# Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

### Submission summary

 As Contributors: Oleksandr Gamayun Arxiv Link: https://arxiv.org/abs/1911.01926v1 Date submitted: 2019-11-12 Submitted by: Gamayun, Oleksandr Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approach: Theoretical

### Abstract

We consider an integrable system of two one-dimensional fermionic chains connected by a link. The hopping constant at the link can be different from that in the bulk. Starting from an initial state in which the left chain is populated while the right is empty, we present time-dependent full counting statistics and the Loschmidt echo in terms of Fredholm determinants. Using this exact representation, we compute the above quantities as well as the current through the link, the shot noise and the entanglement entropy in the large time limit. We find that the physics is strongly affected by the value of the hopping constant at the link. If it is smaller than the hopping constant in the bulk, then a local steady state is established at the link, while in the opposite case all physical quantities studied experience persistent oscillations. In the latter case the frequency of the oscillations is determined by the energy of the bound state and, for the Loschmidt echo, by the bias of chemical potentials.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1911.01926v2 on 4 February 2020
Submission 1911.01926v1 on 12 November 2019

## Reports on this Submission

### Strengths

1-Calculation clearly presented and easy to follow
2-Comprehensive and elegant exposition of many results scattered in the literature

### Weaknesses

1-Some of the results are not new

### Report

The authors revisited the calculation of the particle Full Counting Statistics (FCS), the half chain Entanglement Entropy (EE) and the Loschmidt Echo (LE) for a systems of free fermions with a weak link (parameterized by $0<\varepsilon<1$). In the initial state the right half is empty, while the left half is characterized by temperature, and density (or chemical potential at finite temperature).

The authors derive Fredholm determinant representations for the FCS and the LE, from which in most of the cases they are able to extract exact large-time asymptotic.

Although it is true that most of the results presented here already appeared in some form elsewhere, I found the Fredholm determinant derivation of the FCS and the LE comprehensive, elegant and pedagogically written.
The bound state oscillatory contributions to the current and the entanglement are also clearly emphasized.

I believe that the results presented are ready for publications in their present form.

I spot a few innocent typos, listed below:

-Eq. (7) differential is missing
-Eq. (42), $\varphi\rightarrow\phi$
-Eq. (66), $t\rightarrow\tau$
-Below Eq. (86), I think $N\rightarrow N_f$

### Requested changes

See report

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

### Report

The authors report on the exact computation of the time evolution of various nonequilibrium observables of interest, such as current, shot noise, return amplitude and entanglement entropy, in a chain of hopping fermionic particles, starting from a quite general imbalanced configuration. The topic is timely due to active theoretical and experimental interest in the dynamics of inhomogeneous systems with conservation laws. The main manuscript contribution is a technical innovation in the computation of the full counting statistics of transported charge via Fredholm determinants. Also, the physical effects of a defect at the location of the initial “domain wall” are studied in detail.

Although the paper is extremely technical, the physical content is clearly explained and accessible to a nonspecialist. This study is a useful addition to the literature and the authors perform an excellent job in comparing with previously known results and indicating the original contributions of this study. Calculations are properly presented (I would make the derivation of Eq. (45) more pedagogical, as it is a central point). I have essentially nothing to criticize about this manuscript, and I think it is suitable for publication in SciPost in the present form.

As a side remark, the authors might find it interesting to compare the a.c. current induced by the defect, which they discuss in this manuscript, with that induced by the superficially similar (but seemingly unrelated) effect studied in P.P. Mazza et al., Phys. Rev. B 99, 180302(R).

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -