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The propagator of the finite XXZ spin-$\tfrac{1}{2}$ chain
by G. Z. Fehér, B. Pozsgay
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Submission summary
| Authors (as registered SciPost users): | Balázs Pozsgay |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1808.06279v3 (pdf) |
| Date submitted: | 2019-04-18 02:00 |
| Submitted by: | Pozsgay, Balázs |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We derive contour integral formulas for the real space propagator of the spin-$\tfrac12$ XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained by two different methods. First we compute it through the spectral sum of a deformed model, and as a by-product we also compute the propagator of the XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second way we also compute the propagator through a lattice path integral, which is evaluated exactly utilizing the so-called $F$-basis in the mirror (or quantum) channel. The final expressions are similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. As an application of the propagator we compute the Loschmidt amplitude for the quantum quench from a domain wall state.
Author comments upon resubmission
We are thankful to the referee for the review of the manuscript and the comments. The two main points of critique of the referee were that 1. We did not mention/use an earlier result of Kitanine et. al. (hep-th/0407108) 2. We did not publish the full proof of our formulas, just up to two particles.
We certainly agree with the first point. We did not know the particular paper pointed out by the referee, and indeed the lemmas and theorems there are very useful for our purposes too. Therefore, in addition to properly citing this paper we also included a new derivation of our results using the methods of this paper. So we present now our paper after a major revision.
With regard to the second point we do not completely agree. The full combinatorial proof is long and technical and we did not find an easy way to present it. In our own notes it takes more than 40 pages. And now, with the much more simple proof using the twisted transfer matrix and the kappa->0 limit we do not see reasons to present the considerably longer proof as well.
Furthermore, one could argue that the second part, using the F-basis could be deleted completely from the manuscript. However, we chose to keep it, in order to show an alternative idea, which might be still useful in other circumstances, for other derivations or other models.
List of changes
-We included references to the earlier works dealing with dynamical correlation functions.
-We restructured the manuscript, and included a complete derivation using the eigenstate basis of the twisted transfer matrix.
-We explained that the kappa deformation is not only a neat mathematical trick, but it can also be used to compute the propagator for the XXZ model perturbed by a Dzyaloshinskii–Moriya interaction term.
-We added a new section: as an application we computed the finite volume Loschmidt echo for the quench from the domain wall state. This exact result can be the starting point of later asymptotic analysis.
-We added some references requested by the referee.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2019-5-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1808.06279v3, delivered 2019-05-02, doi: 10.21468/SciPost.Report.928
Strengths
1- The author have followed my previous recommendation to refer to what is now Reference [21], and have built their main proof of the integral representation of the propagator by closely following the procedure given in this reference. This indeed provides a much more straightforward derivation of this representation.
Weaknesses
1- The authors have not given the general proof of the main result by means of their original method using the F-basis. As I was fearing from the two-particle case, it seems that this is not really tractable in the general case. This is a pity since the method was a priori original and ingenious, but I don't insist on this.
2- References on the previous literature are still sometimes treated in a rushed manner.
3- One may wonder whether the obtained formula is really useful: is it really possible, from the final result, i.e. from the multiple integral representation obtained for the propagator, to study the large-size and long-time limits ? It seems to me quite complicated. Some attempts have already been made to study the asymptotic behavior of these kinds of integral representations in the context of correlation functions at equilibrium (based on the work of references [20,21]). In particular, in the couple of references:
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions", Journal of Statistical Mechanics: Theory and Experiment, P04003 (2009)
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications", Commun. Math. Phys. (2009) 291: 691
the large-distance asymptotic expansion was derived in the static case from the multiple integral representation of Reference [20]. But the derivation is quite complicated, and I don't know any work that managed to extract a long-time and large-distance asymptotic behavior directly from the representation obtained in reference [21].
Report
In this revised version, the authors have followed my suggestion to consider the proof of their result through reference [21], so that they now present a short derivation of their main formula. They have moreover added an illustration of their result by presenting a multiple integral representation for the Loschmidt amplitude for the domain wall quench. Unfortunately, the proof by means of the F-basis is still limited to the two-particle case, but it seems to be too complicated to be presented in the general case. The question remains to see whether the integral representation for the propagator is really useful for effective calculations. I think it would at least deserve some more critical discussion, based on what it was possible to do from similar representations in the context of the study of correlation functions.
Requested changes
I think that, with respect to the previous version, the paper has improved, and is now suitable to be published, provided the authors consider the following small changes:
1- Add a few references at some places:
1.1- For instance, in the very beginning, I think the sentence "whereas the largest part of the literature is devoted to the study of the state functions and correlation functions in the ground state or at finite temperatures" would deserve several more references:
- some references about the q-vertex operator approach of the Kyoto group, at least of the book "Algebraic analysis in solvable lattice models" of Jimbo and Miwa.
- some references about the ABA approach of the Lyon group, for instance:
N Kitanine, JM Maillet, V Terras, "Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field", Nuclear Physics B 567, 554-582 (2000)
and some further papers, such as references [20,21]
- some references about the approach of Boos, Jimbo, Miwa, Smirnov and Takeyama, for instance:
H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama, "Algebraic representation of correlation functions in integrable spin chains", Annales Henri Poincare 7: 1395-1428 (2006)
M. Jimbo, T. Miwa, F. Smirnov, "Hidden Grassmann Structure in the XXZ Model III: Introducing Matsubara direction", J. Phys. A 42: 304018 (2009)
and some other papers
- some references about the QTM approach of Goehmann, Klumper et al, for instance:
F. Göhmann, A. Klümper, A. Seel, "Integral representations for correlation functions of the XXZ chain at finite temperature", J.Phys. A37 (2004) 7625-7652
and some further papers
- some references about the analytic derivation of the asymptotic behavior of correlation functions, for instance
N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions", Journal of Statistical Mechanics: Theory and Experiment, P04003 (2009)
N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "A form factor approach to the asymptotic behavior of correlation functions in critical models", Journal of Statistical Mechanics: Theory and Experiment 2011 (12), P12010
as well as the review they already provided.
1.2- Also, when introducing the Dzyaloshinsky-Moriya interaction term, the authors do not provide any reference at all about previous literature on the subject. In addition to the seminal papers, they could also cite:
F. C. Alcaraz and W. F. Wreszinski, "The Heisenberg XXZ Hamiltonian with Dzyaloshinsky-Moriya Interactions", Journal of Statistical Physics, Vol. 58, 45 (1990)
1.3- What the authors call the "Izergin-Korepin determinant" was first derived by Izergin alone in
A.G. Izergin, Sov. Phy. Dokl. 32 878-9 (1987).
This reference should be cited.
1.4- The appendix on multidimensional residues also deserves some reference.
2- In the conclusion, I would like to see a more critical discussion about the possibility to extract in practice the large-size limit, finite-size effects, and long-time limit from such multiple integral representations. As mentioned in the paragraph "Weaknesses", point 3, previous attempts in this direction from similar formulas in the context of correlation functions have shown that it was not so easy. It was shown to be possible in the static case (from the formulas of reference [20]), and I think the works
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions", Journal of Statistical Mechanics: Theory and Experiment, P04003 (2009)
- N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov, V. Terras, "Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications", Commun. Math. Phys. (2009) 291: 691
should be mentioned in that context. However, in the dynamical case, it seems to me that the question still remains open.
3- Finally, since the core of the paper is now based on the proof issued from reference [21] which I have explicitly pointed to the attention of the authors in my previous report, I think it would be correct that the authors thank me for that in the acknowledgments (as the "anonymous referee").