Tuning the phase diagram of a Rosenzweig-Porter model with fractal disorder

Rosenzweig-Porter (RP) model has garnered much attention in the last decade, as it is a simple analytically tractable model showing both ergodic--nonergodic extended and Anderson localization transitions. Thus, it is a good toy model to understand the Hilbert-space structure of many body localization phenomenon. In our study, we present analytical evidence, supported by exact numerical computations, that demonstrates the controllable tuning of the phase diagram in the RP model by employing on-site potentials with a non-trivial fractal dimension instead of the conventional random disorder. We demonstrate that doing so extends the fractal phase and creates unusual dependence of fractal dimensions of the eigenfunctions. Furthermore, we study the fate of level statistics in such a system and analyze the return probability of a wave packet localized at a single site to provide a dynamical test-bed for our theory.

The RP model is given by the Gaussian random-matrix ensemble of size L, where each element is random number obtained from a normal distribution, and the off-diagonal elements are rescaled by a factor of L −γ/2 , where h n = M mn = 0, h 2 n = M 2 mn = 1.It has been shown that [29,30], with increasing γ from 0 to large values, this model, first, exhibits ergodicity, and then undergoes a transition to nonergodic extended (fractal) phase at γ = 1.In the fractal phase the eigenfunction support sets contain extensive number, but measure zero of all the lattice sites, scaling as L D , where D = 2 − γ denotes the second fractal dimension of the eigenfunction (the precise definition of D is provided in the next section).As immediately apparent, D = 1 corresponds to the ergodic phase.Furthermore, at γ = 2, D goes linearly to 0, marking the onset of the Anderson localized phase.Although, this version of the model lacks the genuine multifractality in its eigenfunctions, recent developments [32,[36][37][38][39][40] show that some modified versions of this model may exhibit multifractality.Further studies [41] demonstrated the instability of the nonergodic extended phase in non-Hermitian version of the model.Unlike the latter case, in this work we show how to extend the range and stability of the fractal phase of the RP model by employing a 'fractal' on-site disorder.We also include the possibility of obtaining the nonlinear dependence of fractal dimension on the parameter, γ.
Summary of Results: In our study, we selectively employ a random normal distribution solely for the offdiagonal elements of H, while the diagonal elements (H mm ) are sourced from a 'fractal' disorder distribution with a Hausdorff dimension of d.This implies that there are typically L 1−d×b random diagonal elements, present in an energy window of width L −b , if the total bandwidth is taken to be O(1).One of the well-known examples of such a distribution is the Cantor set with a Hausdorff dimension [42] The main result of our work is shown in Fig. 1, where we represent the extended phase diagram of the RP model in terms of the second fractal dimension obtained from numerical fits of the generalized IPR, for q = 2; where |i⟩ denote computational basis states and |χ j ⟩ denotes an eigenvector with index j.Then IP R (2) ∼ L −D2 , where D 2 is the second fractal dimension, one obtains D 2 from averages over numerical fits from a band of eigenvectors.D 2 can then be used to distinguish between the ergodic (= 1), nonergodic extended, i.e. fractal (0 < D 2 < 1) and Anderson localized phase (= 0).We refer to D 2 as D since the higher moments show the same value i.e.D q = D 2 , q > 1/2, thus indicating that the eigenfunctions are fractal and not multifractal.In Fig. 1, we plot the γ-and d-dependence of D.
We note the following from the plot, 1. d = 1 represents the case for the generic RP model and the ergodic-fractal transition occurs at γ = γ ET = 1 and the Anderson transition occurs at γ = γ AT = 2 replicating the known results [29].
2. As the Hausdorff dimension of the diagonal elements decreases, γ ET is intact, but γ AT monotonically increases, extending the fractal phase in γ.
3. Both the transitions can be very well approximated by perturbative analytical expressions, which becomes exact in the thermodynamic limit, denoted by black dashed lines in the plot.
It is also worth mentioning that, for d > 1, the phase diagram shows similar behaviour as for d = 1.This is because beyond the physical dimension of the diagonal disorder (1D in our Hermitian case), any increase in fractal dimension cannot have an effect [43].
In what follows, we, first, analytically calculate the fractal dimension for eigenfunctions of the fractal RP model with changing γ and d.Then we compare the obtained expressions with exact numerics performed for (i) the commonly studied Cantor set fractal distribution, and then (ii) for a distribution with arbitrary Hausdorff dimensions d, suggested in [42].For completeness, we also discuss the level spacing statistics in such a model, and in the supplementary material [44] discuss the timedependent survival probability of a wave packet, initially localized at a single site.
Analytical phase-diagram calculations: As mentioned before, we consider the h n s to be distributed in a fractal (and later multifractal) manner [45].This implies that the h n 's are distributed such that, the number (#) of h n 's in a given energy inter- with a certain f (b)≤ 1, characterizing the above fractal.We also assume that the overall bandwidth of the h n is ∼ O(1) = L 0 .Thus, f (0) = 0.For any generic fractal with the Hausdorff dimension d we will have, For the special case of the Cantor set, d = ln 2/ ln 3. Note that, in general, f (b) can depend on E, but for the case of the Cantor set E-dependence arises only in 1/ ln L corrections to f (b) beyond the saddle-point expression (3).In contrast, in the case of uniform disorder distribution, the number of h n 's is proportional to the width of the energy interval i.e., f (b) = b.Thus the usual Hermitian case [29] corresponds to d = 1, while the non-Hermitian complex one [41] gives d = 2.The above saddle-point consideration in Eq. ( 3) is valid as soon as the number L 1−f (b) ≫ 1 is large.As we will see below, this corresponds to delocalized phases, where all the energy intervals are much larger than the typical level spacing, δ typ , i.e. the energy interval where one typically finds a single energy level.Indeed, the typical level spacing of the disorder, δ typ is given by, In this work, we focus only on real entries and, thus, work in the scenario 0 < d < 1.The generalization to the non-Hermitian matrices to cover 0 ≤ d ≤ 2 is straightforward.In what follows, we provide a short description of computation of the fractal dimension of a typical eigenstate of this model and thus compute γ AT and γ ET .
Using the standard cavity Green's function method, we can find a self-consistency equation for the level broadening (the imaginary part of the self energy) Γ m as (see [44] and [30,32,33]), where E is the eigenenergy of the corresponding eigenvector and η is a small regularizer.The parameterization Γ = L −a in the limit η → 0 gives the following result from Eq. ( 6) within the saddle-point approximation, (see [44]) This determines Γ ∼ L −a via the parameter γ and works for Γ ≫ δ typ .The corresponding fractal dimension D q ≡ D is determined via the number of levels located in the interval Γ ∼ L −a .This number is related to the fractal dimension as L D .From Eq. ( 3), we know that this is given by L 1−f (a) .Thus, This definition of D is the fractality in the "space" of h n , but for the RP-like fractal phases it is equal to the spatial fractal dimension due to the Lorenzian structure of the eigenstates [32,33,41,46,47]: As by fixing either E or h n , one has the Lorenzian, the fractality over the energy E and over the "space" h n is equivalent to each other.In space n, the above Lorenzian forms a fractal miniband [42] of the width Γ, with the underlying fractal structure h n , living in that miniband, In the fractal case of Eq. ( 4), we obtain for γ > 1 using Eqs.( 7) -( 8), The Anderson transition point corresponds to Γ ≃ δ typ , i.e., D = 0, since in the localized phase the number of energy levels within the Lorenzian bandwidth becomes an intensive quantity.Hence, a = b typ = 1/d and The ergodic transition occurs at Γ ∼ O(1), γ ET = 1.Note that both γ ET and γ AT are continuous transitions, unlike the fat-tailed distributed RP models [36][37][38].
Cantor Set diagonal elements: The first example we consider is when the diagonal elements are represented by Cantor set, C. Cantor set is a set of points lying in a line segment normalized to the interval [0, 1], obtained by removing the middle third of the continuous line segments in a recursive manner.The set generated by the first few iterations of this are, . . .We generate the diagonal elements by choosing the boundary value of each subset at the n = log 2 L iteration.The self similar nature of the Cantor set is evident from the construction and the Hausdorff dimension is calculated to be d = ln 2 ln 3 [48].In Fig. 2(a) we plot the second fractal dimension D 2 = D calculated from the numerical fitting in system size 2 p , p = 7 . . .12, for all the eigenvectors arranged in increasing order of IPR.In Fig. 2(b) we plot the same quantity, but averaged over 60 mid spectrum states.From Fig. 2(a) it can be clearly seen that there is no mobility edge in the spectrum, all the eigenstates show similar fractal dimensions D, hence one can average over them, which is plotted in Fig. 2(b).The point where the system ceases to be ergodic is clearly visible at γ = γ ET = 1.Furthermore the variation of the fractal dimension of the eigenfunctions D matches sufficiently well with the analytically obtained black dashed line, Eq. ( 10), in the γ ET < γ < γ AT regime, thus accurately predicting the γ AT point as well.Finite-size effects, given by 1/ ln L terms in D(L), are maximal close to γ AT and of the magnitude ∼ 0.1.
Generic fractal diagonal elements: Next, we consider the case of generic fractal diagonal elements.The generation of diagonal elements distributed in a generic fractal dimension was introduced recently in Ref. 42, this section also serves as a demonstration of applicability of the technique.Below, we give a short summary of the method.
A random fractal spectrum of Hausdorff dimension d can be generated using i.i.d.non-negative level spacings of ordered which are distributed as a Pareto distribution [49] where δ typ ∼ L −1/d , is the typical level spacing of the model and we omit the subscript n for brevity.Indeed, one can count that for the usual Cantor set with d = ln 2/ ln 3, at n th step one keeps L • P (s) ∼ 2 n levels with the spacings s ∼ 3 −n , leading to the above expression.Due to the formal divergence of the mean level spacing for all d < 1 at large s, for any finite L one should put an upper cutoff s max O(1), given by the entire bandwidth: and consider a typical realization where there is the only s n ≃ s max ≃ O(1), determining the bandwidth.In Fig. 3(a) and (b) we demonstrate how our theoretical predictions of D match with numerical results for d = 0.6 and d = 0.8.We see that even for generic dimensions our analytical predictions match very well with numerics.
Multifractal disorder: As a final example, we consider the more general case of multifractal disorder.Unlike the fractal case, where the scaling behaviour of all the moments of the distribution are the same, in a multifractal they are a nontrivial function of the moment order.Thus, one needs to define the probability distribution of level spacings in an energy window appropriately scaling with system size.In this case the probability distribution of level spacings is given by [44], where g(ν) is a non-linear function of ν.As an example we consider a particular case of the log-normal distribution where, g(ν) = 1 − (ν−ν0) 2 4(ν0−1) .Then we can compute the fractal dimension D (see [44]) as, where the Anderson transition happens at D = 0, i.e., at b = ν 0 and γ = 2ν 0 .The above formula works for 1 < ν 0 < 2. Note that unlike the fractal case, here there are 4 regimes: (i) ergodic phase, Γ ≫ O(1); (ii) usual fractal case, δ ≪ Γ ≪ O(1); (ii) new fractal case, δ typ ≪ Γ ≪ δ; (iv) localized phase, Γ ≪ δ typ .Here the unusual fractal phase (iii) appears only when the mean level spacing δ converges and differs from the typical one, δ typ ≪ δ ≪ O(1).
The results are plotted in Fig. 4 where the predicted fractal dimensions from our saddle point approximation match well with exact numerical results.We clearly see a curvature in D vs γ, a feature absent in the fractal case, which increases with increasing ν 0 .However it seems that finite-size effects are stronger in this case than for fractal disorder, due to the logarithmic dependence of the prefactors in Eq. ( 16), see similar effects in [36,37,39].Indeed, here finite-size effects to D(L) are given by ln ln L/ ln L, that for available system sizes give 2.5 times larger deviations.
Level statistics: Until now, our focus has been exclusively on the properties of the eigenfunctions.To provide a complete analysis, we shall now study the behaviour of a signature of the phase transition in the energy levels, the consecutive level spacing ratio r, defined  16), with ν0 = 1.01 (blue), ν0 = 1.5 (red), ν0 = 1.8 (green).The dotted lines are the analytical predictions of D(γ) from Eq. ( 17).by, where δ n = E n − E n+1 , E n is the n th eigenvalue when they are sorted in the increasing order.In the ergodic phase, it is well known that for the Gaussian Orthogonal Ensemble (GOE) ⟨r⟩ ∼ 0.53, [50,51], where p(r) = In Fig. 5 we plot the variation of ⟨r⟩ with γ for different d.
As expected from our analysis for γ < γ ET = 1, it admits value close to 0.  14), the finite size effects are stronger.
According to Eq. ( 9), γ-dependence of ⟨r⟩ goes to a kink at γ = γ AT in the thermodynamic limit.Interesting to note here that the fractal value r = d/(d + 1) covers the range from 0 (well below Poisson value at d = 0) to 0.5 (rather close to GOE one at d → 1).This means that if in some other models the fractal spectrum emerges, it can be mistakenly associated with the Poisson, Wigner-Dyson or any other statistics, based solely on r-statistics.Another interesting aspect is another 'kink', observed in the plots for d ≳ 0.9.While the first kink is due to breakdown of level repulsion, the second kink occurs due to the fat tail of P (r) in the localized phase for d ∼ 1.When the weight of large r values for non-hybridized eigenstates deep inside the localized phase become significantly larger than what it was in the ergodic or fractal phase, it shows up as a slight increase in ⟨r⟩.(Also see [44]) Discussion: In this work, we have demonstrated that making the distribution of the diagonal elements to be fractal in the RP model allows one to adjust the phase diagram and change the location of the Anderson localization transition γ AT .We have derived an analytical expression Eq. ( 10) that relates the Hausdorff dimension of the disorder to the fractal dimension of the eigenstates in the RP Hamiltonian, and have confirmed our findings through exact numerical computations.Furthermore, we have shown that one can manipulate the disorder dependence of the fractal dimension by utilizing a multifractal disorder.Finally, we have evaluated the implications of our modification on the eigenspectrum through level spacing ratio.This work gives the first step in the direction of usage of the fractal disorder for the controllable tunability of the phase diagrams of various disordered models.
In particular, this work opens the way to study, whether such fractal diagonal disorder enhances fractality of wave functions in other long-range models, such as the power law banded models [52], Burin-Maksimov model [53][54][55][56], some Bethe-ansatz integrable ones [57][58][59], on the random graphs [60,61], or even in the interacting disordered models [4].In all these cases (especially in the latter two), the fractal disorder may open a room for non-ergodic spatially extended phase of matter, intensively discussed and highly relevant for quantum algorithms [62] and machine learning [63].The analysis of spectral statistics for d ∼ 1 using spectral form factor can also show interesting behaviour at different timescales near γ AT , which can help to identify more clearly the origin of the sudden dip in the ⟨r⟩ statistics and shed light on spectral distribution in the critical (fractal) regime of ergodic-localized phase transitions and structure of fractal minibands [42].
where we introduced the notations ν q for the moments to which the main contribution is given by s q ∼ N −νq .δ typ ∼ N −ν0 is the typical (most probable) value of s, while the mean-level spacing, given by the first moment (if it converges) 3) In the last equality we assume a bandwidth E BW ≡ N δ ∼ N 0 to be finite.
Note that for a smooth function g(ν) the Legendre transform in Eq. (B.1) gives the condition while the prefactor in Eq. (B.1) within a saddle-point approximation is given by where the smoothness of g(ν) at ν = ν 0 guarantees finiteness of the second derivative.a. Correspondence between fractality of spectrum f (b) and level spacing distribution g(ν).In order to understand how the levels ζ n are distributed according to Eq. ( 3), one should take the extensive number of them m ≡ N 1−f (b) and calculate the distance between which gives an estimate of the number of levels in between.Please note that here, unlike Eq. (3), our control parameter will be the scaling of m, but not N −b .In order to find the correspondence between f (b) and g(ν) we will use the method, developed in [7] to find a distribution of the extensive sums of multifractal i.i.d.random numbers.Indeed, for any f (b) (or m) there can be two types of main contributions to Eq. (B.6): the individual one and the collective one.In order to understand it, we will separate the sum of m elements into the "bins", close to a certain ν, s ∈ [N −ν−dν , N −ν ).For each of these bins, the number of elements in the corresponding part of the sum is given by Within the saddle-point approximation, the parameter b is determined by the maximum of the collective and individual contributions.The individual contribution is given by a maximal s k ∼ N −ν * which appears in the above sum Eq. (B.6) at least once, M ν * ∼ N 0 .Thus, ν * is given by the smaller solution of the equation Thus, the amplitude of the sum is at least given by N −b ≳ N −ν * .The saturation of the latter inequality, b ≤ ν * , gives f (b) = g(b).
Summarizing both cases, one obtains This result is quite straightforward as we know from the properties of f (b) that its derivative cannot be larger than 1.The latter case corresponds to d = 1 in the fractal regime Eq. ( 4).Hence, correspondence between f (b) and g(ν) can be found to be b. Log normal distribution as an example Let's consider a log-normal distribution of s, then P (ν) should be Gaussian, i.e., g(ν) is a parabola where we have parameterized it with the location of the maximum (typical s typ ∼ N −ν0 ) and used the normalization condition g(ν 0 ) = 1 from Eq. (B.1).For this choice the prefactor in Eq. ( 16) is exact.Taking into account also Eqs. (B.3) and (B.4), one obtains (B.13) Next, using Eqs.( 7) and ( 8 16) leading to Eq. ( 17) in the main text.

C. FURTHER DATA ABOUT r STATISTICS
In Fig. 1 we show the histogram of probability density function of r defined in Eq. (18) for different γ to shed some light on the source of the second kink.From Fig. 5, we see that the second kink occurs inside the localized phase (for d = 0.9 at γ > γ AT ∼ 2.2).Looking at Fig. 1 we can verify that indeed already at γ = 2.4, the level repulsion is sufficiently weak.However the distribution undergoes further changes as we increase γ not only in the regime of r ∼ 0, where we have increasing weight, but also in the region r ∼ 1 due our choice of diagonal disorder.Deep inside the localized phase the energy levels are completely non-hybridized and the eigenenergies would be given by unperturbed diagonal elements.Since P (r) ∼ 1/r 1−d , for d ∼ 1 this indicates large weight at r ∼ 1 which is seen for the γ = 3.2 line in the plot, which has a larger weight in that regime than γ ∼ 2.4, 2.8.Hence the mean, ⟨r⟩ shows a rise at large values of γ for d ∼ 1.

D. SURVIVAL PROBABILITY:
We conclude our analysis by studying the dynamical behaviour of this model.Specifically, we consider the time dependence of the survival (or return) probability, R(t) [8], defined by, Like in the standard RP-model [8], in the NEE phase R(t) ∼ e −Γt , where Γ = L − γ−1 2−d , Eq. (10).In Fig. 2 we see that the rescaling of time by the analytical expression Γ collapses the curves for different L up to small values of R(t) (clearly there is no collapse if the time is not rescaled).Discrepancy at later times might be given by non-self-averaging nature of the fractal spectrum.The exponential behavior of the return probability is given by the Lorenzian wave-function profile, Eq. ( 9), in the energy domain.Similarly to the standard RP case [8], one can calculate the return probability from its Fourier transform K(ω), which also takes the Lorenzian form.Here Γ plays the role of the energy/frequency scale, below which levels are Wigner-Dyson correlated, while otherwise their statistics is given by the one of the diagonal elements (fractal in this case).

RFIG. 2 .
FIG. 2. Survival probability of a single-site initial wavepacket in the NEE phase for γ = 2 and fractal disorder with (a) d = 0.6 and (b) d = 0.8 vs time t rescaled by Γ.The raw data with no rescaling of t is plotted in (c) and (d) showing clear separation between different L.
regime, and the span in γ where such values are observed increases with smaller d, consistent with our previous results.As the smaller d-values corresponds to the fatter distribution tail Eq. ( 53, while at large γ > γ AT = 2/d, it settles at ∼ d d+1 , Eq. (19).It admits intermediate values in the fractal