# Exact Thermal Properties of Integrable Spin Chains

### Submission summary

 As Contributors: Michał Białończyk Preprint link: scipost_202104_00014v1 Date submitted: 2021-04-09 19:12 Submitted by: Białończyk, Michał Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Quantum Physics Statistical and Soft Matter Physics 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:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202104_00014v1 on 9 April 2021

## Reports on this Submission

### 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.

• validity: good
• significance: good
• originality: ok
• clarity: high
• formatting: perfect
• grammar: excellent

### 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.

• validity: good
• significance: good
• originality: good
• clarity: top
• formatting: perfect
• grammar: excellent