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Statistical mechanics of integrable quantum spin systems
by Frank Göhmann
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Submission summary
| Authors (as registered SciPost users): | Frank Göhmann |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202004_00045v1 (pdf) |
| Date submitted: | 2020-04-13 02:00 |
| Submitted by: | Göhmann, Frank |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. The lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. The author was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. He was further asked to explain the graphical language of vertex models and to introduce the concepts of the Trotter decomposition and the quantum transfer matrix. This was basically the contents of the first four lectures presented at the school. In the remaining two lectures these notions were filled with life by deriving an integral representation of the free energy per lattice site for the Heisenberg-Ising chain (alias XXZ model) using techniques based on non-linear integral equations.
Current status:
Reports on this Submission
Anonymous Report 1 on 2020-5-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202004_00045v1, delivered 2020-05-15, doi: 10.21468/SciPost.Report.1672
Strengths
1. Pedagogical quality of the lecture course
2. Good overview of the subject
Weaknesses
None
Report
This is an excellent lecture course perfectly corresponding to the ``les Hoches series'' requirement. It gives rather large overview of the subject while remaining quite pedagogical and accessible for graduate students in statistical mechanics and quantum field theory. The lectures are well written and in my opinion should be published after some minor additions and corrections lister bellow.
Requested changes
1. I think there is a misprint in the introduction (end of page 4) the word quantum (in quantum transfer matrix approach) should be rather inside the quotation marks than outside.
2. In the beginning of the subsection 1.2.2 the number of lattice sites is denoted $N$ this notation can lead to a confusion with the Trotter number denoted also by $N$ throughout the lectures.
3. Misprint in the eq. 1.22 (the density matrix is non-negatively definite)
4. In the remark to the section 4.1 the paper
J. M. Maillet and V. Terras, Nucl. Phys. B 575, 627 (2000)
should be cited together with [16] and [17]
5. There is a misprint in the first phrase of the section 4.4, should be probably has to be used?.
6. In the section 5.1 the original paper
L. D. Faddeev, E. K. Sklyanin and L. A. Takhtajan, Theor. Math. Phys. 40, 688 (1979)
should be cited for the Algebraic Bethe Ansatz
7. For the eqs. (5.129), (6.148) and (6.152) illustrations of the integration contours can be useful for a reader
8. More details of the computations in (6.152) can be useful (like it was done for example for the computation in (5.129)).
Anonymous Report 2 on 2020-5-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202004_00045v1, delivered 2020-05-10, doi: 10.21468/SciPost.Report.1674
Strengths
1- Well written introductory text, good summary.
Weaknesses
no significant weaknesses
Report
This is a well written introductory material. It is supposed to be a Lecture Notes. The author was asked to explain some part of the literature where he contributed significantly with original research works.
I only have minor comments.
Requested changes
1. Short typo: ''an inaccuracies''
2. When the density matrix is introduced, I find that paragraph too quick and short. Perhaps the author could mention, that if the chain is coupled with the environment, and the whole system is in a ''pure state'', then the correlations of the chain can be described by the density matrix. Perhaps this is what the author meant with mentioning experiments, but it is useful to spell this out.
3. When the GGE is mentioned the ref [8] is cited. This is certainly an important paper, but I would not give only this as further reading to students, because it is a very brief 4 page paper for PRL. I would add the reviews
https://arxiv.org/abs/1603.06452
https://arxiv.org/abs/1604.03990
or anything similar.
4. In 2.45 I think the partial transpose is not explained.
5. The proof of the RTT relations for the QTM is very a difficult for a student who sees this first. I don't think that this should be left as an exercise. Perhaps some comments or instructions, or references to other lecture notes should be given.
6. Typo: ''has be used''
Author: Frank Göhmann on 2020-06-06 [id 850]
(in reply to Report 2 on 2020-05-10)Thank you for your positive and helpful report. I have changed the manuscript according to your suggestion. Just point 4 confused me as the partial transposed is explained right after 2.45 in equation (2.46).
Author: Frank Göhmann on 2020-06-06 [id 849]
(in reply to Report 1 on 2020-05-15)I would like to thank the referees for their constructive criticism. I have included all suggested references and tried to comply with all other suggestions. I did not take up point 2 and point 8 of the first report. I think that there is very little danger of confusion with the letter "N" in subsection 1.2.2, since it comes well before the Trotter number is introduced. And for point 8 I simply did not know what to add. In my eyes the presentation is already rather detailed.