On the difference between thermalization in open and isolated quantum systems: a case study

1. Well-posed problem, satisfying main criterion 2.
2. Generally well written manuscript, however with occasional slips (see below)

1. Unreliable numerical analysis of "nonintegrability"
2. Exaggerated claims
3. Chaotic and incomplete references, several missing, several redundant
4. No system size analysis

The manuscript aims to compare two scenarios of thermalization: an isolated system approach and an open system approach. To this end, a very idealized system is considered in which a fermionic noninteracting lead is coupled to a double quantum dot envisioned as two interacting fermionic sites.
The lead is assumed to form a bath for the double quantum dot. Such a simple model allows for a rather detailed analysis. The authors made several claims about their results:
1. they identify a time scale before which thermalization may take place according to an open system approach;
2. they claim that this time scale is confirmed numerically both for noninteracting and interacting (within the dot) cases.
3. They perform analysis of spectra form factor and spacing distributions claiming that for the interacting dot the system has an intermediate statistics and is ``a partially non-integrable model''
4. They analyze the dynamics for long times stating the differences between ``open'' and ``isolated'' approaches linking that to the order of limits taken for time and system size.

Unfortunately, while a comparison between "open" and "isolated" approaches is interesting and has not been studied in such detail before, some of the claims seem either almost trivial or based on insufficient or poorly analyzed evidence.

The "first result" of the paper (p.12) determination of time $t_oqs$ seems a trivial consequence of Lieb-Robinson bounds (citation needed). $L_0$ is set arbitrarily.

Consider the analysis presented in Section 3.4. I do not understand the choice of citations to formulae quoted in pp.(14)-(16) - formulae often originating from either random matrix theory or standard quantum chaos treatments. The reader should be referred to existing good textbooks on quantum chaos (Haake, Stoeckmann) instead of some, rather erratically selected original papers. While these books mainly address single-particle physics, tools such as random matrix theory are universal and were first applied to many-body physics - excited states of nuclei. This physics is almost 70 years old. It is known, in particular, that random matrix theory describes fluctuations around mean values. To get meaningful results one should carefully unfold the spectra (see books above), a procedure which may be tricky (see e.g. PhysRevE.66.036209). There is no mention of the unfolding in Sec.3.4 making the results in Figs. 3-4 doubtful. The mean level spacing appears in (38) but using the argument in (37) x=Wt/2piD where W is the eigenvalues span and D the Hilbert space dimension suggests that the density of states is assumed to be uniform and no unfolding is done. If it is so then the results have to be reevaluated properly. The application of ad-hoc Brody distribution also requires a comment. Again from earlier quantum studies, it is known that it works well for mixed phase space dynamics if a relatively small number of low-lying states are taken into account. Brody fails for highly excited states (see papers by T. Prosen). Why not use Lenz-Haake (PRL 1991) distribution evaluated for 2x2 matrices as Wigner distribution is?

I find the Authors "third" claim "identifying the non-integrable* nature of the interacting DQD is the third result of the paper" as insufficiently proven. They themselves quote papers of Modak and collaborators [47,48] who claimed that a "partially nonintegrable" behavior (and e.g. Brody-like statistics) was a finite system size effect and that with increasing size many-body interacting systems become ergodic. Thus a claim of the present authors that their interacting DQD enjoys intermediate statistics and is a first example of such a behavior,
should be supported by analysis of different system sizes (different L_B) to show that the observed behavior is not a finite size effect. The work in Section 3.4 is restricted instead to L_B+2=18 sites and no size effects are discussed. Let us note that already Modak et al. considered comparable systems of size 22 and it was ten years ago!. The authors concentrate instead on dependence on V/g which again may be affected by a finite size.
Importantly the authors are not aware of or ignore a vast literature on the subject. Here on one side PRB105, 214308 should be consulted both for its contents and for earlier references. Also relevant are studies of ``central spin'' models SciPost Physics 15, 030 (2023) as well as that of impurity induced interactions Phys. Rev. Lett. 126, 030603 ; Phys. Rev. B 105, L220203; Phys. Rev. B 105, 224208; Phys. Rev. B 107, 144201 (2023). Those are in the presence of disorder but for tilted lattices see Phys. Rev. B108 134201. In fact, in the presence of disorder the intermediate statistics appears in MBL transition so a vast number of systems (at finite size) shows the non-integrable* nature. What is really the outcome in the thermodynamic limit? - see the recent review arXiv:2403.07111.

Consider now Sec.3.5. While Fig.5 shows a profound difference in the dynamics of noninteracting and interacting systems, no real explanation for that behavior is given. What are the dashed lines in Fig 5(a) and (b)? Is the correlation function in fig.5(b) decaying exponentially or via a power law? What is the dependence on the interaction strength? As it is Fig.5 shows just a profound difference between a noninteracting and interacting cases which is obvious.

Sometimes I simply do not understand what the authors are meaning. In the description of an isolated system thermalization:
1/Why the initial state should fulfil O(L_B^-1) in (14)? What doe it mean usually? May be some reference?
2/I do not understand the statement (same page, just after (16)). " In the microcanonical picture setting the chemical potential to zero corresponds to a half-filling of the entire chain.". The reviewer does not know the notion of the chemical potential in microcanonical ensemble. Similarly the link of the chemical potential (whatever that means) value to the filling.
3/ The next sentence "Convergence to the grand canonical ensemble with zero chemical potential is guaranteed if the state phi is sharply peaked around the half-filled sector." is similarly problematic. If fermions considered are massive then their state psi should correspond to fixed (precisely) particle number and not be "sharply peaked". This is superselection principle.

Improvement of the analysis presented in Section 3. Clear statement of results. Further changes as apparent from the report.

Please read and correct occasional slips. E.g. p.18 "For late times t_1>t*(L_B), some time t*(L_B), the value of ..." t*(L_B) seems nothing else than the Heisenberg time. Please discuss and explain. In p.17 you write "The situation is different for free (integrable) DQD case, where oscillations persist and thermalization occurs." while in p.19 (bottom) "the non-interacting DQD does not thermalize in the IQS sense." As I understand in p.17 it was also IQS approach so these two sentences contradict each other.

Several short forms are used DQD, OQS, IQS,SFF,MGS,ETH,WD. While some of them are well accepted, I would suggest to change OQS thermalization to OST
(remove quantum since anyway everything is quantum in this paper) and similarly IQS to IST. This will help in avoiding statements as (l.6. p7) "analytical proofs that OQS occurs are lacking". I believe the authors meant OQS thermalization as OQS, I believe, exist in Nature without any proof. CPTP is just used once.

Ask for major revision

The authors address differences between open system vs. closed system perspectives on thermalization. The work is primarily of numerical nature. These numerical investigations are performed for a quantum dot coupled to a lead, the former taking the role of the system the latter that of the bath. It may also be described as a 1-d lattice system of spinless fermions, in which only fermions on the first two sites may interact. If this interaction is tuned to zero, the whole system is integrable. First some theory on open systems and eigenstate thermalization is summarized. Then four results are presented:

1. There is a first temporal regime which is more or less proportional to the length of the bath in which the open system approach is applicable.

2. In this first temporal regime the dynamics of the quantum dot are independent of the size of the bath. If the bath is large enough the mean force Gibbs state is reached. This holds regardless of the system being integrable or nonintegrable.

3. According to the Brody parameter the considered system is never fully chaotic regardless of the interaction strength.

4. In a second temporal regime (basically at all times after the first one) a behavior which the authors associate with isolated system thermalization is found for the interacting case but not for the noninteracting case.

This work is (as the title says) a case study. As such it does not focus on new concepts but brings different concepts together in one example and focuses in the interplay of these concepts. In principle this may warrant publication, however the following points should be satisfactorily addressed prior to possible publication:

1. The title is misleading. The work only focuses on systems that allow for a natural system-bath partition. However, thermalization in isolated quantum systems may also (possibly primarily) apply to systems without any bath. In this case, obviously, open system thermalization simply does not exist/apply Consider e.g. the spreading of heat in a solid. This important difference is not reflected in the title or anywhere else in the paper. Thus the title makes the paper appear more encompassing than it really is.

2. Often the paper emphasizes a contradiction in isolated system equilibration between systems being integrable and nevertheless being equilibrating. This contradiction is over-emphasized. The authors themselves list some counterexamples. However, the discussion of this (at the very end of the paper) is a bit odd: The statement that a form of isolated system equilibration in free fermion systems (as observed in another paper) should be considered as open system equilibration is unjustified, since the free fermion systems do not even have a bath. Attacking the statement "only nonintegrable systems thermalize" is not reasonable, since this is just a flawed oversimplified version of the original eigenstate thermalization as suggested by Srednicki / Deutsch

3. The first result (see 1. above) is not really a result. It is a simple consequence of Lieb-Robinson bounds. It has been already discussed in the same context elsewhere (e.g. papers by Takahiro Sagawa et al.)

4. The third result (see 3. above) may not be too surprising. It is clear that both limits V/g -> 0 and V/g -> \infty are integrable. A system as small as chain length 18 may not reach the fully chaotic regime in between.

5. Specifically interesting is the transition between the two temporal regimes. From the data shown, it appears as if the fluctuations at the (beginning of the) second regime are substantially larger than those at the end of the first regime. To display this clearer it would be helpful to show a plot that focuses on this transition. More concretely, plots that show the transition from the behavior shown in Fig. 2a to the behavior shown in Fig. 5a and the same for Fig. 2d to 5b, would be very valuable.

Ask for major revision