Information theory bounds on randomness-based phase transitions

1) Information-theoretic approach to Harris criterion
2) Succesfull application to a variety of one-particle Anderson models (known results)

1) Apparently it does not apply to many-body localization
2) They did not try to apply it to the Anderson model on the regular random graph which is a toy model for the MBL transition, if it exists

The paper revisits the well-known Harris bound using an information theoretic treatment.

The argument is not that different from the usual one: one must have a sufficiently large system/sample size to distinguish two probability distributions which are close to each other. The differentiation is interpreted in terms of "calls" of a classical computer which runs a dynamics. Although the dynamics is quantum, so one should really talk about a quantum computer.

Nonetheless, the authors discuss the Anderson model and the "Fock space localization," which is another name for the MBL phase and transition.
Here they find a discrepancy between their treatment and the numerics. This is also well known (at least since their ref. [87] was published). They suspect this is due to the fact that the MBL transition is not a vanilla second order phase transition. They might be right (this was also stated in many previous papers, see the discussion in [D]) but they fail to prove their statement.

One way to do it would be to study the one case in which the Anderson transition itself is not a simple second order phase transitions: this is the limit d\to\infty where it becomes Kosterlitz-Thouless like (see Refs. [A,C]). The authors do not seem to be aware of these papers which instead are central to understand the difference between localization in finite and infinite dimensions, while they cite other papers which have made similar claims but do not present a clear picture of the why a KT transition is present here.

If, instead of looking at the leading order term in S_q, like they do in their eq. (11), they looked at the derivative with respect to the logarithm of the system size, they would turn the problem into distinguishing different orbits of the fractal dimensions D_q which evolve as a function of \ln N. Some orbits end on the localized region, some will bounce back on the saddle point/critical point and flow to the ergodic fixed point. Look at [C] to see what I mean.

The question now is how to distinguish the finite-size flow. This is the problem which one needs to put an information theoretic/resource bound on.

A general comment on the references. The authors have been very generous with citations (a 7 pages manuscript with more than 120 references is hard to come by). They however managed to avoid mentioning all the works by Boris Altshuler and collaborators, including the paper which started the whole MBL field. I hope this was not intentional, and was just poor knowledge of the literature on the topic of MBL.

Based on the above facts, I cannot suggest the publication of the paper in the present form.

A considerably revised version of the paper can be reconsidered for publication.

Other notes:

Eq. (3) and (A6) make no sense. The authors might want to use the O(\delta) notation like in the rest of the paper. I think the discussion in Append A should really be, in a shorter version, in the main text.

Bibliography

[A] Universality in Anderson localization on random graphs with varying connectivity
P Sierant, M Lewenstein, A Scardicchio
SciPost Physics 15 (2), 045

[C] Renormalization group analysis of the Anderson model on random regular graphs
C Vanoni, BL Altshuler, VE Kravtsov, A Scardicchio
Proceedings of the National Academy of Sciences 121 (29), e2401955121

[D] Many-body localization in the age of classical computing
P Sierant, M Lewenstein, A Scardicchio, L Vidmar, J Zakrzewski
Reports on Progress in Physics 88 (2), 026502

Apply their reasoning beyond the known, finite-dimensional Anderson transition to obtain some new results on infinite-dimensional, Bethe lattice/Regular Random Graph or MBL.

Ask for major revision

Dear Editor,

The work by Noa Feldman and collaborators discusses information-theory-based bounds on critical exponents for disordered systems. Overall, the arguments presented are concise and clear: the method elegantly derives Harris bounds and generalizations to dynamical settings, supporting conjectured hyperscaling relations for the dynamical critical exponent.

This paper also raises significant challenges for the field of many-body localization (MBL). The authors highlight an inconsistency between their analytically motivated arguments and numerical finite-size scaling (FSS) results for Hilbert space delocalization, as measured by the participation entropy in the many-body computational basis. Indeed, the numerical results appear to violate the authors’ information-theoretic bounds, which could serve as a sine qua non criterion for the validity of FSS in ergodicity-breaking problems.

The authors also discuss the case of measurement-induced phase transitions (MIPTs) in random circuits, employing arguments based on entanglement Rényi entropies. I would encourage the authors to extend their analysis to the case of Hilbert space delocalization (see https://doi.org/10.1103/PhysRevLett.128.130605), where significantly larger system sizes are accessible for Clifford circuits. This setting, distinct from MBL, features a transition driven by the subleading coefficient $c_q$ rather than the (multi)fractal dimension $ D_q$. Including this analysis would further strengthen the authors’ results and align the presentation of their findings more cohesively.

A noteworthy figure of merit not addressed in the manuscript is that the argument extends to noisy (and open) dynamical systems (see https://doi.org/10.1103/PhysRevX.11.031066, https://doi.org/10.1103/PhysRevLett.132.140401, https://doi.org/10.1103/PRXQuantum.5.030327). In such cases, where $z \to \infty$ , the bound on $\nu$ remains valid.

Overall, with these minor inclusions, I strongly support the publication of this paper in SciPost Physics.

see report

Publish (surpasses expectations and criteria for this Journal; among top 10%)