# Transport in one-dimensional integrable quantum systems

### Submission summary

 As Contributors: Jesko Sirker Preprint link: scipost_201910_00039v4 Date submitted: 2020-04-24 Submitted by: Sirker, Jesko Submitted to: SciPost Physics Lecture Notes Discipline: Physics Subject area: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

These notes are based on a series of three lectures given at the Les Houches summer school on 'Integrability in Atomic and Condensed Matter Physics' in August 2018. They provide an introduction into the unusual transport properties of integrable models in the linear response regime focussing, in particular, on the spin-1/2 XXZ spin chain.

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

### List of changes

1) Outline: I added the sentence 'Furthermore, I also note that a
different approach to transport---generalized hydrodynamics---has been
discussed in a separate series of lectures and will not be covered
here.'

2) End of chapter 1: ' and discuss the general picture emerging from
these calculations as well as their limitations.'

3) End of introduction to chapter 2: 'While we concentrate on these
specific anisotropies in the following, we note that the results can
be generalized to all commensurate anisotropies $\gamma = n\pi/m$ with
$n,m$ coprime. On the other hand, it has been argued that ballistic
transport possibly coexists with superdiffusion at incommensurate
anisotropies while transport is entirely superdiffusive at the
isotropic point, $\Delta=1$ [34].

4) Below Eq. (2.28): 'We note that we have assumed here that the integral in
Eq.~(2.28) is convergent. If this is not the case, then the
additional channel is superdiffusive. As indicated before, this
possibly happens at incommensurate anisotropies but will not be
discussed here further.'

5) Below Eq. (2.28): 'It
provides a strict lower bound---possibly even exhaustive---for
rational $\gamma/\pi$ and thus proof that ballistic transport for
these anisotropies indeed persists at finite temperatures.'

6) Reference to Louis, Gros below Eq. (3.9) added.

7) Experimental reference add below Eq. (3.9).

### Submission & Refereeing History

Resubmission scipost_201910_00039v5 on 31 May 2020
Resubmission scipost_201910_00039v4 on 24 April 2020
Submission scipost_201910_00039v1 on 27 October 2019

## Reports on this Submission

See 1st report

See 1st report

### Report

In the revised version, the author has implemented some but not all of the suggestions and
requests from the previous report.

One should stress that the points from the previous report could have been implemented
without much effort, by adding or modifying a few sentences and adding references.

The author dismisses the suggestions using these arguments, paraphrased in my own words:

(i) in the author’s view, the notes should only describe the actual contents
of the author's lecture
(ii) some aspects are mentioned in other lectures of the same school (e.g., GHD) and hence do
not require mentioning
(iii) a fair sampling of e.g. numerics would require "to list many more paper than the three
mentioned by the referee."
(iv) certain topics/methods were not discussed in the lectures.

These arguments are not convincing. Point (i): we are discussing a manuscript that is accessible individually in SciPost and on the arXiv and that can and should be viewed as a stand-alone
article. There is no reason why a lecture and a manuscript should be their exact mirror: one is
accessible to a limited audience, the manuscript is an article available to the public and if
peer-reviewed, subject to the usual standards.

Point (ii): While GHD is covered in other parts of the school, other aspects (e.g., contributions
from numerics) were maybe not covered there. We are not debating adding extensive
discussions, but only references that allow the reader to explore topics on their own that are not covered in the lecture notes.

(iii) If it takes a long list of reference to provide a fair sampling of contributions from numerical methods, then so be it. My comment from the first report did not imply exclusiveness of the
suggested references.

(iv) Precisely because the contents of the lectures cannot cover the whole field, it is essential
that references to those parts that were not covered are provided in the text, in particular, for
students and researchers from outside of the field.

Perhaps the guidelines of the Physical Review on Referencing might be helpful here
(see https://journals.aps.org/prb/authors/editorial-policies-practices):

"Manuscripts must provide proper citations to pertinent earlier work and credit significant contributions by non-authors. Readers benefit from complete referencing, which correctly contextualizes the work in regards to related research. Authors should make every effort to ensure that their citations to previously published work are comprehensive at the time of submission. These citations can include references to books and references to published conference proceedings that contain more than abstracts. Prior to publication, authors should add citations to works published during the course of the review process."

In my understanding, these are generally and widely accepted criteria and are of particular
significance for a review or introductory article such as this manuscript.

The author is therefore advised to implement the remaining suggestions 1), 4), 5), 6), 7) and 8) from the first report – see below for further details.

Furthermore, the author is encouraged the to re-examine the whole manuscript in view of the
guidelines quoted above.

### Requested changes

More detailed comments on the points from the first report that were not implemented:

Ad 1) Numerical methods are actually mentioned and discussed in the manuscript, see the bottom of page 14:

"Numerically, these predictions can be tested by calculating the diffusion constant directly from the current-current correlation function, see Eq. (2.28). In such numerical calculations, the
main problem is to reach sufficiently long times to obtain reliable results for the integral over
the time-dependent current-current correlation function."

Without proper references, this text is useless to readers not familiar with the field or the
literature. The author must provide references to pertinent literature.

In this context, see also page 4: where Ref 34 is cited: Super-diffusion at the isotropic point was first suggested by Znidaric, Phys. Rev. Lett. 106, 220601 (2011).

Ad 4) Discussion of experiments, pages 9/10: Why not cite the most recent review by Hess here? See the first report.

Also, on page 9: "Obtaining a detailed understanding of the heat transport as measured experimentally is, however, a complex and still somewhat open issue. It requires an identification of the dominant relaxation processes and a formalism to incorporate such scattering mechanisms in the calculation of the thermal conductivity."

There is pertinent theoretical literature here as well that should be cited, by Rosch, Chernyshev, Rozhkov, and others. The statement is a dead end for the reader if left without references.

Ad 5) Page 14: The author should add the references mentioned in the first report about Delta >1, Otherwise, the history and contributions of a number of researchers are not properly accounted for. See the sentence

"The quasi-local charges which protect part of the spin current, on the other hand, become non-local for Delta > 1 and the spin transport becomes diffusive"

Ad 6) See the previous report.

Ad 7) The key (and seminal) contribution of the work of Prosen et al. Ref. 7-9 was to discover the quasi-local charges and to relate them to the Drude weight. This contribution should be acknowledged.

For instance, see page 6: Page 6: "These charges are sometimes called quasi-local and play an important role in understanding the spin transport properties of the XXZ chain."

Here, the author should state that these charges were discovered in Prosen PRL 2011 Prosen and Ilievski PRL 2013 (Refs. 7, 8).

Ad 8) See the previous report.

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

Author Jesko Sirker on 2020-05-31
(in reply to Report 1 on 2020-05-08)

In the second report, the referee repeats the points from the first
report. While it is clear that the referee is very passionate about
the requested changes, the report, in my view, does cross a line which
is not helpful. Besides, the report misses that many of the requested
changes have actually been implemented. For the remaining points where
we do disagree, it would have been appropriate, in my view, to reflect
on my arguments instead of dismissing them outright. It is possible to
disagree in a respectful manner.

Since I have already replied to all the points raised in my first
in order to move forward. One the one hand, I have some general
remarks about lecture notes which are perhaps worth considering for
the future. On the other hand, I suggest a compromise for this
particular case which, hopefully, allows to bring this long process to
a conclusion.

Let me start with my general remarks: The Les Houches Lecture Notes
are a well established format which is different from a review
journal. Being pedagogical and also reflecting the personal views of a
researcher of a particular research field are, in my view, what makes
them valuable and is why they are often recommended to graduate
students. For the graduate students which attended the school it is,
furthermore, important that the notes do reflect what was actually
taught. For this reason it should also be clear that lecture notes can
never be fully up-to-date. While this school (held in 2018) might be
an extreme case, there will likely always be some delay between the
actual lectures and the date when the corresponding notes are
published. I do not think that there should be any pressure on the
lecturers to include material published after the school was
concluded. I personally feel that it would be a shame to move away
from the established, successful format and to effectively turn the
Les Houches Lecture Notes into yet another review journal.

Coming back to the manuscript at hand: I am facing a bit of a
conundrum here because I am convinced that my lecture notes are
completely in line with the established format. A check of a sampling
notes from previous schools on related topics confirms this. In
particular, I note that most full lecture courses (mine was only a
half a course) - resulting in lecture notes of 30-80 pages - list
40-50 references (of course, there are some exceptions with much fewer
or many more as well). In the latest version of my notes, 41
references are included which therefore is already near the upper end
of the spectrum.

In the main text, I do not want to move away from presenting the
material in a pedagogical way and from selecting the references
accordingly. I also believe that it would be counterproductive to
extend the notes to topics which were not covered in the actual
lectures. As a compromise and in order to move forward, I have now
for topics not or only very briefly discussed in my lectures are
given. This includes, in particular, numerical studies of transport in
integrable models and studies of the regime with anisotropy Delta > 1.

I hope that the manuscript in its current form can now be accepted and
published without further delay.