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Random matrix universality in dynamical correlation functions at late times
by Oscar Bouverot-Dupuis, Silvia Pappalardi, Jorge Kurchan, Anatoli Polkovnikov, Laura Foini
Submission summary
| Authors (as registered SciPost users): | Oscar Bouverot-Dupuis |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2407.12103v2 (pdf) |
| Date submitted: | 2024-09-10 20:13 |
| Submitted by: | Bouverot-Dupuis, Oscar |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study the behavior of two-time correlation functions at late times for finite system sizes considering observables whose (one-point) average value does not depend on energy. In the long time limit, we show that such correlation functions display a ramp and a plateau determined by the correlations of energy levels, similar to what is already known for the spectral form factor. The plateau value is determined, in absence of degenerate energy levels, by the fluctuations of diagonal matrix elements, which highlights differences between different symmetry classes. We show this behavior analytically by employing results from Random Matrix Theory and the Eigenstate Thermalisation Hypothesis, and numerically by exact diagonalization in the toy example of a Hamiltonian drawn from a Random Matrix ensemble and in a more realistic example of disordered spin glasses at high temperature. Importantly, correlation functions in the ramp regime do not show self-averaging behaviour, and, at difference with the spectral form factor the time average does not coincide with the ensemble average.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1. The results are clearly and concisely presented.
2. The analytical calculations are very detailed, and extensive numerical calculations have been carried out to illustrate the ramp and plateau in correlation functions.
Weaknesses
1. The setting that the authors focus on, namely observables whose diagonal matrix elements do not depend on energy, is very restricted. In Hamiltonian systems, these matrix elements generically depend on energy.
2. The ramp and plateau in autocorrelation functions are exponentially small in the system size, so it is not clear that these features are observable in an experiment taking less than exponential time.
Report
Thank you for sending this manuscript. The authors study the behavior of autocorrelation functions of local observables in chaotic many-body quantum systems. They explore a late-time ramp and plateau in the time-dependence of these autocorrelation functions. Focusing on situations where the matrix elements of the observable are described by random matrix theory, and so where the time dependence is controlled by spectral statistics, it is shown that the well-known ramp and plateau in the spectral form factor imply similar features in autocorrelation functions. Understanding when such a feature should arise in autocorrelation functions is an interesting question. The ramp and plateau behavior was identified for Sachdev-Ye-Kitaev models in Ref. [7] and in chaotic Floquet systems in Ref. [25]; in the latter setting the analysis is simplified by the fact that the diagonal matrix elements of the observable do not depend on (quasi)energy.
Here, the authors also focus on the setting where diagonal matrix elements do not depend on the energy, although this situation is atypical for Hamiltonian systems. A key aim of the present work is to justify when the ramp and plateau arise, but by focusing on such a restricted setting it is not clear whether that aim has been achieved. A basic question is the degree by which diagonal matrix elements can vary while still preserving a ramp and plateau. For example, if the diagonal matrix elements vary, on average, by an amount of order unity from one end of the spectrum to the other, what happens to the ramp and plateau? A simple model in the spirit of ETH would be e.g. diagonal elements which vary linearly with energy.
The ramp and plateau in autocorrelation functions have amplitudes that are exponentially small in system size, and so it is difficult to judge their significance. Within the framework of systems described by the ETH, are there any situations where the ramp and plateau become observable (in an experiment which does not require exponential time)?
Above Eq. (30), the authors write ‘we assume for simplicity that the system is put at infinite temperature (beta=0) since using a finite temperature is a straightforward generalization’. Although it may be straightforward in the situation where the diagonal matrix elements do not depend on energy, this assumption is masking the complexity of the problem in Hamiltonian systems. It would be useful to see explicitly how e.g. Eq. (45) is modified at finite beta, at least for a simple model where the diagonal matrix elements have a simple (but nontrivial) energy dependence.
Due to the issues above, for now I will not recommend publication in SciPost. However, a revised version, which addresses when the ramp and plateau are expected (beyond the one setting of observables whose diagonal matrix elements do not depend on energy), could be suitable for publication.
Requested changes
1. Provide additional details indicating when the ramp and plateau are expected. How are these features of autocorrelation functions suppressed if diagonal matrix elements depend on energy?
2. Elaborate on the amplitude of the ramp and plateau, and how this amplitude depends on temperature. When are these features not exponentially small in system size?
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