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

by Philip J. D. Crowley, Anushya Chandran

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Submission summary

As Contributors: Philip Crowley
Preprint link: scipost_202106_00009v2
Date accepted: 2022-02-22
Date submitted: 2022-01-04 23:13
Submitted by: Crowley, Philip
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


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.

Published as SciPost Phys. 12, 103 (2022)

List of changes

In the response to the referees comments we have:

- Included discussion of the connection to the locator expansion (Sec 3.1 p.8, Sec 3.2.2 p.11)
- Re-written discussion point "Connections to the many-body localisation finite-size crossover" to clarify the points discussed in our response to the referee's comments (Sec. 8 p.29).
- Corrected typos.

Reports on this Submission

Report 2 by Marco Rossi on 2022-1-25 (Invited Report)

  • Cite as: Marco Rossi, Report on arXiv:scipost_202106_00009v2, delivered 2022-01-25, doi: 10.21468/SciPost.Report.4232


The paper concerns a detailed study of the interaction of a two level system with a bath.
The discussion is very general: various coupling regimes and different types of baths are studied, as the first referee remarks.
In particular the main novelty is the discussion of the weak and intermediate coupling regimes,
with the development of an ETH-like ansatz.

The paper is quite technical, but I appreciate that the results are proved and discussed
with a remarkable accuracy. Non trivial checks of the statements are provided by use of numerical
techniques and are illustrated in several useful pictures.

Therefore, in my opinion the paper deserves publication in the present form.

I just noticed some misprints:

1) at the beginning of page 6 should the open boundary condition involve \sigma ^x (and not
\sigma ^z)?
2) five lines after formula (34): relates -->related
3) five lines before formula (53): The the --->The

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2022-1-6 (Invited Report)


The authors have improved the text and have followed all the suggestions given in the previous report. Therefore I think the paper can be published as it is.

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

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