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Transport in one-dimensional integrable quantum systems

by J. Sirker

Submission summary

As Contributors: Jesko Sirker
Preprint link: scipost_201910_00039v1
Date submitted: 2019-10-27
Submitted by: Sirker, Jesko
Submitted to: SciPost Physics Lecture Notes
Discipline: Physics
Subject area: 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.

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Editor-in-charge assigned

Reports on this Submission

Anonymous Report 2 on 2019-11-6 Invited Report


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

Anonymous Report 1 on 2019-10-31 Invited Report


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

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