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Fluctuations of linear statistics of free fermions in a harmonic trap at finite temperature
by Aurélien Grabsch, Satya N. Majumdar, Grégory Schehr, Christophe Texier
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Authors (as registered SciPost users):  Aurélien Grabsch · Gregory Schehr 
Submission information  

Preprint Link:  http://arxiv.org/abs/1711.07770v1 (pdf) 
Date submitted:  20171122 01:00 
Submitted by:  Grabsch, Aurélien 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study a system of 1D non interacting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of linear statistics $\mathcal{L}=\sum_n h(x_n)$, where $h$ is an arbitrary function of the fermion coordinates $\{ x_n \}$. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations.
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Anonymous Report 2 on 201819 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1711.07770v1, delivered 20180109, doi: 10.21468/SciPost.Report.320
Strengths
1) Very neat presentation of original results
2) Calculations and logic easy to follow
Weaknesses
1) Focus only on the harmonic potential
Report
The authors develop a general method to calculate the variance of linear statistics in a free onedimensional Fermi gas, with a confining potential (Eq.(38) in particular). They apply the formalism to determine the variance of the number of fermions in the positive semiline (index) both in the canonical and grand canonical ensembles. The analysis is limited to the harmonic potential.
The authors also recover the variance of the potential energy previously derived in Ref. [27]. They conclude with a numerical simulation of the determinantal process associated to the fermion positions in an harmonic potential that further confirms their findings in the grand canonical ensemble.
I found the paper pedagogically written and interesting. Since moreover to my best knowledge the results presented in Sec.2.2, 3.2.1 and 45 are original and correct, I recommend the paper to be published after the authors will address the minor points below.
Requested changes
1)The formalism allows calculating in particular the variance of N_+ in the Ground State for an arbitrary potential. Does this show a Log(N) term as in Eq.(5)? If yes, is this term universal (i.e. potential independent)?
2)At pag. 13: There is a typo in the penultimate sentence of sec. 3.1
3)At pag. 22: kinetic>potential, (although they are equivalent seems more consistent with the text to continue discussing potential energy)
4)At pag. 22: formula (113) an "f_y" is missing after the second sum
5)At pag. 23: Formulas (119)(120). I think "c" has to be replaced with "g", saying perhaps that what is calculated in (120) is the variance in the canonical ensemble with N replaced by \bar{N}_g. Correct?
6)It seems that the Bose factor that appears in Eq.(110) is independent from the choice of the potential. Perhaps the authors can stress this fact and its implications for the asymptotics (111) in the quantum regime. Are then these asymptotics expected to be potential independent?
Anonymous Report 1 on 201818 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1711.07770v1, delivered 20180108, doi: 10.21468/SciPost.Report.319
Strengths
1. simple problem with physically interesting question
2. clear presentation
Weaknesses
1. results could have been further discussed
Report
The authors study fluctuations properties of a free Fermi gas
trapped in a harmonic potential at finite temperature.
More precisely, they consider the socalled linear statistics,
i.e. the sum of an arbitrary function of the fermion coordinates.
The main result of the manuscript is an explicit formula for
the variance of the linear statistics, Eq. (38), which is valid
for both canonical and grand canonical ensembles.
This result is then applied to calculate the fluctuations
of the particle number in the right half of the trap.
The scaling functions in the two different, quantum and thermal
regimes are explicitly calculated. In the quantum regime the
analytical result is compared against numerical simulations
with a good agreement.
I believe that, despite the simple, textbooklike calculations,
the manuscript deals with a physically interesting problem.
In particular, the finite temperature results on the particle
fluctuation are new, and the general result on linear statistics
possibly has a broader range of applicability.
Therefore I recommend the publication of the manuscript after the
authors have considered the issue raised below.
Requested changes
What I was missing a bit is a discussion about the role of the
potential. In fact, one could have equally well asked about the
fluctuations in one half of a finite box with no external potential.
At zero temperature this question has already been studied, see
EPL 98, 20003 (2012) and Phys. Rev. B 82, 012405 (2010).
Remarkably, it turns out that the boxresult at $T=0$ is
identical to Eq. (5) of the manuscript, i.e. the role of the
potential is irrelevant. This naturally raises the question,
what is the situation at finite temperature? Since the precise
form of the eigenfunctions enters only through integrals in
Eqs. (9091), one could expect some universality also at finite T.
It would be nice, if the authors could comment about this.
Author: Aurélien Grabsch on 20180209 [id 210]
(in reply to Report 1 on 20180108)
We are thankful to the referee for his/her careful reading of the manuscript and his/her positive comments.
The referee has raised an interesting question about the universality of our results, which were derived for a harmonic confinement.
I) Concerning the zero temperature result: although the referee is right about the universality of the dominant logarithmic term of the variance, we stress that the subleading constant is potential dependent. For a hard box confinement, the zero temperature result for $\mathcal{N}_+$ can be found for example in Eq. (40) of EPL 98, 20003 (2012) and reads
\[ \mathrm{Var}(\mathcal{N}_+) _{T=0}^\text{Box} = \frac{1}{2\pi^2} \ln N + \frac{1+\gamma+2\ln 2}{2\pi^2} \:, \]which differs from Eq. (5) by a constant term $\ln 2/(2\pi^2)$.
II) Nevertheless, at finite temperature, stimulated by the question of the referee we have investigated further the universality of our formulae. We have now added a new section (5.5) and extended the discussion of Appendix A. The main outcome is:  In the quantum regime the thermal fluctuations involve excitations around the Fermi level. For a one dimensional smooth confining potential, the spectrum is regular and can be linearlised near the Fermi level. The fluctuations are thus described by the same function $F_Q$ as for the harmonic oscillator.  In the thermal regime all the spectrum contributes and therefore the results are not universal.
Author: Aurélien Grabsch on 20180209 [id 211]
(in reply to Report 2 on 20180109)We are grateful to the referee for his/her very careful reading of the manuscript and his/her positive report.
1) and 6) The referee has raised the same interesting points as Referee 1. Here is our answer: I) Concerning the zero temperature result: although the referee is right about the universality of the dominant logarithmic term of the variance, we stress that the subleading constant is potential dependent. For a hard box confinement, the zero temperature result for $\mathcal{N}_+$ can be found for example in Eq. (40) of EPL 98, 20003 (2012) and reads
\[ \mathrm{Var}(\mathcal{N}_+) _{T=0}^\text{Box} = \frac{1}{2\pi^2} \ln N + \frac{1+\gamma+2\ln 2}{2\pi^2} \:, \]which differs from Eq. (5) by a constant term $\ln 2/(2\pi^2)$.
II) Nevertheless, at finite temperature, stimulated by the question of the referee we have investigated further the universality of our formulae. We have now added a new section (5.5) and extended the discussion of Appendix A. The main outcome is:  In the quantum regime the thermal fluctuations involve excitations around the Fermi level. For a one dimensional smooth confining potential, the spectrum is regular and can be linearlised near the Fermi level. The fluctuations are thus described by the same function $F_Q$ as for the harmonic oscillator.  In the thermal regime all the spectrum contributes and therefore the results are not universal.
25) We thank the referee for pointing out these typos.