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Non-equilibrium quantum transport in presence of a defect: the non-interacting case

by Marko Ljubotina, Spyros Sotiriadis, Tomaž Prosen

This is not the current version.

Submission summary

As Contributors: Spyros Sotiriadis
Arxiv Link:
Date submitted: 2018-11-08
Submitted by: Sotiriadis, Spyros
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Mathematical Physics


We study quantum transport after an inhomogeneous quantum quench in a free fermion lattice system in the presence of a localised defect. Using a new rigorous analytical approach for the calculation of large time and distance asymptotics of physical observables, we derive the exact profiles of particle density and current. Our analysis shows that the predictions of a semiclassical approach that has been extensively applied in similar problems match exactly with the correct asymptotics, except for possible finite distance corrections close to the defect. We generalise our formulas to an arbitrary non-interacting particle-conserving defect, expressing them in terms of its scattering properties.

Current status:
Has been resubmitted

Ontology / Topics

See full Ontology or Topics database.

Free fermions Localized defects Quantum quenches

Author comments upon resubmission

Following the referees' suggestions, we resubmit a new version of our paper with an improved abstract and introduction that clarify better the main goal of our paper, that is the rigorous derivation of the exact asymptotics after an inhomogeneous quench in presence of a defect, which allows us to demonstrate the validity of the semiclassical approach and identify potential corrections. We have also included a comparison with Landauer's theory and corrected the discussion on numerics as well as several typos and other issues raised by the referees.

List of changes

1. Improved abstract and extensively modified the introduction (mainly in p.3) to clarify the goal of the paper, also added further information on earlier work including additional citations (on integrability, CFT or Landauer-theory based techniques)
2. added subsection 2.3 on comparison with Landauer’s theory in the special case of thermal baths
3. added discussion on numerical methods, explaining that exact diagonalisation is more efficient for this problem

4. added citations on use of Landauer’s theory to similar problems
5. changed citation for ray dependence of NESS to (Bertini Fagotti, PRL 117(13) 130402 (2016)),
6. added citation [36] (Bernard, Doyon, Viti, J. Phys. A 48 (2015) 05FT01)
7. added colour scale in fig.1
8. fixed rescaling issue in figures 5,6,8
9. corrected mistyped eqs.(55-58) for density/current asymptotics for general defect
10. fixed sign typo in eq.(9)
11. corrected confusing wording in p.25 and elsewhere

Reports on this Submission

Anonymous Report 2 on 2018-11-25 Invited Report


1. Highly-detailed discussion of a particular physical situation using analytic approximations and numerical methods.

2. Validation of semi-classical approximation in highly non-equilibrium setting with inhomogeneous initial state-an interesting result.

3. Authors have improved the manuscript based on the feedback of previous referee reports


1. Some unclear statements remain


The authors consider non-interacting tight-binding dynamics in a system of fermions with a centralized defect which separates two semi-infinite regions of constant particle density. Using semi-classical arguments, the particle density and current are calculated at long times. Numerical results are obtained using tDMRG, ultimately showing that the simple analytic approach captures the long-time limit of simple observables quite well.

The comments from my previous review were taken into full consideration, and I find that the authors have also made revisions in accordance with the suggestions put forth by other reviewers. At the level of my own understanding of the criticisms raised by other reviewers, I am satisfied with the responses and changes made in the revised manuscript.

I have only a few minor suggested changes which are included below. These remaining points mainly concern the wording of certain statements and some seemingly simple generalizations of the the results presented. In particular, Ref. [5] includes results for domain walls which are of arbitrary "size," while the authors only consider a maximal jump from a fermion density of 0 to density 1. This is understandable in the tight-binding language (as opposed to the equivalent "spin language") because the linearly-varying chemical potential naturally leads to this maximal state, regardless of the slope. However, I am curious as to whether generalization of the present results are straightforward or hindered by the complicated nature of the generalized domain wall state presented in Ref. [5].

Requested changes

1. The phrase "It has been shown that an inhomogeneous quench starting from a step profile allows for an exact solution" appears in the introduction. I'd argue that any type of chemical potential allows for an "exact" solution in the sense that the problem can be formally diagonalized. A linearly-varying potential can be exactly diagonalized for any slope (T. Hartmann et al, New J. Phys. 6, 2 (2004), and many others). But I would agree with the more careful statement that the expressions are cleaner and perhaps more useful for the sharp step profile (caused by an infinite field gradient).

2. In the introduction, the phrase "KAM" is used without explanation. It's likely clear to many that this refers to the Komolgorov-Arnold-Moser theorem, but I think the acronym should be introduced.

3. The conclusion contained the statement “Lastly, it is worth to stress the analogy between an inhomogeneous quench and the Landauer problem....” I feel this should should read something like “lastly, it is worth emphasizing the analogy between….” (minor point of grammar)

4. If tDMRG is being used to treat a truly non-interacting problem, I would like to see that the results are as general as possible. Concerning the last paragraph in my report, is it possible to consider more general domain wall states (as considered in Ref. [5]) in this framework using your tool kit, or does something make this technically difficult?

  • validity: high
  • significance: good
  • originality: high
  • clarity: good
  • formatting: perfect
  • grammar: excellent

Author Spyros Sotiriadis on 2018-12-04 (in reply to Report 2 on 2018-11-25)

We would like to thank the referee for his comments, which we have taken into account in the new version. In more detail:

1. We agree that the full dynamics resulting from any initial density profile in this free system can be solved formally by exact diagonalisation. What we meant instead was "exact derivation of the asymptotics in closed-form expressions". In fact we should also clarify that closed-form expressions are possible also for smooth steps and some of the references cited at that point refer indeed to smooth profiles. We have changed the sentence accordingly.

2. Indeed, KAM should have been written explicitly as Kolmogorov-Arnold-Moser. We have replaced the acronym.

3. We have corrected this sentence.

4. This is an interesting point: Even though we focused our analytical proof on initial states corresponding to a density step with values $\mu\pm\nu$ changing sharply between the two middle sites 0 and 1, our method applies to any other initial state that is Gaussian and characterised by different asymptotics far on the left and right side, independently of the behaviour in the middle. This class of states includes the partitioning protocol of ref.[5]. We have added two paragraphs at the end of section 5 to comment on this. The main idea is that the asymptotic formulas are derived from the “kinematical” pole at equal momenta of the expression for the correlation function after a quench, which in turn is determined only by the large distance asymptotics of the initial state: e.g. if we consider a density profile given by a Heaviside step function or by $\tanh(ax)$ (or any other smooth profile), then we can see that in both cases the Fourier transform has the same pole $i/k$ as $k\to 0$, independently of the slope parameter $a$ in the smooth case. More details will be given in future work.

Anonymous Report 1 on 2018-11-24 Invited Report


1. Analysis is carried out by detailed first principle computations.
2. The paper provides the unifying framework to study transport in non-interacting impurity systems.
3. Exact asymptotics of the NESS current and density are obtained.


The methods developed in the paper are not applicable to interacting impurity models.


I do acknowledge the weaknesses pointed out by other referees in the first version, but I think the authors managed to overcome these criticisms by addressing the comments raised. The goal of this paper, the exact determination of the asymptotics of the NESS current and density, is now clearly stated in the introduction. I therefore recommend for the publication in SciPost as it is.

Requested changes

1. I think eq. (24) is not valid at $x=0$, only for $x>0$.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: good

Author Spyros Sotiriadis on 2018-12-04 (in reply to Report 1 on 2018-11-24)

We would like to thank the referee for his comments.

Indeed eq.(24) as introduced (general solution of the recursive equation in the right half side) holds only for x>0 not for x=0. We have corrected this in eq.(24) and everywhere else in the paper. Note that this doesn't affect the analytical derivation of the asymptotics at the middle because the latter was based on the reflection symmetric expressions in full space.

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