Many-body localization in presence of cavity mediated long-range interactions

- Timeliness and experimental relevance

- Technicality
- insufficient discussion of the continuous measurement of the scattered light

This paper presents a numerical study of Many-Body Localization (MBL) in an extended one dimensional Hubbard model. The main originality is the inclusion of infinite-range interactions, motivated by the experimental progresses with quantum gases in optical cavities, where such an interaction has been demonstrated.

The core of the paper is dedicated to the study of the consequences of this particular interaction on the MBL phase, showing that (i) it shifts the transition to higher disorder strength and (ii) the MBL phase still exists even for moderate strength for this interaction. The observables used are the long time dynamics of density correlations and the growth of entanglement entropy, which are the canonical tools for characterizing MBL.

The paper is timely, since it deals with one of the most active research topic in quantum gases and quantum physics in general. It proposes the combination of two experimentally accessible configurations, the production of disorder and controlled interactions, and the use of a high finesse cavity. While no experimental setup exists so far that combines all of these, it is reasonable to expect that a new generation of experiments could have the capabilities to realize the proposal, and the present paper may be motivation for such an experimental development. I appreciate the efforts done the authors to treat the cases closest to the experiments, in particular the quasi-random lattice case.

As an experimentalist, I cannot judge the validity of the numerical methods or approximations used, but I find the conclusions on the persistence of MBL as well as the extra delocalization induced by the infinite range interaction convincing (in particular in figure 6).

The aspect I found most interesting is the existence of a new observable, the cavity field leaking from the cavity, as a way to further characterize the MBL state and transition. I am however quite disappointed by the treatment of this part. The premise that the measurement is non-destructive is obviously wrong: the operator $I(t)$ does not commute with the Hamiltonian. Therefore, the measurement will induce a new dynamics, which the authors miss. A very important question arises: does the MBL phase indeed survives the continuous measurement process, or does the measurement brings in new states ? I understand that the theoretical treatment of such an open system would be much more involved and probably beyond the scope of the study, but without any discussion of this aspect, I am not convinced that indeed the cavity field leakage can indeed be interpreted as indicating anything regarding MBL.

I would recommend that the authors discuss this point, and leave out for a possible further study the exact interpretation of the cavity leakage. I also think that the material provided by the authors in the other sections is enough to support the main claims of the paper. Provided that the technical aspects of the paper are confirmed by a theoretician, I can propose publication in Scipost when the following changes are made.

1. figure 1: \kappa and \Omega_z are introduced but nowhere used in the text nor in the discussion. I have the feeling that the processes yielding the long range interaction (photon exchanges with the cavity field) could be better described, even though this has been covered by the literature. I suggest to add an appendix with a summary of the derivation.
2. Eq (6): it is not clear over which ensemble the minimum and maximum is supposed to be taken. Is it over a range of energies, over disorder realizations ? Please make this definition more explicit.
3. page 4, last paragraph: the calculation is done for exact commensurability, which is a very special case strongly favoring super radiance. Is there any reason to think that this is representative of the more general case ? What should one qualitatively expect in other situations ? This is of interest since any experimental realization will have inhomogeneities in the filling.
4. page 5: please define the acronyms PETSc/SLEPs or remove them from the text.
5. page 6: missing ’s’ in the sentence ‘’So far…. indicates…’’
6. page 6, last paragraph of section 3: this is quite confusing: which non ergodicity are the authors talking about ? I suggest to enhance this paragraph in order to include a short physical discussion of this other regime at large interactions, in particular to convince the reader that it cannot be favored by disorder and explain the observations.
7. page 7 last sentence of the first paragraph: typo ‘regime where’
8. page 6 last sentence of the page: I assume that the lattice is split in two subsystems separated in space. Considering the peculiar interaction pattern, I have a naive question: would it make sense to split the system in even/odd sites ? That would not have the interpretation of localization in space, but could help representing the memory of the initial state.
9. page 8, figure 4: the persistence of size effects in the entanglement entropy for U_1=1 while the density correlations do not depend on size could simply to suggest that there are other degrees of freedom (e.g. phases) which contribute to entanglement. Please comment.
10. figure 7: letters designating the sub panels should be mentioned in the caption.
11. page 13: the measurement is not ‘non destructive’. Please comment on this point (see above).
12. page13, Eq 14: this quantity is measured in principle for one given realization of the disorder (and of the experiment). Here, it is not clear whether the authors consider the average (and which average). Please comment if this is still to be used.
13. Figure 9: The superposition of both C(t) and I^2(t) on the same graph brings more confusion than clarity. Is the scale for the two the same ?
14. page 13-14: its seems that figure 9a shows that even in the ergodic phase, I^2(t) can reach a non zero steady state. I agree that this most likely means that the state stays close to the super radiant phase, but is casts doubts on the possibility to use I^2(t) as unambiguously indicating ergodicity. In general, I find the interpretation of these results confusing.
15. page14, last paragraph: an off-resonant laser with random intensity would not be quasi-random but truly random (as is typical for speckle patterns. see for exemple the work of the Aspect group).

1- Experimental relevance of the results.

I have two main concerns outlined in the report.

1) Is the saturation of $S_P$ sufficient to claim MBL, in particular since the $S_C$ is growing linearly at long-times (assuming the thermodynamic limit)? How are these contributions behaving in systems with long-range interactions? Can the authors observe a transition between the discussed phases and the phase with dominant long-range interactions?

2) In systems with long-range interactions, the convergence of results with system size is typically very slow. Can the effects of infinite-range interactions be distinguished from the ones obtained by finite range interactions over several sites for systems with 20 sites? Several other approaches can be used to calculate the time evolution for larger systems. For example, using the time-dependent variational principle, or by transforming the model into a short range model with a self-consistent population imbalance which enables the use of standard methods for simulations of 1D short-range systems.

Besides above it would be interesting to see if there is any qualitative difference considering U1=8 in figures 4 and 5 which is in the ergodic regime for the considered system size and disorder strength?

- Use a consistent reference to sub figures. Sometimes left/right is used and sometimes a),b).

1- the use of cavity QED to study MBL physics is very interesting, since the CQED could prepare fancy interaction form but not ground state, and the MBL physics explores properties not only about ground state. The detection method is very novel.
2- the phenomenological scaling effect produces very convincing results in Fig.3.

1- the effects of cavity is not only providing this long-range interaction. The quantum optical properties such as photon induced contrast lost are not discussed. These properties, however, should be a very significant limitation for optical-related detecting method as suggested by authors.
2- authors should cite more related articles such as those experiments using cavity-feedback interaction. Examples include but not limited to [1-2] for Boson and [3] for Fermion.

[1] M.H. Schleier-Smith, I.D. Leroux, and V. Vuletić, Phys. Rev. A 81, 021804(R) (2010)
[2] E. Davis, G. Bentsen, L. Homeier, T. Li, and M. Schleier-Smith. "Photon-mediated spin-exchange dynamics of spin-1 atoms," Phys. Rev. Lett. 122, 010405 (2019)
[3] Boris Braverman, et al. "Near-Unitary Spin Squeezing in 171Yb." arXiv preprint arXiv:1901.10499 (2019)

The proposal to use cavity feedback long-range interaction to study MBL physics under Hubbard Hamiltonian is both physically interesting and feasible. The long-range all-to-all interaction provides an experimental platform for unexplored MBL phases, and the detection method is also innovative. The paper is already in a good shape despite few places needed to be improved as I list in the requested changes.

The system contains a lot of complicated parameters, such as all-to-all energy scale, on-site energy scale, disorder scale, etc. In simplified system with random all-to-all and disorder, according to [1], sec III.C.1, an MBL phase is impossible in the infinite range of interactions setup. In this work, however, the authors observed this MBL behavior with strong long-range interaction, which is quite interesting. Though, it should be careful that whether this is MBL or MBL spin glass phase.

However, the explanation for eigenstate statistics is not enough to show the MBL, and the authors enhance the argument by providing dynamical investigations. The ergodicity breaking observed proves the MBL phase.

In general this is a good paper and I recommend for minor revision after answering and solving the weaknesses and requested changes.

[1] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, Maksym Serbyn, "Many-body localization, thermalization, and entanglement" arXiv preprint arXiv:1804.11065 (2018)

1- should cite more related experimental articles for this cavity QED system, especially those with optical lattice (say, arXiv:1901.10499).
2- should discuss about the mean gap ratio's fluctuation. In Fig.2, the range spans only from 0.40 to 0.52. The necessary uncertainty analysis is needed to determine whether this phase diagram has enough signal-to-noise ratio.
3- the scaling exponents, in the main text, is $\nu(U=1)=1, \nu(U=4)=1.5$. However in Fig.3, they're $1.3, 1.8$ respectively. I believe this is a typo that should be fixed.
4- discuss possibilities that this is a MBL spin glass phase. Not necessary but if possible, the authors should try to distinguish these by using the spin glass order as introduced in [1].

[1] Jonas A. Kjäll, Jens H. Bardarson, Frank Pollmann, "Many-body localization in a disordered quantum Ising chain" Phys. Rev. Lett. 113, 107204 (2014)