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.