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Impurities in systems of noninteracting trapped fermions

by David S. Dean, Pierre Le Doussal, Satya N. Majumdar, Gregory Schehr

Submission summary

As Contributors: David Dean · Gregory Schehr
Arxiv Link: https://arxiv.org/abs/2012.13958v2 (pdf)
Date accepted: 2021-03-30
Date submitted: 2021-02-25 08:25
Submitted by: Schehr, Gregory
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Statistical and Soft Matter Physics
Approach: Theoretical

Abstract

We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-P\'ech\'e (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.

Published as SciPost Phys. 10, 082 (2021)



Author comments upon resubmission

Dear Editor,

We thank you and the referees for the reports on our paper. As you point out, despite a globally positive impression, there are some concerns about the relation of some of our results to those in the existing literature.

While the referees acknowledge the originality and significance of our results on the filling transitions and at the edge, the earlier part of the paper is somewhat closer to the previous studies.
The reference of Kohn and Sham (KS) 1965, pointed out by referee 1, is perhaps the most important case.
We agree that our results in the bulk are fully consistent with the results already derived in KS. However,
our paper has the novelty of deriving the bulk modification of the density and of the kernel for a delta function potential {\it at all distances, and all interaction strengths}.
This allows the computation of the effective potential at all distances which is a new result.

We also added the additional references suggested by the referees, and discussed our results in the context of this literature.

Best regards,

The authors.

List of changes

Response to referee 1

“In this theoretical work, the authors pursue the study of the one-dimensional non-interacting Fermi gas (this is the continuation of a series of at least seven papers). Their main focus is on the gas being trapped by an external potential and containing one or two delta-impurities that modify the density. At long distance, the impurities give rise to Friedel oscillations in the density, at short distance they deplete (repulsive) or enhance (attractive) the local density. When there are two impurities, they interact via a kind of Casimir interaction. There is an interesting filling transition when a sufficiently attractive impurity is placed outside of the support of the gas’ density.
The article is well and carefully written. The work is mainly technical and a bit academic considering the venerable age of the subject (almost a century, starting with Sommerfeld 1927).
Do we learn something about non-interacting fermions that we did not know before? Maybe the filling transition. The rest, I believe, is mostly well-known, even if the exact formulas were maybe not obtained before. If it is really original, then the authors should show that they have carefully explored the literature. I suggest digging seriously in the following fields: solid-state physics (the electron gas., but also the XX spin ? chain that, via Jordan-Wigner transformation, is equivalent to non-interacting fermions) and in cold atomic gases (either with spin-polarized fermionic atoms or with impenetrable bosonic atoms, the so-called Tonks-Girardeau gas).
Examples of relevant papers:
- on the inhomogeneous electron gas: Kohn and Sham, Phys. Rev. 137, A1697 (1965). See also Kohn and Majumdar, Phys. Rev. 138, A1617 (1965).
- on the Tonks-Girardeau gas with an impurity (repulsive or attractive): Goold, Krych, Idziaszek, Fogarty and Busch, New J. Phys. 12, 093041 (2010). Fu and Rojo, PRA 74, 013620 (2006).
But there are probably much more relevant references.”

---> We have explored these papers and the associated literature and thank the referee for pointing them out. We have cited and discussed them in the MS. (See the changes in blue in the introduction and section 3).

Regarding the paper of KS, we agree that this is an important paper which we missed,
and we thank the referee for pointing it out. We have now discussed in detail the KS paper in the
introduction. We agree that our results in the bulk are fully consistent with the results already derived in KS. However,
our paper has the novelty of deriving the bulk modification of the density and of the kernel for a delta function potential {\it at all distances, and all interaction strengths}.
This allows the computation of the effective potential at all distances which is a new result.

In the paper by Kohn and Majumdar they show in a very general sense that the addition of an attractive impurity which can lead to the formation of a bound state gives a kernel which is analytic in the impurity strength. This is an interesting observation as the appearance of a bound state is often referred to as a transition but the transition is not associated with any thermodynamic singularity. This general result is confirmed by our explicit computation for delta function potentials and this fact is mentioned in the new version -in blue after equation (6).

The paper by Goold et al 2010 and a number of other papers by the same group consider an impurity in a harmonic potential. One can compute the corresponding wave functions analytically but in these papers the kernel sum is performed numerically for typically the order of 20 particles. One of the points of our paper was to explore the problem in a more general potential (bulk and edge) and it the scaling limits where analytical results can be obtained. The paper by Fu and Rojo is more concerned with the regime when there is a BEC. The work of Goold et al has been discussed in the Introduction and and after equation (72) (again in blue).

“Questions/remarks:

- in the title or in the abstract it is not mentioned that the study is restricted to one dimension. Although this becomes obvious in the core of the article, it should be specified in the title or at least in the abstract.”

---> We have made this clear in the new version of the abstract - in blue.

“- what is called the kernel in the present work is usually called the reduced single-particle density matrix in the solid-state literature on the Fermi gas (e.g. in Kohn and Majumdar, Phys Rev 1965).”

---> We have added this fact before equation (6) which defines the kernel.

“- the notation lambda introduced near equation (9) is a bit strange for a quantity which is the inverse of a length (usually lambda stands for a wavelength and k or q or kappa for an inverse length). “

---> We agree with this point, however in the literature on this problem the use of lambda is standard and maintaining this notation in the current paper makes it easier to compare with the papers we have cited.

“In the same vein, the authors use k to label their eigenvalues and eigenvectors but here k is not the momentum which is not conserved due both to the potential and to the impurities. Nevertheless, they use k_F (which is the standard notation) to denote the Fermi momentum. This is a bit confusing. It would be better to have another label for quantum numbers k that are not momenta.”

---> We have changed the sum variable to j, and as we are on one dimension hope that this will remove any potential for confusion.

“- It would have been interesting to discuss the phase-shifts in the Friedel oscillations contained in equation (69).”

---> It is an interesting question, but we feel that it is beyond the scope of the present paper.

- Equation (82) is simply the standard mean-field result: interaction strength times local density = g rho.”

---> We have mentioned this after and also its relation to the Hellmann-Feynman theorem.

Requested Changes

“1 - Authors should make clearer what was known before and what is truly new. This is especially important for a subject with such a long history. In particular, they should make a detailed comparison with Kohn and Sham 1965 and with results for delta-impurities in the Tonks-Girardeau gas.”

---> Please see the comments above.

“2 - Experiments are mentioned in the conclusion but no reference is given. It would be interesting for the reader to have such an experimental motivation. For example, Paredes et al. Nature 2004; Wenz et al., Science 2013. But there are probably better references.”

---> We have included references to experiments in the introduction of the MS, including the references mentioned by the referee.

Response to referee 2

“In this paper, the authors study non-interacting fermions in a potential, and the effects of one or several impurities on the density profile. They take a first principle microscopic approach based on Green's functions.
Overall the paper is a pleasant read, more on the technical side but with enough details to be able to reproduce the computations. The results on impurities near the edge are interesting and to my knowledge, new. The paper would be useful to a more mathematically minded audience with an interest in condensed matter physics. For this reason it deserves publication.
I'm less sure it would be that useful to a condensed matter theory audience, though. Most of the physical results presented in this paper would be considered well-known, especially for the free Fermi gas. While the authors correctly acknowledge the relation with Friedel oscillations, they do not do such a good job of explaining their results in a broader context. In particular, there is already a considerable literature on impurity effects even in interacting models, where some of what the authors get is modified (see for example the 1992 paper by Kane and Fisher). The authors also insist several times that their short distance results have not been computed before, which is probably true. However, there is little hope to find universality in such short distance physics. The point of Friedel oscillations being that you do not really need an exact delta potential to observe those. Thus, they should make clear how much of what they are computing depends on the precise potential used to model the impurity.”

---> Indeed we agree it is crucial from a technical point of view that we use a delta function potential in order to get an exact expression for the Green’s function in the bulk and at the edge. However as we are also interested in the effective potential we needed a problem where we could compute the density exactly at the site of the impurity. Concerning the filling transition, it is possible that the method of Kohn and Sham could be adapted to give a universal result in terms of the scattering coefficients of the impurity.

“I have other minor comments:

1) In figure 1, show also V_eff”

---> As the effective potential is a function of k_F(x) and lambda one can either plot it in the form give in Fig. 1 , otherwise it can be written as V_eff(x) = k_^2(x) W_2(\gamma), this will give a continuous function at the origin, however we think the first form is more logical as lambda is a constant.

“2) Near (23), (24). Wouldn't it be simpler to say that the difference (23)-(24) is zero because there are no poles below the contour?”

---> This is true as long as the integral on the contour 3 is zero and the extra discussion is to discuss his point.

3) At the end of section 5. Since the joint PDF for random matrices is identical to the joint PDF for the fermions in a harmonic potential, what would be the fermionic analog of the BBP setup?

---> We say in that section that it is a single particle bound state which is filled.
we explain that the position of the fermion is fixed by the position of
the impurity, while in the BBP setting the eigenvalue is not fixed. This shows that
it is an analogy but not a strict mapping between the two systems.

4) After (136). 'formula' should be plural.

---> This has been corrected.

Additional point:

Reference 8 has been updated to the published version, we have also made a reference to a paper on the Thermal Casimir effect which exhibits similar oscillatory behavior to that found for the quantum case studied here [44].


Reports on this Submission

Anonymous Report 2 on 2021-3-15 Invited Report

Report

The changes made to the manuscript are appropriate, I recommend publication.

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

Anonymous Report 1 on 2021-3-7 Invited Report

  • Cite as: Anonymous, Report on arXiv:2012.13958v2, delivered 2021-03-07, doi: 10.21468/SciPost.Report.2659

Report

The authors answered in an appropriate manner to all the requested changes (both by the other referee and myself). They made substantial changes expanding the manuscript by a few pages and adding many references. They also took most remarks into account and carefully answered questions. I think the paper should now be published.

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

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