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Exact Thermal Properties of Integrable Spin Chains
by Michał Białończyk, Fernando Javier Gómez-Ruiz, Adolfo del Campo
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Submission summary
| Authors (as registered SciPost users): | Michał Białończyk · Fernando Gómez-Ruiz |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202104_00014v1 (pdf) |
| Date submitted: | 2021-04-09 19:12 |
| Submitted by: | Białończyk, Michał |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.
Current status:
Reports on this Submission
Report #2 by Ning Wu (Referee 1) on 2021-5-11 (Invited Report)
- Cite as: Ning Wu, Report on arXiv:scipost_202104_00014v1, delivered 2021-05-11, doi: 10.21468/SciPost.Report.2900
Weaknesses
The expressions for the obtained exact partition function seem already exist in previous literature.
Report
The authors focus on thermal properties of one-dimensional quantum spin chains that admit a free-fermion representation via the Jordan-Wigner transformation. Explicitly, they consider a finite-size quantum XY spin chain under periodic boundary conditions. The resulting even and odd fermion parity sectors are carefully treated to obtain explicit expression of the exact partition function, which is compared with the one in the even parity sector only for different temperatures and transverse fields. The discrepancy between the two in the low-temperature regime is explained by a two-level approximation. In addition, the full counting statistic of observables that preserves the fermion parity (such as the transverse magnetization and the number of kinks) is provided.
Investigation of finite-temperature and non-equilibrium properties of finite-size integrable spin chains in a mathematically rigorous way is important to the understanding of various concepts in statistical mechanics and mathematical physics. The manuscript is clearly written and the results are reliable. I believe the paper is worth being published in SciPost Physics, though have several comments the authors may wish to address.
1. The derived exact partition function for the spin-1/2 XY chain given by Eq. (46) seems quite similar to Eq. (1.46) in Minoru Takahashi's book [M. Takahashi, Thermodynamics of one-dimensional solvable models (Cambridge University Press, Cambridge, 1999)]. The authors my wish to point out the connection/difference between the two results.
2. On page 15 it is mentioned that the treatment of observables having components linear in fermion operators is beyond the scope of the present paper. The authors should comment further on the possible difficulties in obtaining the full counting statistics of such kind of operators.
Report #1 by Anonymous (Referee 2) on 2021-4-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202104_00014v1, delivered 2021-04-19, doi: 10.21468/SciPost.Report.2805
Report
1) Overall, it is certainly an illuminating exercise to be able to work our thermal states of quadratic fermionic systems but these thermal states are not as interesting as they might seem to be because these integrable systems do not thermalize. At best they can dephase towards a generalized Gibbs ensemble which should not be mistaken for the traditional Gibbs state considered in this work.
2) It seems that a ``positive parity approximation'' is being criticised both in the abstract and the introduction. However, the parity is a good quantum number so if the system is initialized in, say, a positive parity state then it will remain in the positive parity subspace during its evolution and it is no approximation at all. Therefore, one should be very carefull not to criticise works where it is not an approximation but a simplification thanks to the conservation law.
The main added value of the manuscript is the formalism that allows to work out the thermal distributions which is a non-trivial task. For this reason it is worth publication in SciPost after the authors carefully rewrite the introduction in order to avoid misleading comments.
Author: Michał Białończyk on 2021-05-21 [id 1448]
(in reply to Report 1 on 2021-04-19)
,,Overall, it is certainly an illuminating exercise to be able to work our thermal states of quadratic fermionic systems but these thermal states are not as interesting as they might seem to be because these integrable systems do not thermalize. At best they can dephase towards a generalized Gibbs ensemble which should not be mistaken for the traditional Gibbs state considered in this work.''
Reply: We thank the referee for emphasizing the relevance of our study.
We agree with the referee that the generalized Gibbs state is the relevant ensemble in many scenarios. This is the case when the time-evolution preserves the conserved quantities so that these are indeed invariants of motion. In fermionic models, this is certainly the case under unitary evolution, in the absence of coupling to a surrounding environment.
However, the canonical Gibbs state we consider is the equilibrium state that results from the coupling of the Ising chain to a thermal environment (in weak-coupling) when the dynamics is not unitary and does not preserve the integrals of motion. Such thermal state has manifold applications indicated in the manuscript. Quantities that are conserved under unitary dynamics (fermionic occupation numbers, that become integrals of motion in this scenario) are not conserved under generic open quantum dynamics (nonunitary evolution), in the presence of a surrounding environment.
In short, the relevant ensemble can be the generalized or the standard Gibbs ensemble, depending on whether the thermalizing time-evolution preserves integrals of motion other than the energy or not.
We have added the following sentence to the end of the Discussion section:
“As a prospect, it is interesting to extend our results to the generalized Gibbs state whenever the relaxing dynamics of an initial state preserves a set of integrals of motion.”
,,It seems that a positive parity approximation is being criticised both in the abstract and the introduction. However, the parity is a good quantum number so if the system is initialized in, say, a positive parity state then it will remain in the positive parity subspace during its evolution and it is no approximation at all. Therefore, one should be very carefull not to criticise works where it is not an approximation but a simplification thanks to the conservation law.''
Reply: We have been careful in this regard. All the cited references (“[1–3, 43, 58–62]”) used approximate results and either consider the partition function using the PPA [1-3] or canonical thermal states using the PPA or both [43]. The exact treatment in all these cases involves the consideration of both subspaces, as no particular simplification occurs in the considered scenarios. We have taken the chance to add other seminal works that use the PPA in the thermodynamic limit and emphasized that not even then the PPA is exact. We have emphasized classic references such as the work by Pfeuty and the textbooks by Sachdev and Suzuki et al. to give a better account of the widespread use of this approximation.
,,The main added value of the manuscript is the formalism that allows to work out the thermal distributions which is a non-trivial task. For this reason it is worth publication in SciPost after the authors carefully rewrite the introduction in order to avoid misleading comments.''
Reply: We thank the referee for the positive assessment of our work.
Anonymous on 2021-05-22 [id 1456]
(in reply to Michał Białończyk on 2021-05-21 [id 1448])As I already wrote in my first report, this is a piece of solid work that definitely is worth publication. My only critical comments concerned introduction/motivation. Now that they have been satisfactorily fixed I have no longer any objections. I am impressed by the in depth response to Referee 2. I would suggest publication without necessity for any further ammendmends.
Author: Michał Białończyk on 2021-05-21 [id 1447]
(in reply to Report 2 by Ning Wu on 2021-05-11)Due to the significant number of accompanying graphs, we submit the full response in one .pdf file.
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scipost_202104_00014_v1.pdf