were not aligned but one could always choose 'smart' axes where sigma was diagonal and the pole structure was clearly separate. Is this what happens in this theory? Why or why not?

5) I strongly object to calling this "Hall response" or "Hall-like response", it is simply anisotropic conductivity which is well-established in anisotropic materials. I also don't know why this response would be considered "anomalous" in a tilted Dirac material which is anisotropic?

6) I believe the argument at the top of page 12 is almost certainly wrong: if there were an emergent B-field that was giving rise to Hall-like transport, then the conductivity would become antisymmetric?

7) I worry the fact that the system becomes heated uniformly in a uniform temperature gradient in Eq. (54) might be a consequence of an incorrect implementation of temperature gradient in the earlier section 3.1, as per my previous point above. Typically *within linear response* one can always find steady-state solutions to the equations, and that seems not to be true here, which is a bit concerning!

Requested changes

Please address the points in the report.

  • validity: ok
  • significance: good
  • originality: high
  • clarity: low
  • formatting: acceptable
  • grammar: acceptable

Author:  Seyed Akbar Jafari  on 2022-05-17  [id 2488]

(in reply to Report 3 on 2022-03-06)
answer to question

We thank the present referee for insisting on the points that leads to improvement of our paper. Please see the attached file for a detailed response to your comments/criticisms.



Anonymous Report 2 on 2022-3-3 (Invited Report)


The current version further elaborate on the key assumptions on the applicability of hydrodynamics with nontrivial metric in material with tilted Dirac cone. This resolve the main concern raised in the previous version.

Moreover, the computational details are now much more readable. A few strange terminologies (e.g. "Hall" conductivity in the previous version) and notations inconsistencies are removed.

Overall, this is a much better manuscript than in the previous version.


I think there are still a few typos and some terminology which are not very precise. These are minor details but should be properly fixed.


I think that the assumptions stated in the current version by the authors are reasonable and, as already stated in the previous report, that the phenomena pointed out in the manuscript are interesting.

Requested changes

There are still minor wordings and typos that I found, for example
- In the last two sentence of the abstract, I'm not sure what the term "electric energy" means. Perhaps, the authors would like to use a more precise term such as electric current.
- On paragraph below equation (1), the hyperlink is broken i.e. " are fermionic or bosonic [?, 20]"
- On the second paragraph of Section 2, "The later is valid for the emergent spacetime" where it should be "The latter..."

  • validity: ok
  • significance: good
  • originality: good
  • clarity: good
  • formatting: reasonable
  • grammar: reasonable

Author:  Seyed Akbar Jafari  on 2022-05-17  [id 2487]

(in reply to Report 2 on 2022-03-03)

We thank the present referee for his/her positive evaluation of the current version of our paper. In the revised version
(1) We have replaced the "electric energy" with "electric current".
(2) We have fixed broken links to references and have updated some other references.
(3) We have fixed other typos.

We hope that the current version will satisfy the present referee.

Anonymous Report 1 on 2022-2-28 (Invited Report)


1. They detail a novel mechanism to generate electrical/thermal currents via temporal fluctuations of temperature by exploiting tilted Dirac fermions in 2+1D.


1. While they argue for experimental realizability, it is not yet fleshed out in this paper.


This is my second pass of this paper, and in its current state I believe it meets the SciPost criteria for publication.

As before, this paper primarily outlines a novel procedure to generate electrical/thermal currents from temporal fluctuations of temperature by utilizing tilted Dirac fermions in 2+1D; the tilt provides an effective metric which mixes spatial and temporal coordinates, realizing this effect.

Most of the previous issues have been sufficiently addressed or corrected in their updated paper. However, I still find the discussion on experimental detection lacking. While in their submission form, they state that "the vector transport coefficients ... will be comparable to thermoelectric coefficients," I still do not see numerical estimates of this in the paper. What *is* present are estimates of temperature gradients, which are (in theory) controlled by the experimenter. In other words, for a given temperature gradient/temperature fluctuation, how much current do I expect to measure in these tilted Dirac fermion systems? Without such estimates, I find claims of experimental feasibility hard to substantiate.

  • validity: good
  • significance: ok
  • originality: good
  • clarity: good
  • formatting: reasonable
  • grammar: reasonable

Author:  Seyed Akbar Jafari  on 2022-05-17  [id 2486]

(in reply to Report 1 on 2022-02-28)
answer to question

We thank the present referee for his/her positive evaluation of the current version of our paper. We have revised the text by adding texts around Equations (65)-(68). Estimates are based on the thermal conductivity measurements performed on graphene samples.

The actual numbers required by the referee are as follows: For industrially available heating rates on the sale of 1000 K/s, samples of the width ~1cm are expected to give ~4nA currents. Larger samples can provide larger currents. Details are given in the revised text.

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