# Partial thermalisation of a two-state system coupled to a finite quantum bath

### Submission summary

 As Contributors: Philip Crowley Preprint link: scipost_202106_00009v1 Date submitted: 2021-06-04 19:03 Submitted by: Crowley, Philip Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

The eigenstate thermalisation hypothesis (ETH) is a statistical characterisation of eigen-energies, eigenstates and matrix elements of local operators in thermalising quantum systems. We develop an ETH-like ansatz of a partially thermalising system composed of a spin-1/2 coupled to a finite quantum bath. The spin-bath coupling is sufficiently weak that ETH does not apply, but sufficiently strong that perturbation theory fails. We calculate (i) the distribution of fidelity susceptibilities, which takes a broadly distributed form, (ii) the distribution of spin eigenstate entropies, which takes a bi-modal form, (iii) infinite time memory of spin observables, (iv) the distribution of matrix elements of local operators on the bath, which is non-Gaussian, and (v) the intermediate entropic enhancement of the bath, which interpolates smoothly between zero and the ETH value of log 2. The enhancement is a consequence of rare many-body resonances, and is asymptotically larger than the typical eigenstate entanglement entropy. We verify these results numerically and discuss their connections to the many-body localisation transition.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202106_00009v1 on 4 June 2021

## Reports on this Submission

### Strengths

1- Detailed analysis of the distribution of the susceptibility
2- Basing the analysis of other observables (entropy, asymptotic values of spin evolution) on the distribution of the susceptibility
3- Finite size scaling discussion allows comparison with numerics and experiments

### Weaknesses

1- Too short discussion of the implications for the MBL phase/transition
2- (minor point) Very technical on some derivations, whose analysis feels like it could have been simplified

### Report

Referee report on "Partial thermalisation of a two-state system coupled to a
finite quantum bath" by P. J. D. Crowley, A. Chandran.

The paper studies the different coupling regimes of a single spin to a bath, the latter described by either one of the Gaussian ensembles or a random matrix with Poisson statistics or a realistic model given by an ergodic spin chain.

The main object of study is the susceptibility $\chi$ in Eq. (3) or (28).

This is a random variable good to study the response of a system to the introduction of a local operator, which has been studied by several people before. In the specific case at hand, I think the authors fail to recognize that this is exactly the first term in a locator expansion in the spirit of [1] and [2] and more recently in several other papers (see the more recent [3]).
In [1-3] the denominators in the energy are uncorrelated, while in principle $E_a, E_b$ in this paper are, for some ensembles, correlated. But the long tail
$$f\sim \chi^{-3/2},$$
which is a main feature of the distribution, follows from small denominators. The small denominators come from pairs $a,b$ for which $E_a-E_b\sim h_s=O(1),$ but since the level spacing $\delta \ll 1\sim h_S$ (which is the scale on which $E_a,E_b$ correlate), this means that $E_a$ and $E_b$ are uncorrelated to a very good approximation. So we fall back in the conditions of [1,2], and the tail follows from the discussion after eq. (5.1) in [2]. Analogous results are in the denominator analysis in [4,5].

I think the authors should recognize this fact, which does not subtract to their analysis (which goes beyond this point, significantly) but it connects with a set of papers which the authors have not involved in their discussion. I was in particular surprised of the absence of reference to [4] in their otherwise very generous bibliography, considering this work has significant implications for MBL, for which the authors have a separate sub-section.

Regarding the issue of whether ETH should be modified or not in the thermalizing region preceding the MBL transition is not clear to me if the authors agree or not with their references (in the paper [32,53]) and if they agree or disagree with [6] (in this report) which seems to find that if one goes to sufficiently large system sizes, the distribution of the off-diagonal matrix elements return gaussian and that this is not in contradiction with having subdiffusive transport.

For the rest of the paper I have no objections or comments. I like the treatment of the off-diagonal elements of the operators for the intermediate values of couplings, I think it is an original and nice addition to the ETH vulgata and it deserves to be published. The paper as a whole deserves publication, once the authors fix the discussion above.

I caught only one typo:

Fig. 12. Upper and lower panel should be left and right panel in the caption.

[1] Anderson, P. W. (1958). Absence of diffusion in certain random lattices. Physical review, 109(5), 1492.
[2] Abou-Chacra, R., Thouless, D. J., & Anderson, P. W. (1973). A selfconsistent theory of localization. Journal of Physics C: Solid State Physics, 6(10), 1734.
[3] Pietracaprina, F., Ros, V., & Scardicchio, A. (2016). Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions. Physical Review B, 93(5), 054201.
[4] Basko, D. M., Aleiner, I. L., & Altshuler, B. L. (2006). Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of physics, 321(5), 1126-1205.
[5] Ros, V., Müller, M., & Scardicchio, A. (2015). Integrals of motion in the many-body localized phase. Nuclear Physics B, 891, 420-465.
[6] Panda, R. K., et al. (2020). Can we study the many-body localisation transition?. EPL (Europhysics Letters), 128(6), 67003.

### Requested changes

1- Connect with the locator expansion
2- Clarify and/or extend the discussion in the subsection "Connections to the many-body localisation finite-size crossover"

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