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Full Counting Statistics in the Transverse Field Ising Chain

by Stefan Groha, Fabian H. L. Essler, Pasquale Calabrese

Submission summary

As Contributors: Pasquale Calabrese · Fabian Essler · Stefan Groha
Arxiv Link: https://arxiv.org/abs/1803.09755v3
Date accepted: 2018-06-17
Date submitted: 2018-06-13
Submitted by: Groha, Stefan
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
Approach: Theoretical

Abstract

We consider the full probability distribution for the transverse magnetization of a finite subsystem in the transverse field Ising chain. We derive a determinant representation of the corresponding characteristic function for general Gaussian states. We consider applications to the full counting statistics in the ground state, finite temperature equilibrium states, non-equilibrium steady states and time evolution after global quantum quenches. We derive an analytical expression for the time and subsystem size dependence of the characteristic function at sufficiently late times after a quantum quench. This expression features an interesting multiple light-cone structure.

Current status:

Ontology / Topics

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Full counting statistics Quantum quenches Steady states Transverse-field Ising chain

Author comments upon resubmission

Referee 1:

We thank the referee for their careful reading of our manuscript and for the constructive comments. In the following we reply to the questions s/he raised.

1. "The definitions of some relevant object are scattered in different parts of the paper, which makes the reading a bit uneasy. In particular, the notion of symbol for Toeplitz matrices and the associated winding number is used in eq. (47), but defined only in Appendix A, which is refered to afterwards. I would suggest to define those in a clearer fashion, and for generic Toeplitz matrices, in the bulk of the paper."

We thank the referee for pointing out the missing definition of the symbol of Toepliz matrices. We added this definition as Eq. (47).

2. "In section III, the authors introduce a representation of the lattice spin operators in terms of a set of Majorana fermions. It would be useful to have the relation between those and the Jordan-Wigner fermions of section II.A clarified."

We thank the referee for pointing out the missing definition of the relation between the complex fermions and Majorana fermions. We added it in the text after Eq. (18).

3. "The multiple light-cone structure exhibited in section VI deserves to be commented further. Is there an interpretation of the associated velocities in terms of quasiparticles, for instance ?"

We agree with the referee that a comment on the light-cone structure in the discussion of the result is in place and have added a paragraph at the end of chapter VI.B.
We agree that finding a quasi-particle interpretation would be interesting, but as this does not seem a trivial task we leave it for future work.

4. "Some minor typos and comments :
- section III, after eq. (16): double occurrence of "the" in the sentence "Hence they are univocally determined by the the correlation matrices...."
- section V.B : "We now turn to the time evolution of P_w^(u)(m,t): P_w^(s)(m,t) should be mentioned as well, since it is studied in this section
-conclusion : missing word in the sentence "A more straightforward but interesting extension would be to certain observables...." "

We thank the referee for pointing out the typos and further comments. We have incorporated the changes in the current version.

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Referee 2:

The referee requested the following changes:


(1) "Address the work on FCS of charge transport. It would be nice if the authors could compare the finite time corrections in the IRLM with the findings in the current manuscript."

While we share the referee's opinion that the FCS of charge transport in quantum impurity problems is very interesting, we fail to see its relevance to the problem considered in our work. We determine the FCS of the transverse magnetization in a large, finite subsystem after a global quantum quench in a homogeneous system, which in our understanding is an altogether different problem. We therefore
see no reason to either discuss the IRLM or attempt to provide a list of references for the analysis of FCS in such kinds of problems.

(2) "Possibly providing an independent numerical check."

We are extremely puzzled by this comment. As stated in the paper we have conducted extensive numerical tests of the analytic results, and some representative comparisons are in fact shown in the manuscript. For example Fig. 22 shows a comparison between the analytic calculation, which is based on an asymptotic expansion, to numerically exact results (no approximations) in the thermodynamic limit (L=\infty).

(3) "Correcting the typos."

We are grateful to the referee for pointing out these typos, which we have corrected.

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Referee 3:

We thank the referee for his/her helpful comments.


1. "Optional change: Given the length of the manuscript, a short summary of the main results might be helpful for some readers (although this is a matter of taste)."

We have extended the summary of the paper along the lines suggested by the referee.

List of changes

- Added definition of symbol of Toeplitz matrices (Eq. 47)
- Added relation between complex fermions and Majoranas
- Added paragraph with comment on multiple lightcone structure at end of chapter VI.B
- Extended summary
- Fixed typos

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