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Transport in one-dimensional integrable quantum systems

by J. Sirker

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Authors (as registered SciPost users): Jesko Sirker
Submission information
Preprint Link: scipost_201910_00039v1  (pdf)
Date submitted: 2019-10-27 02:00
Submitted by: Sirker, Jesko
Submitted to: SciPost Physics Lecture Notes
 for consideration in Collection:
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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

Reports on this Submission

Anonymous Report 2 on 2019-11-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_201910_00039v1, delivered 2019-11-06, doi: 10.21468/SciPost.Report.1299


Very pedagogical
Very good account of basic formalism, TBA, and field theoretical treatment


Incomplete account of related literature (which, in particular, for students and newcomers to the field) is important to know about)

Without actual references, some statements about, e.g., the shortcomings of numerical
methods cannot be understood by non-experts.

The presented picture for transport is not up to date with the latest developments.


The manuscript provides an introduction into the theoretical description
of transport in the spin-1/2 XXZ chains, addressing its unusual
transport properties. It focuses on the basics (i.e., general linear
response expressions, Kubo formula and the Mazur inequality) and
then covers results from Bethe-ansatz methods such as TBA and then field theory.

The text is quite well written and will certainly be of use to its target
audience (graduate students, postdocs and newcomers to the field).
Therefore, I recommend its publication in SciPost Physics
Lecture Notes, provided the following criticism is appropriately addressed.

Requested changes

1) While it is quite reasonable to restrict the presented material to specific
methods and results (also considering that this is based on actual lectures),
the author nevertheless needs to give a balanced account of related literature, in particular,
whenever the picture presented here is not complete or conflicts with other

For instance, the author is advised to account for the existence of numerical methods
and their relevance by adding proper citations. A fair sampling of relevant
papers should include: Znidaric, Phys. Rev. Lett. 106, 220601 (2011), Steinigeweg et al
Phys. Rev. Lett. 116, 017202 (2016), Karrasch et al. Phys. Rev. Lett. 108, 227206 (2012).

2) The author correctly states that the field theory provides the generic
picture in nonintegrable models, i.e., diffusion at T>0.

For integrable models, it makes a prediction that there is either diffusion
or diffusion is the subleading correction next to the Drude weight (which itself
cannot be obtained from field theory independently). This is summarized in Fig. 2.

The full picture, established from exact lower bounds to the diffusion constant
and from generalized hydrodynamics, is more complex, though.

It is now understood that the subleading correction is superdiffusive almost
everywhere in the regime Delta <1, with the exception of the so-called
commensurate points.

Since the article aims at introducing the "unusual transport" in this model,
these very important insights should be mentioned. Also, it is worth mentioning
that so far, field theory fails to predict superdiffusion anywhere in the model (also at Delta=1),
thus pointing the reader to another interesting open problem.

The relevant papers are: Ilievski et al., Phys. Rev. Lett. 121, 230602 (2018)
(see footnote 67), Agrawal et al. arXiv:1909.05263

In the same context, the statement before Eq. 2.28 is not general enough:
diffusion only results if the integral converges, superdiffusion, if it
diverges. For completeness, subdiffusion should also be mentioned (although it
is not believed to occur in clean spin chains).

3) Below Eq. (3.9): The author should also cite Louis and Gros, who first
studied the term magnetothermal correction in Phys. Rev. B 67, 224410 (2003)
for the spin-1/2 XXZ chain.

4) In the discussion of the experiments on page 11, one should clearly state
that the notion of integrability is not needed to explain the experiments
on the current level of experimental data. Moreover, it would help the
reader to be directed to comprehensive reviews on the experiments,
such as Physics Reports 811, 1 (2019). The largest thermal conductivities
in spin chain materials were actually reported in Phys. Rev. B 81, 020405(R) (2010).

5) References for diffusion for Delta >1: The authors should also cite:
Phys. Rev. Lett. 106, 220601 (2011), Phys. Rev. B 89, 075139 (2014),
Phys. Rev. B 80, 184402 (2009), Phys. Rev. Lett. 107, 250602 (2011).

6) References to the current understanding of spin transport at Delta=1
should be added, in particular those also pointing to superdiffusion:
Phys. Rev. Lett. 122, 127202 (2019), Phys. Rev. Lett. 122, 210602 (2019),
as well as : J. Stat. Mech. 2011, P12008 (2011).

7) In particular for a text for students, it is my impression that the seminal
importance of Ref. 7-9 is underrepresented in this article. These papers
solved the long-standing question of whether there is a spin Drude weight at T>0
at all (in the affirmative for Delta <1) and introduced the notion of
quasi-local charges, leading to (one might even say, revolutionary) new insights
into the theory of Bethe ansatz integrable models. I suggest that this should be stated,
where the quasi-local charges are mentioned for the first time (i.e., below Eq. 2.16).

8) The GHD result for the spin Drude weight also agrees precisely with the
TBA formula and the lower bound, Eq. 4.3, see Phys. Rev. Lett. 119, 020602. This could be mentioned on page 12.

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

Author:  Jesko Sirker  on 2020-04-24  [id 807]

(in reply to Report 2 on 2019-11-06)

I thank both referees for their reports.

Referee 1: Before replying to each of the points raised, let me stress again that these are Lecture Notes, i.e., they provide a written account of my lectures at the Les Houches summer school. The manuscript is not intended, and is also certainly not in practice, a full review of this field of research. I believe I have made this clear in the outline of my lecture notes. I have now also 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.' in the outline. These lecture notes will be published together so it makes little sense in my view to now include a discussion of a topic with many further references which was not at all discussed in my lectures at Les Houches but was rather covered in depth in a separate series of lectures at the same summer school.

1) For the first part, see above. I have not added any references to numerical papers because i) I never discussed numerical methods at all during the lectures so giving references for a topic not covered makes little sense in my view, and ii) I believe that if I would try to add some discussion about numerical results then a fair sampling would then require to list many more paper than the three mentioned by the referee.

2) The article Phys. Rev. Lett. 121, 230602 (2018) is already cited and discussed in the conclusions. Furthermore, it is mentioned several times that only anisotropies Delta=cos(pi/m) are considered. I now refer to the results away from the commensurate points a few more times throughout the notes, see list of changes.

Below Eq. (2.28), in particular, I added a sentence about superdiffusion and the divergence of the integral.

3) The reference has been added.

4) The reference Phys. Rev. B 81, 020405(R) (2010) has been added. I note that the authors do interpret their results as clear indication of ballistic transport in the underlying spin-1/2 Heisenberg model. We know that many properties of these systems are very well described by the spin-1/2 Heisenberg model. It is therefore also a natural starting point to discuss the transport properties. I have made it clear, however, that a detailed understanding is still lacking.

5) The case Delta>1 is not part of the lectures. It is briefly discussed in the conclusions and some references are given.

6) Again, this case was not part of the lectures. It is briefly discussed in the conclusions and some references are given.

7) The results in [7-9] do confirm the results derived by Zotos in 1999. They have been without a doubt very important for our understanding of transport in the XXZ chain but I would not subscribe to the referee's point that they have solved the question whether or not there is a spin Drude weight at T>0. They are partly independent from the approach used by Zotos while both share some other possible issues. So either both approaches are correct in which case the spin Drude weight for T>0 was already established by Zotos or they are both incorrect (which seems much less likely) in which case the problem would still be open. In any case, the references are cited and the results discussed.

8) GHD is discussed in a separate series of lectures in the same volume.

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)

Anonymous Report 1 on 2019-10-31 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_201910_00039v1, delivered 2019-10-31, doi: 10.21468/SciPost.Report.1281


These lectures are on a topic of actual theoretical interest, transport in integrable systems at finite temperatures, where recently significant developments are taking place.
They are also complementary to other lectures in this summer school.
The notes are well written and the presentation systematic and
well organized with enough details for a student to follow.
The key notions, thermal and spin Drude weight, Mazur inequality,
Kubo formalism are carefully introduced and the main known results criticaly discussed.
Of course the whole machinery of the TBA method is not presented, but I suppose it was discussed in other lectures.
Finally the notes are complemented with a bosonization approach
so that a complete picture of the frequency dependent
spin conductivity emerges.

I recommend publication of this manuscript as is.

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

Author:  Jesko Sirker  on 2020-04-24  [id 808]

(in reply to Report 1 on 2019-10-31)

I thank the referee for their report. Basics of Bethe ansatz methods were indeed covered in other earlier lectures at the same summer school.

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