Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree

1 - Interesting topic
2 - Combination of matching analytical and numerical results

1 - Hard to follow the logic of the text,
2 - One have to address Appendix during reading,
3 - Possible mapping to free-fermion models?
4 - Unclear finite-size effects

The authors address a random transverse field Ising model on a Bethe lattice and investigate the Anderson and ergodicity breaking phase transitions, using travelling/non-travelling phase transition as an analogue.

Concerning the manuscript I have several questions and comments:

1) On of the main concerns is related to the fact that Jordan-Wigner transformation for the random transverse field Ising model in 1d is known to be mapped to the free-fermion system with some induced superconductivity (a quadratic Hamiltonian, conserving only particle-number parity).
From the first glance, it seems that the same construction should be valid also for the Bethe lattice, as there are no loops on such graph and, thus, one can easily sort the sites in a certain 1d ordering.
As soon as one can map the system to the non-interacting fermionic one, it seems natural to expect that the model will be in the same universality class as the Anderson model on the Bethe lattice.
Please comment on whether it is the case and if so, what you predict about the critical exponents for the corresponding Anderson model on the Bethe lattice.

2) If the authors still claim that their model is interacting and cannot be mapped to any non-interacting model, then there appears another question, related to the boundary conditions (bare regularizer $B_0$).
A mentioned in [30] for the Rosenzweig-Porter model, by scaling $B_0 \sim N^{-\phi}$, with the number of sites on the graph, one can address not only Anderson localization transition, but also the ergodicity-breaking one.
Why does the same scaling procedure work for the interacting model in focus?

3) As a follow-up comment, I would like to emphasize that in [30] the scaling of the regularizer with $N$ was important in order to find the scaling of the mini-band width (Thouless energy), which in the non-ergodic extended phase scaled down with a certain $\phi>0$. At the same time it is known that in the Anderson model on Bethe lattice and/or on the random regular graph, the Thouless energy does not scale with $N$ and stays finite (but small with respect to the bandwidth) up to the Anderson transition.
Please comment on why do you need to take $B_0\sim N^{-\phi}$? Do you expect to have the corresponding Thouless energy scaling down with $N$?

4) If the authors claim that the $B_0\sim N^{-\phi}$ in the non-ergodic extended phase is related to the $N$-scaling of the Thouless energy, they should consider either the overlap correlation function $K(\omega)$ or the local density of states, showing the corresponding miniband structure, like in the Rosenzweig-Porter.

5) The random regular graph is known to have drastic finite-size effects, while in the current model (which seemed to be mapped to a very similar model) the authors seem to overcome this issue. Please comment on the finite-size effects in the model in focus and on the possible influence of them on the (numerical) results.
Especially this question should be asked to the claimed presence of the non-ergodic extended phase.
In order to clarify this, please show the drift of the extrapolated fractal dimension $D_1$ in Fig. 10, fitted from a sliding window over system sizes.

6) The Anderson model on the random regular graph is known to show the mobility edge behavior. Please comment on the presence of the mobility edge in the model in focus and, in the case of its presence, please clarify which averaging over the eigenenergies has been taken to calculate fractal dimensions and other measures of the transitions.
It might happen that the energy-resolved measures are needed for the case in focus.
This is especially important, taking into account spatial inhomogeneity of the model, discussed in Sec. 7, as it can imply some spectral inhomogeneities as well.

7) In Figure 3 deeply in the ordered (delocalized) phase, there is an apparent bimodal distribution $P(\ln B)$. Please comment on the origin of this bi-modality.

8) In addition to the previous questions and comments, I would like to draw the authors' attention to different values of the branching number $K$, especially to the small-world networks, considered for the case of the Anderson model on the random regular graph by some of the authors of this manuscript.
What do you expect to see for $1<K<2$ in the considered model? What are the peculiarities of this model?

To sum up, I find the topic of the manuscript rather interesting and valuable for the community, but the current version of the manuscript should be significantly improved in order to reach high standards of SciPost Physics.

Please answer all my questions and address all comments and suggestions. After that, if I am asked, I will provide my final opinion on the manuscript.

Main requested changes: please see the report above.

Minor changes:
a - Please clarify in the abstract what "constant and algebraically vanishing boundary conditions" mean: it is unclear, while reading for the first time.

b-References [30] and [55] are mostly devoted to the Rosenzweig-Porter model, but not to the Bethe lattice or random regular graph: I am not sure that [30] is correctly cited in several places, as well as [55].

c - The reference list on a non-ergodic delocalized phase are far from being complete:
the works on Gaussian Rosenzweig-Porter model contain not only [30] and [55], but also
- mathematical proof of it https://doi.org/10.1007/s11005-018-1131-7
- further investigations in statics
https://doi.org/10.1209/0295-5075/116/37002
https://doi.org/10.1103/PhysRevE.98.032139
and dynamics
https://doi.org/10.1088/1751-8121/aa77e1 - including subdiffusive behavior
https://doi.org/10.1209/0295-5075/117/30003
https://doi.org/10.21468/SciPostPhys.6.1.014

There are some (multifractal) generalizations of the Rosenzweig-Porter models with fat tailed distributions of off-diagonal elements:
starting from Levy-Rosenzweig-Porter:
https://doi.org/10.1088/1751-8121/aa77e1 - also mentioned above in the dynamics
https://doi.org/10.1103/PhysRevB.103.104205
-Log-normal Rosenzweig-Porter:
https://arxiv.org/abs/2002.02979
https://doi.org/10.1103/PhysRevResearch.2.043346 = [48]
https://doi.org/10.21468/SciPostPhys.11.2.045 - including the subdiffusive dynamics

Even in short-range Floquet-driven systems one can observe multifractal phases:
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.81.066212
https://doi.org/10.1103/PhysRevE.97.010101
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.184309
https://doi.org/10.21468/SciPostPhys.4.5.025
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.104504
https://doi.org/10.1103/PhysRevB.106.L020201
https://scipost.org/SciPostPhys.12.3.082

In addition, in the correlated setting of the on-site disorder with short-range hopping, there is a whole bunch of works on Aubry-Andre model with p-wave superconducting pairing, showing a fractal phase.
This wave has probably started with two works
http://dx.doi.org/10.1103/PhysRevLett.110.176403
http://dx.doi.org/10.1103/PhysRevLett.110.146404,
followed by the phase diagram calculation of the fractal phase in
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.104504
and now has quite a number of publications (please see the works citing the latter one).
Please consider to cite some of the representative papers in your work.

d - It is rather hard to go back and forth in reading the numerical part of the manuscript as it refers to the analytical part quite heavily. Please consider to re-arrange the manuscript in such a way to make it readable without massive back-and-forth scrolling.

e - The same is true about the location of the numerical figures: please place them in the corresponding places, where you discuss them, but not a couple of pages before.

f - The usage of the notion of the inverse participation ratio in (31) and Fig. 12 is very confusing as it is related usually to the fractal dimension $D_2$. Please call $I_2$ in (31) the second moment in order to avoid this confusion.

All the results are very interesting and explained in detail with many figures,
while the Appendix is very useful to make the traveling-wave analysis self-contained.

None

I strongly recommend the publication of this preprint in the present form.
Congratulations to the authors for their great work !

No requested changes