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Partial thermalisation of a twostate system coupled to a finite quantum bath
by Philip J. D. Crowley, Anushya Chandran
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Authors (as registered SciPost users):  Philip Crowley 
Submission information  

Preprint Link:  scipost_202106_00009v1 (pdf) 
Date submitted:  20210604 19:03 
Submitted by:  Crowley, Philip 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The eigenstate thermalisation hypothesis (ETH) is a statistical characterisation of eigenenergies, eigenstates and matrix elements of local operators in thermalising quantum systems. We develop an ETHlike ansatz of a partially thermalising system composed of a spin1/2 coupled to a finite quantum bath. The spinbath 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 bimodal form, (iii) infinite time memory of spin observables, (iv) the distribution of matrix elements of local operators on the bath, which is nonGaussian, 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 manybody resonances, and is asymptotically larger than the typical eigenstate entanglement entropy. We verify these results numerically and discuss their connections to the manybody localisation transition.
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Reports on this Submission
Anonymous Report 1 on 202178 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202106_00009v1, delivered 20210708, doi: 10.21468/SciPost.Report.3214
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 twostate 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 [13] 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 $E_aE_b\sim h_s=O(1),$ 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 subsection.
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 offdiagonal 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 offdiagonal 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] AbouChacra, 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 meanfield approximation for the Anderson and manybody 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 manyelectron system with localized singleparticle states. Annals of physics, 321(5), 11261205.
[5] Ros, V., Müller, M., & Scardicchio, A. (2015). Integrals of motion in the manybody localized phase. Nuclear Physics B, 891, 420465.
[6] Panda, R. K., et al. (2020). Can we study the manybody 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 manybody localisation finitesize crossover"
Author: Philip Crowley on 20220104 [id 2067]
(in reply to Report 1 on 20210708)We thank the referee for their time, their positive report, and their recommendation to publish the manuscript with minor amendments.
We respond to the referee's comments inline, and include with this report, an amended manuscript with the major alterations highlighted.
"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 [13] 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 subsection."
We thank the referee for raising this connection. The referee is correct about the close connection between our analyses to the locator expansion, and the relation of our results to those previously obtained in this literature. We are indeed clear in Sec 3.1 that we are performing an expansion in the coupling between two otherwise decoupled subsystems. However, in order to further make the connection explicit in the revised manuscript we have included a discussion of this point at the end of Sec 3.1, and after point (i) in Sec 3.2.2, where we emphasise that the $f \sim \chi^{3/2}$ result can be understood in this context of the locator expansion. In each of these places we have included the references suggested by the referee.
"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 offdiagonal matrix elements return gaussian and that this is not in contradiction with having subdiffusive transport."
The referee has asked for clarification regarding whether our results are in contradiction or agreement with Refs. [32,58] (reference numbers updated to correspond to most recent draft).
We first summarise the relevant findings of Refs. [32,58]. These studies focus on two distinct features of the offdiagonal matrix elements on the thermal side of the MBL transition in random Heisenberg type models, both of which the authors argue are consequences of a putative subdiffusive thermal phase. The first observation concerns changes to the $L$ dependence of the spectral function, and the second regards deviations from Gaussianity in the distribution of offdiagonal matrix elements. We focus on the latter point, which we contest, and provide a different explanation of the observed nonGaussianity.
Our analyses does not provide a complete model of an interacting many body system. Our results do nevertheless allow us to understand on the origin of the nonGaussian distributions of off diagonal matrix elements in numerically accesible spin chains. Non Gaussianity results from the presence of spins which are insufficiently strongly coupled to their environmentirrespective of the level statistics of their environment. This explanation does not make reference to the presence, or otherwise, of subdiffusion. Specifically, in sufficiently small systems, where spins are found in the intermediate regime ($J \sqrt{\chi_\star} \lesssim 1$), we predict off diagonal element distributions of the form of Fig. 11. This nonGaussianity goes handinhand with other features of the intermediate coupling regime, such as the spins having eigenstate entanglement entropies which are bimodal, being either close to the thermal ($S= \log 2$) or localised ($S=0$) value. The presence of bimodal spin entropies is inconsistent with a thermal phase, subdiffusive or otherwise, in contrary to the second claim of Refs. [32,58]. Such bimodal distributions have been observed numerically [66] at the same system sizes as Refs. [32,58]. Finally, Ref.[6](of the referees report)pointed out by the refereeshows these distributions flow towards Gaussianity upon increasing the system size. The SpinETH model explains this flow via an increasing value of $J \sqrt{\chi_\star}$ felt by each of the spins in the chain.
Ref. [6] pointed out by the referee indicates, in agreement with our claim, that even in the putative subdiffusive phase, the distribution of offdiagonal elements becomes Gaussian at sufficiently large system size.
In response to the referees comments we have expanded and clarified this point in the discussion section of the revised manuscript to avoid any confusion for the reader.
"For the rest of the paper I have no objections or comments. I like the treatment of the offdiagonal 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."
We thank the referee for their positive recommendation.
"I caught only one typo: Fig. 12. Upper and lower panel should be left and right panel in the caption."
We thank the referee's keen eye, and have corrected this typo, amongst others.