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Spin conductivity of the XXZ chain in the antiferromagnetic massive regime

by Frank Göhmann, Karol K. Kozlowski, Jesko Sirker, Junji Suzuki

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Submission summary

Authors (as Contributors): Frank Göhmann
Submission information
Arxiv Link: (pdf)
Date accepted: 2022-04-26
Date submitted: 2022-04-14 11:52
Submitted by: Göhmann, Frank
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We present a series representation for the dynamical two-point function of the local spin current for the XXZ chain in the antiferromagnetic massive regime at zero temperature. From this series we can compute the correlation function with very high accuracy up to very long times and large distances. Each term in the series corresponds to the contribution of all scattering states of an even number of excitations. These excitations can be interpreted in terms of an equal number of particles and holes. The lowest term in the series comprises all scattering states of one hole and one particle. This term determines the long-time large-distance asymptotic behaviour which can be obtained explicitly from a saddle-point analysis. The space-time Fourier transform of the two-point function of currents at zero momentum gives the optical spin conductivity of the model. We obtain highly accurate numerical estimates for this quantity by numerically Fourier transforming our data. For the one-particle, one-hole contribution, equivalently interpreted as a two-spinon contribution, we obtain an exact and explicit expression in terms of known special functions. For large enough anisotropy, the two-spinon contribution carries most of the spectral weight, as can be seen by calculating the f-sum rule.

Published as SciPost Phys. 12, 158 (2022)

Author comments upon resubmission

Following the editor's recommendation the manuscript underwent
a major revision. The main change concerns the introduction which
is now considerably enlarged such as to allow for the citation of more
background material. In addition, we have addressed all points that
were explicitly raised by the referees and some points that were implicit
in their statements.

Here are our comments to the points raised by the referees:

Referee 1:

Minor comments:

1 - No exact error estimates can be obtained at long times in time-dependent DMRG calculations. As a substitute, one uses, on the one hand, the truncation error - given by the sum of the eigenvalues belonging to the eigenstates of the reduced density matrix which are truncated in a single renormalization group step - and, on the other hand, a comparison of simulations where a different number of states have been kept. The truncation error and the number of states kept are given in the manuscript. For the longest simulation times, we do indeed only claim an estimated accuracy of the order of the symbol sizes in the figure which are ~10^{-1}. This is spelled out explicitly as well. We believe that our results for the achieved simulation times and accuracies are in-line with the best available tDMRG codes. tDMRG calculations at long times are not quasi-exact and it is not possible to estimate the error to a precision of 10^{-6}.

2 - Thank you for raising this point. The relation with Section 2.3
was not clearly explained. The velocity v was actually taken as
v_2 from Section 3.2. We have revised the text at the bottom
of page 7 and the caption of the figure showing Table 1, where we
now refer to Eq. (2.17).

3 - The analytical result for the two-spinon contribution is indeed zero for omega->0. The purpose of Fig. 4 was to show that the obtained results for the time-dependent current-current correlation function are so precise that even a direct Fourier transform without any filters or windowing functions yields only very small deviations from zero at zero frequency. This has been further clarified in the amended manuscript.

Comments on the text:

" ... attempted to go beyond the framework of linear response"
-> We have removed this sentence as we understand it can be
mistakable here.

" ... requires and honest calculation of ..."
-> This sentence has been removed as well as we understand that
it may appear slightly polemic to some readers.

"So far, the most successful attempts ..."
-> This sentence refers to exact results at finite frequency for
the XXZ chain. Here we have referred to everything relevant of that
kind we were aware of. Most of the articles cited here are not
our attempts but belong to other authors. We think it would be
misleading to list works for zero frequency here.

We understand, however, that it might be helpful for the readers
to place our work into the context of those works as was also
suggested by the third referee.

In order to comply with this suggestion we have split the
paragraph following Eq. (1.1) and have considerably extended the
discussion of spin transport. Now the ballistic component and
the Drude weight are discussed and reference to the corresponding
literature is made (see bottom part of page 2 and top part of
page 3).

"The Lorentzian peak around ..."

We are not aware of any previous exact results for the spin conductivity at finite frequencies.

Referee 2:

We have clarified in the abstract that we are discussing the
limit of zero temperature. Thank you for this suggestion.

Referee 3:

1 - done (see reply to second referee)
2 - done (see reply to first referee)
3 - we have changed the text accordingly and hope that this
is clearer now
4 - corrected, thank you!
5 - We do not think that these formula are singular, but the
case omega = 0 would require a separate consideration. In
Appendix B we show that the function sigma' is even in omega.
For this reason we concentrate on omega > 0, which in the
revised version is stated in several place in the text e.g. on
pages 10 and 11.
6 - We think that the massive nature of the excitations is
responsible for the fast convergence at T = 0. The true
advantages of the method will become apparent at T > 0, where
there is indeed a sum less, which is replaced by a projection
onto a single dominant state, just like in case of the thermodynamics
and of the static correlation functions. Amazingly, in the
limit T -> 0 another advantage appears. It does not affect
the relative size of the terms in the series but the structure
of the integrands in the multiple integrals. In our understanding
this is an important technical achievement on the way
from [46] (which was [27] in the previous version) to [53,54]:
As compared to previous methods the integrand is explicit in every
term and no additional integrals or sums appear in the definitions
of the integrands. It is this feature which makes the integrals
computable and gives us hope that we can prove the convergence
of the series and derive explicit error estimates. We have tried
to make this more clear by extending the second paragraph on
page 3 and the Summary section.

We have also included a number of remarks on how to derive our main
formulae (2.11), (2.12) in the last paragraph of the introduction.
Some experts may now understand this point. We hope that this
may easy, at least partially, the dissatisfaction uttered by the referee,
when he/she says that "this paper is not really self-contained". In
fact, the derivation of the main formulae will be presented elsewhere,
but the methods required to derive them were fully laid out in our
previous work.

List of changes

1) "at zero temperature" appended to the first sentence of the abstract.

2) We replaced the term "dynamical conductivity" by "optical conductivity" as this seems more common in the physics literature.

3) Second sentence of the first paragraph of the introduction "The theoretical emphasis ..." deleted and first and second paragraph merged.

4) Former third paragraph split into a part on the thermal Drude weight and another one on the spin Drude weight.

5) Text starting from "Still, it may have..." on the bottom of page 2 of the revised manuscript to "... does not involve any kind of string hypothesis" in the middle
of page 3 is new. In this new part of the text we now discuss ballistic and diffusive spin transport at T > 0 and place our work into the context of the cited results.

6) Phrase "... and can be interpreted as a resummation of the latter" added in the third paragraph on page 3.

7) Sentences from "This has to be contrasted with..." to "... massive nature of the excitations" at the end of the third paragraph on page 3 added. We added this in order to comply with the third referee's suggestion to explain more details of the method.

8) For the same reason we added "Although the general formula is easy to guess ..."
and the sentences from "For this special case the result may be obtained ..." to the first paragraph on page 4.

9) On page 6 we added the second paragraph of section 2.2 starting with "In order to estimate the truncation error ...". We did this to clarify the question about the numerical error raised by the first referee.

10) Typo in the second line of third paragraph of section 2.2 corrected.

11) End of the third paragraph of section 2.2 starting from "... in agreement with common belief ..." modified and amended.

12) Last paragraph of section 2.2 and Table 1 modified and amended "$v$" changed into "$v_2$" reference to section 2.3 added.

13) At two places on page 11 "$\omega > 0$" added.

14) Text of the second paragraph below equation (3.8) starting with "Note that we perform ..." amended.

15) Last sentence "We attribute the fast convergence ..." added to the first paragraph of section 4.

16) Last paragraph starting with "Further future goals ..." added to section 4.

17) References [17-35] and [69] added.

18) In addition some minor typos corrected and graphical improvements performed.

Reports on this Submission

Anonymous Report 3 on 2022-4-15 (Invited Report)


the paper can now be published as is

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Anonymous Report 2 on 2022-4-14 (Invited Report)


I thank the authors for the changes brought to the manuscript, I now recommend submission in the present form.

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Anonymous Report 1 on 2022-4-14 (Invited Report)


The paper is now ready to be published.

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