Emergent fractal phase in energy stratified random models

1. Useful technical results in a new family of models, interpolating between well known canonical models.
2. New analytic technique presented.
3. Writing is well paced, and with an explanatory style.

1. Results are not clearly organised/communicated, in particular no intuitive understanding of the results is given.

In this paper the authors define a family of models which interpolate between Richardson’s Model (RM), and a translationally invariant form of the Rosenzweig-Porter model, (dubbed the TIRP).

They present analytic calculations of the fractal dimension using a technique which is developed in the manuscript, and provide some numerical evidence that shows agreement with these calculations. In particular authors explore how the principle of cooperative shielding, which is known to dictate the behaviour of the RM, breaks down as one tunes towards the TIRP.

The manuscript is well paced, and adopts a useful pedagogic style. However, despite this, it is unnecessarily difficult to follow, primarily due to (i) the lack of explicit definitions of the quantities of interest, and (ii) the lack of explanation of the physics underlying results.

However, my concerns about the manuscript are easily addressed, and overall I feel the results represent a useful addition to the literature. As a result recommend the manuscript for publication with corrections.

Major recommended change:

Organisation of results: My main concern with the paper is that the main results can be understood with simple physical intuitive arguments which are not clearly presented. I recommend that such short arguments should be included early in the paper, before delving into technical analysis, as they help guide the reader, and make the results accessible. In particular to readers who may be less interested in the details of the technical machinery which the authors develop.

I am sure the authors are aware of these simple arguments, but in the current version of the manuscript they are not highlighted in an accessible way.

Specifically, I refer to the fact the model has two regimes, which are distinguished by the hierarchy of the bandwidth of T ( = N^0) and the bandwidth of V ( = N^{1 - \gamma/2 - \beta / 2} ). In each of these regimes the fractal dimension can be found according to simple arguments (specifically: Energy stratification, and Fermi's golden rule).

1. \beta + \gamma < 2: In this regime the bandwidth of V >> the bandwidth of T, and we consider the action of T on the diagonal orbitals of V. Due to its much smaller bandwidth T is unable to mix all of the diagonal orbitals of V, it is however able to mix the entropically dominant fraction of the diagonal orbitals of V, that is the N – N^\beta levels at zero energy. Naively one might expect that these degenerate orbitals would be localised by the potential T. However, (as the authors point out at the bottom of page 13) it is not possible to define a localised basis over the N – N^\beta remaining levels, as the N^\beta energy stratified levels are delocalised. The best one can do is define a basis in which the diagonal orbitals of H are spread over M = N^\beta of the diagonal orbitals of T. i.e. the fractal dimension of the diagonal orbitals of H (defined by M ~ N^D) is given by D = \beta. This obtains the phase diagram (Fig 4) to the left of the diagonal line.

2. \beta + \gamma > 2: In this regime the fractal dimension may be calculated from Fermi’s golden rule (FGR). Specifically, as the bandwidth of V << bandwidth of T, and we may treat V as a perturbation which endows a finite lifetime to the diagonal orbitals of T. The FGR decay rate is the \Gamma = \rho_T |V_{ij} |^2 ~ N^{-1} |N^{-\gamma/2}|^2 = N^{1 - \gamma}. It follows that diagonal orbitals of T are superpositions of diagonal orbitals of H drawn from an energy window of width \Gamma ~ N^{1 - \gamma}, i.e. a number of orbitals M = \Gamma \rho_T = ~ N^{2 - \gamma}. By reciprocity it follows that the diagonal orbitals of H are superpositions of M ~ N^{2 - \gamma} diagonal orbitals of T, i.e. the diagonal orbitals have a fractal dimension D (defined by M ~ N^D) of D = 2 - \gamma. The only additional wrinkle is to note that M cannot exceed N or fall below 1, thus D = 1 if \gamma < 1 and D = 0 is \gamma > 2. This obtains the phase diagram (Fig 4) to the right of the diagonal line.

Minor changes:

1. The authors should explicitly define the fractal dimension at least once, preferably early in the paper. There are many ways to define fractal dimensions, and while one may be considered a standard default definition in the analysis of wavefunctions defined on a lattice, the manuscript should be explicit.

2. “However this guess is quite surprisingly fails …” -> “However, surprisingly, this guess fails …”

3. “given by a simply looking formula” -> “given by the simple formula”

4. Missing power of two in the denominator of the Lorentzian factor in Eq. 10

5. I did not understand the footnote 9 (bottom of page 13). The authors are correct in their primary point: that an orthonormalised basis of orbitals defined over the N - N^\beta non-stratified levels typically consists of orbitals spread over at-least M = O(\beta) sites. However it is not true that a basis of N orthogonal orbitals projected onto N - N^\beta non-stratified levels is spread in the same way (i.e. it seems the orthonormality is an important ingredient). In particular it is easily checked that the plane waves given in the footnote are spread over O(1) sites in the limit of large N, specifically they have participation ratio PR = 1 + O(N^{\beta-1}).

6. As a final comment, I am curious to ask, was it deliberate to use T to denote a potential in the localised basis, and V to denote a kinetic term, contravening the standard convention (T for kinetic energy, V for potential energy)?

1- The study of the properties is very detailed and the results look reliable.
2- The model is new and instructive.
3- The paper is very clearly written.

In this paper the authors study the effect of correlations in the off-diagonal terms on the spectral properties of long-range disordered random matrix models. In particular they consider a set of disordered random matrix models with translationally-invariant correlations in the hopping which interpolate continuously from the fully correlated Richardson's model to the Rosenzweig-Porter ensemble with translation-invariant correlations. Such set of models is constructed via an energy-stratified spectral structure of the hopping that allows one to gradually reduce the range of correlations.

In the completely correlated Richardson's limit there is measure zero of high-energy states that take the most spectral weight of the hopping and effectively screen the bulk states. Due to this effect, called cooperative shielding, the majority of states cannot be delocalized by the hopping terms, despite their long-range nature.

The main result of the paper is to convincingly show, both analytically and numerically, that (in contradiction with simple cooperative shielding arguments) as soon as the number of energy stratified states starts to grow with the system size, the bulk spectrum becomes delocalized but fractal.

Since all previously known analytical methods to tackle these kinds of problems are limited to the case of fully uncorrelated models, in order to calculate the fractal dimension of the bulk states the authors put forward a generalization of the cavity method and of the Dyson Brownian motion based on the spectral decomposition of the kinetic term and the Sherman-Morrison formula for the Green's function.

These methods yield the complete phase diagram of the model and demonstrate the existence of non-ergodic extended states as soon as one deviates from the completely correlated case. The authors also perform numerical simulations that confirm the validity of the analytical results.

The study of the properties is very detailed and the results look reliable. The model is new and interesting. Moreover the methods developed in the paper could be applied to a broader class of models. The paper is also very clearly written, very pedagogical, and pleasant to read.

Based on all this, I think that the paper is suitable for publication in SciPost.