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Timescales of quantum and classical chaotic spin models evolving toward equilibrium
by Fausto Borgonovi, Felix M Izrailev, Lea F Santos
Submission summary
| Authors (as registered SciPost users): | Lea Santos |
| Submission information | |
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| Preprint Link: | scipost_202402_00033v1 (pdf) |
| Date submitted: | 2024-02-22 19:01 |
| Submitted by: | Santos, Lea |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the quantum and classical quench dynamics of a one-dimensional strongly chaotic lattice with $L$ interacting spins. By analyzing the classical dynamics, we identify and elucidate the two mechanisms of the relaxation process of these systems: one arises from linear parametric instability and the other from nonlinearity. We demonstrate that the relaxation of the single-particles energies (global quantity) and of the onsite magnetization (local observable) is primarily due to the first mechanism, referred to as linear chaos. Our analytical findings indicate that both quantities, in the classical and quantum domain, relax at the same timescale, which is independent of the system size. The physical explanation for this behavior lies in the conservation of the $L$ spin angular momenta. We argue that observables with a well-defined classical limit should conform to this picture and exhibit a finite relaxation time in the thermodynamic limit. In contrast, the evolution of the participation ratio, which measures how the initial state spreads in the many-body Hilbert space and has no classical limit, indicates absence of relaxation in the thermodynamic limit.
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In this manuscript, the authors studied the post-quench dynamics for a chaotic one-dimensional lattice spin model. They investigated the “quantum-classical correspondence” of this model and numerically showed that its classical version captures the time scales of quantum dynamics in the chaotic regime they focused on. Then, they used this correspondence to understand the relaxation dynamics especially in the quantum case. One focus of their discussion is the relaxation of a global observable (single particle energy spread), where they find the ballistic and diffusion timescales are both independent of system size. They also analyzed the system size independent diffusion timescale with a local observable (Sz for a local spin). In the last portion of the paper, they study the relaxation of participation ratio, which they claim is an inherently quantum metric that has a diverging relaxation timescale in the thermodynamic limit. Overall, the manuscript is well written and contains both numerical and semi-analytical explanations to support their arguments. It should be of interest to readers working in the broad field of out-of-equilibrium quantum many body systems. We recommend publication after the authors address the following questions/comments.
1. It is not clear that the interaction contribution to the Hamiltonian [Eq. (3)] provides a well-defined limit of large S, in the sense it is not extensive with spin length. If we normalize the interaction amplitude as J0/\sqrt{S(S+1)}, the \sqrt{S(S+1)} in the denominator of Eq. (36) now cancels, and the contrasting behaviour of \tau_N with system size L in the thermodynamic limit vs spin length S in the classical limit is removed. Can the authors comment on that?
2. In Fig.3 (b), it seems with the increase of L, the curves shift to left. But L=200 seems to shift in the opposite direction compared to L=100, can the author comment on that?
3. Is there a factor of 1/9 missing in the first line of Eq.(13)?
4. In Fig. 4, can the authors comment why Delta_Estat seems to go to zero in the integrable limit J0=0? It is not clear from their prior analysis, which focused on the ergodic case.
5. In the explanation of Fig. 5, the authors mention that the Gaussian should have a variance proportional to Dt. Can the authors explain what is the t in this expression? Right now, it may be confusing to the reader that this variance changes with time.
6. In Fig.10, it seems the L=4 data in panel (b) have more significant noises compare to (a), can the authors comment on that?
7. There is a typo B=0=1 at the caption of Fig2.
8. In Eq. (16), it appears the first term should be B0^2 instead of 1
9. The abbreviation ‘QCC’ is used in the Sec I but defined in Sec VII.
10. Should the “participation ratio” at the end of last paragraph of Sec VII be “inverse participation ratio”? It seems that is the quantity discussed in that reference.
11. A general comment, can the authors specify what is the S used in each plot?