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Dynamical phases in a ``multifractal'' RosenzweigPorter model
by I. M. Khaymovich and V. E. Kravtsov
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Submission summary
Authors (as registered SciPost users):  Ivan Khaymovich 
Submission information  

Preprint Link:  scipost_202106_00010v1 (pdf) 
Date accepted:  20210817 
Date submitted:  20210607 16:15 
Submitted by:  Khaymovich, Ivan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We consider the static and the dynamic phases in a RosenzweigPorter (RP) random matrix ensemble with a distribution of offdiagonal matrix elements of the form of the largedeviation ansatz. We present a general theory of survival probability in such a randommatrix model and show that the {\it averaged} survival probability may decay with time as a simple exponent, as a stretchexponent and as a powerlaw or slower. Correspondingly, we identify the exponential, the stretchexponential and the frozendynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretchexponent $\kappa$ in the thermodynamic limit. As another example we consider the logarithmicallynormal RP random matrix ensemble and find analytically its phase diagram and the exponent $\kappa$. Our theory allows to describe analytically the finitesize multifractality and to compute the critical length with the exponent $\nu_{MF}=1$ associated with it.
Published as SciPost Phys. 11, 045 (2021)
Reports on this Submission
Anonymous Report 2 on 2021726 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202106_00010v1, delivered 20210726, doi: 10.21468/SciPost.Report.3295
Strengths
This paper provides a rather detailed treatment of the localization properties of a class of random matrix models. More specifically, the models analysed are Rosenzweig Porter models in which the offdiagonal matrix elements have a broad distribution. The phase diagram for the models is established, both according to static criteria (are eigenstates localized, multifractal, weakly or fully ergodic?) and according to dynamic criteria (how does a wavepacket  initially concentrated on one basis state  spread in time?).
The strengths of the paper are:
1. Explicit and complete results for the RosenzweigPorter model with a lognormal distribution of offdiagonal matrix elements, including some finitesize effects.
2. Clear presentation of the background for the work, including an approximate mapping to the RP model from the problem of Anderson localization on random regular graphs.
3. Clear presentation of the essentials of calculations.
Weaknesses
The main weakness of the paper, as I see it, is that the problem treated is a very specialized one.
Report
While the problem treated is, as I have noted, a specialized one, the results are important in relation to an extensive body of past work. This work is concerned with viewing manybody localization in Fock space, making links to Anderson localization on the random regular graph, and with localization of the random regular graph. On those grounds, the paper can be argued to meet the acceptance criteria by "Opening a new pathway in an existing research direction ...".
In my opinion the general acceptance criteria are also met.
Anonymous Report 1 on 202174 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202106_00010v1, delivered 20210704, doi: 10.21468/SciPost.Report.3179
Report
The authors convincingly argue in the introduction that
the PorterRozenzweig type random matrices
with heavytail (''multifractal'' or ''large deviation type'')
distributed offdiagonal entries
may serve as a valid model for grasping essential features of
multifractal eigenstates in various regimes/phases
typical for MBL systems.
The most essential new result is a careful dynamical analysis
of survival probability showing that
in a broad region of parameters
the stretchexponential behavior of such object is generic,
implying subdiffusion. Moreover, the authors
show that there is a genuine (in the thermodynamic limit)
phase transition between phases with exponential and
stretchexponential decay. In fact dynamical approach
reveals new phases which can not be detected by simply looking at
eigenfunctions in static approach. Thus dynamical phases reflect
effects different from those detected by ergodicity violation criteria.
The analysis is based on carefully explored
WignerWeiskopff approximation which allows to take into account
the finitesize effects which turn out to be of crucial importance
for correctly interpreting the earlier numerical results and
resolving the longstanding controversies in a convincing way.
A crucial feature not much discussed before is the revealed existence of a tricritical point. This point seems to be very essential for reliable interpretation of numerics as in its vicinity a new, parametrically different correlation length arises not seen in previous analysis.
In summary, this is a high quality paper of broad interest, well written and nontrivially contributing to a topic of active research interest.
I suggest it is published after the authors consider the minor remarks below.
1) page 13, after eq.(30):
'and unity with a polynomial correction'
Sounds cryptic for me, please reformulate
2) Is (33) a definition of the exponent \Delta?
3) Eq.(34) seems to be the definition of \tau_*,
which should be clearly stated
(mentioned in words in fig 4
but better to repeat it in the text)
4) when discussing the analysis of the Fourier transform (52)(55) for this class of models I believe it is appropriate to mention the paper
Y.V. Fyodorov and A.D. Mirlin, Phys. Rev. B, vol. 55, R16001 (1997)
where this type of correlator was addressed
for sparse random matrix ensemble closely related to RRG.