Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model

In this paper we suggest an extension of the Gaussian Rosenzweig-Porter (GRP) model, the LN-RP model, by adopting a logarithmically-normal distribution of off-diagonal matrix elements. We show that rare large matrix elements from the tail of this distribution give rise to a peculiar weakly-ergodic phase that replaces both the multifractal and the fully-ergodic phases present in GRP ensemble. A new phase is characterized by the broken basis-rotation symmetry which the fully-ergodic phase respects. Thus in addition to the localization and ergodic transitions in LN-RP model there exists also the transition between the two ergodic phases (FWE transition). We formulate the criteria of the localization, ergodic and FWE transitions and obtain the phase diagram of the model. We show that truncation of the log-normal tail shrinks the region of weakly-ergodic phase and restores the multifractal and the fully-ergodic phases. We also suggest a new criteria of stability of the non-ergodic phases and prove that the Anderson transition in LN-RP model is discontinuous, in contrast to its GRP counterpart.


Introduction
The structure of many body wave function is important for a variety of problems that range from many body localization (MBL) Basko et al. (1) to quantum computation. It was recently realized that in many of these problems the wave function is neither localized nor completely ergodic, i.e. spreads uniformly over whole Hilbert space, but has a dimension that is proportional to the dimension of the full space. These fractal wave functions were reported and intensively discussed in the physical problems of localization on random regular graphs (2)(3)(4)(5)(6)(7)(8)(9)(10)(11), the Josephson junction chains (12), the random energy model (13,14) and even in the Sachdev-Ye-Kitaev model of quantum gravity (15)(16)(17).
In quantum computation similar fractal wave functions appear in the search algorithms based on the efficient population transfer and it is believed that the appearance of the fractal dimensions is linked with quantum supremacy (18). Generally, a wave function of the fault-tolerant quantum computer is confined to the computational space which dimension is much smaller than that of the full Hilbert space, so the wave function of a fault tolerant-computer must be fractal. However, despite the apparent importance of this phenomena, its understanding and analytic description is still in its infancy.
Generally, one expects that fractal wave function might appear in the intermediate regime sandwitched between fully ergodic and fully localized states. However, the only solvable model that shows the appearance of such a regime is the simplistic Gaussian Rosenzweig-Potter (GRP) model (19)(20)(21)(22)(23)(24)(25). In this paper we introduce a natural generalization of this model and show that it displays a much richer phase diagram.
In GRP model every site of the Hilbert space is connected to every other site with the transition amplitude distributed according to the Gaussian law. Such model occurs as the effective decription of the systems without internal structure, in which transition between resonance sites is due to a small number of hops, such as random energy model (13,14). In more realistic models delocalization of the wave function is due to a long series of quantum transitions. Each transition has a random amplitude, so their product is characterized by the log-normal distribution, rather than the Gaussian one as in GRP model. Inspired by this argument in this paper we introduce and study the generalization of RP model in which the transition amplitude between sites has a small typical value, as in RP model, but with much wider, log-normal distribution function that we define in Section 2 (see upper-left panel of Fig.1).
It appears that the rare large hopping matrix elements from the tail of this distribution may significantly alter the phase diagram of the system by considerably shrinking the region of multifractal phase as the parameter p that controls the weight in the tail, increases. For large enough p the multifractal phase is totally replaced by an ergodic one (see upper-right panel of Fig. 1).
However, a careful look at the random matrix models with long-range hopping in the presence of correlations in the offdiagonal matrix elements has lead to a conclusion that the mere statement that the eigenfunction fractal dimensions Dq = 1 are all equal to one is not sufficient for complete characterization of the ergodic phase. As was shown in Ref. (26), in certain cases of matrix elements correlated along the (non-principal) diagonals, all fractal dimensions Dq = 1 in the reference basis, yet in the Fourier-transformed 'momentum' basis all eigenvectors are localized. Consequently, the eigenvalue statistics is Poisson, despite extended character of wave functions in the reference basis. This example represents the case of an ergodic (Dq = 1) phase in the reference basis which is severely non-invariant under the basis rotation. On the other hand, the ergodic phase in the GRP model with totally uncorrelated off-diagonal entries remains ergodic in any basis. This observation urged us to distinguish between the fully-ergodic (FE) phase where: (i) the fraction of populated sites in an eigenfunction is f = 1, (ii) the eigenvalue statistics is Wigner-Dyson (WD) and (iii) eigenfunction statistics is invariant under basis rotation, and the weakly-ergodic (WE) phase (27) where this invariance is broken together with the WD eigenvalue statistics, and f < 1. Furthermore, since the so defined two ergodic phases differ by the symmetry with respect to basis rotation, there should be a transition and not a crossover between them. We will refer to this transition between the fully-and weakly ergodic phases as FWE transition.
It is probably this WE phase which is responsible for a socalled "bad metal" phase on the ergodic side of the localization transition, where all the fractal dimensions are already ergodic, Dq = 1. In such a phase, both many-body systems (28) and hierarchical structures like RRG (8,9) have been shown to demonstrate the anomalous sub-diffusive transport.
Surprisingly, the ergodic phase in LN-RP model which emerges and proliferates as the weight p of the fat tail increases appears to be the weakly-ergodic one. We show in this paper that this phase is separated by a new FWE phase transition from the fully-ergodic phase existent at smaller disorder (see lower-left panel of Fig.1). The weakly-ergodic phase in many respects can be considered as the under-developed multifractal phase, as at large but finite system sizes it shows a quasimultifractal behavior. Furthermore, this WE phase is fragile. If the LN tail is cut off, the WE phase shrinks dramatically (see lower-right panel of Fig. 1), and the MF and FE phases are restored, with MF phase even extending its spread compared to that in GRP case at moderate values of the cut-off onset.
The analytical theory of the Ergodic (ET), Localization (AT), and FWE transitions developed in this paper is verified by extensive numerics based on the Kullback-Leibler divergence (29,30) of certain correlation functions KL1 and KL2 of wave function coefficients (31) and on numerical investiga-tion of the typical (ρtyp) and the mean (ρav) Local Density of States (LDoS). The quantity φ = 1 − ρtyp/ρav is an order parameter for the FWE transition, with φ = 0 in FE phase and φ > 0 in WE phase (see lower-left panel of Fig. 1), while the onset of divergence (with the system size N ) of KL1 and KL2 marks the AT and ET transitions, respectively.

Log-Normal Roseizweig-Porter model
We introduce a modification of the RP random matrix ensemble (19,20) in which the Gaussian distribution of independent, identically distributed (i..i.d.) off-diagonal real entries Hnm = U is replaced by the logarithmically-normal one: It is characterized by two parameters: the disorder-parameter γ which determines the scaling of the typical off-diagonal matrix element with the matrix size N and the parameter p that controls the weight of the tail. The i.i.d. diagonal entries are supposed to remain Gaussian distributed, as in the original RP model: This LN-RP model is principally different from the Lévy random matrix models (see, e.g., (32,33)

Criteria of Localization, Ergodic and FWE transitions for RP random matrices
In this section we consider simple criteria of localization, ergodic and FWE transitions for random N × N matrices with the long-range uncorrelated random hopping Hnm = 0 and diagonal disorder ∼ O(1). More general picture and examples of systems are presented in Refs. (26,37). The first criterion, which is referred to as the Anderson localization criterion, states that if the sum: converges in the limit N → ∞ then the states are Anderson localized. Here .. W stands for the disorder averaging, where the subscript W implies that the tails of the distribution of off-diagonal entries U should be cut off at |U | > W ∼ O(1) greater than the typical value of diagonal on-site energies (26,34).
The physical meaning of this criterion is that the number is sites in resonance with a given site n is finite. Indeed, consider for simplicity the box-shaped distribution F (ε) of on-site energies. The probability that two sites n and m are in resonance is: (Hnm). [5] Then integration over (εn + εm)/2 and integration by parts over ω = εn − εm gives: [6] where U = |Hnm|.
One can easily see that at Utyp ∼ N −γ/2 O(1) the last integral in Eq. (6) is always small and the second term in the first integral is at most 1/2 of the first term. Thus with the accuracy up to a constant of order O(1) the number of sites in resonance with the given site, m Pn→m, coincides with Eq. (4) up to a pre-factor of order O(1). We presented this standard derivation in order to show that the localization criterion involves only |U | < W , and thus imposes an extrinsic cut-off on the distribution P (U ).
The second criterion referred to as the the Mott's criterion is a sufficient criterion of ergodicity. It states that if the sum diverges in the limit N → ∞ then the system is in the one of the ergodic phases (26). Note that similar to Eq. (4), the averaging in Eq. (7) should be done with the distribution truncated at Umax ∼ W ∼ O(1), the on-site disorder spread. The reason for that is that rare large matrix elements |Hnm| O(1) split the resonance pair of levels so much that they are pushed at the Lifshitz tail of the spectrum and do not affect statistics of states in the body of spectrum that we are studying (38).
The physical meaning of Eq. (7) is that the Breit-Wigner width Γ ∼ √ S2 that quantifies the escape rate of a particle created at a given site n, is much larger than the spread of energy levels W ∼ O(1) due to disorder. In other words, the fulfillment of the Mott's criterion implies that the width Γ is of the same order as the total spectral bandwidth and thus there are no mini-bands (which width is Γ) in the local spectrum. As the presence of such mini-bands is suggested (12, 39, 40) as a "smoking gun" evidence of the non-ergodic extended (e.g. multifractal) phase, the fulfillment of the Mott's criterion Eq. (7) immediately implies that the system is in the ergodic extended phase.
Eq. (9) coincides with Eq. (7) for non-tailed distributions of U and involves the non-zero moments of Hnm of minimal order q → 0 and q = 2. The physical meaning of Eq. (9) is directly related with the rare large hopping matrix elements from the tail of the distribution. Indeed, according to Eq. (3) such rare events make S3 = N U 2 much larger than the typical value N U 2 typ and tend to violate the condition Eq. (9) under which the fully-ergodic phase may exist.

Phase diagram
For the log-normal distribution Eq. (1) one easily computes the moments |U | q W truncated at Umax ∼ O(1): and finds using Eq. (3), Eq. (4), Eq. (7), Eq. (9) and Utyp = N −γ/2 the following critical points of the localization (γAT ), ergodic (γET ) and FWE (γF W E ) transitions: γF W E = 1 1 + p . [13] The phase diagram for the log-normal Rosenzweig-Porter ensemble, Eq. (1), resulting from Eq. (11)-Eq. (13) is presented in the upper-right panel of Fig. 1. The main conclusion we may draw from this phase diagram is the emergence and proliferation of the weakly-ergodic phase that pushes away both the multifractal (MF) phase and the fully ergodic phase, as the strength of the tail p in the distribution Eq. (1) increases. For p > 1 the MF phase is completely gone replaced by the weakly ergodic one. However, WE phase is fragile. Truncation of the tail of this distribution such that |U | < N −γ tr , γtr > 0, eliminates the WE phase and restores the MF phase, as well as increases the range of the fully-ergodic one (see the lower-right panel of Fig. 1 and Appendix A for details).

Δ(α)/γ+α
Intervals of α = D1/γ with different functional dependence are shown by dashed vertical lines. The Anderson localization transition corresponds to the lower of the blue and orange curves equal to 2/γ at α = 0. This transition is always determined by the orange curve representing the log-normal part of the distribution P (V ). On the contrary, the stable fractal dimension D1(γ) = 2 − γ/γ ET (p) for γ ≤ γ AT is always determined by the blue curve representing the Gaussian part of the distribution P (V ). The Anderson transition in all cases but p = 0 is discontinuous, with the minimal stable fractal dimension of the support set being D min

Stability of non-ergodic states against hybridization
In this section we consider the stability of non-ergodic (multifractal and localized) states against hybridization. It allows us not only to derive expressions, Eq. (4) and Eq. (7), for the Anderson localization and ergodic transitions in a different way but also find the fractal dimension D1(p, γ) of the multifractal support set. Furthermore, the new method presented below is physically transparent and generic enough to be applied to analysis of the multifractal states in other systems.
Let us consider two states ψµ(i) and ψν (i) on different fractal support sets as it is shown in Fig. 2(a) and (b). We assume that both states are multifractal with m ∼ N D 1 sites on a fractal support set where |ψ(i) Here we apply a usual Mott's argument for hybridization of states when the disorder realization, in this case the offdiagonal matrix element, changes from Hij to H ij = Hij + δ Hij. The key new element in the theory we are introducing here is the hopping matrix element Vµ,ν between the states and not between the sites as is customary: [14] Here ψµ(i) is the eigenfunction of the µ-th state of Hij, and δ Hij = H ij − Hij, where H ij is drawn from the same lognormal distribution as Hij.
Introducing gij = − ln δ Hij/ ln N and suppressing the indices i, j for brevity we conveniently rewrite Eq. (1) as follows * : . [15] By the constraint g ≥ 0 we implemented the cutoff at |U | ∼ O(1) discussed in Sec.3, and ε is absorbed by the normalization constant.
The typical number of terms in the sum Eq. [16] If σ(g, D1) < 0, the sum, Eq. (14), is dominated by a single term with the largest |Gij|. For positive σ(g, D1) > 0, * Here we omit the small deviations from the log-normal distribution for g ij = − ln |H ij − H ij |/ ln N > γ/2 which are not important in the current setting. For the details please see Appendix C.
many terms contribute to this sum and the distribution P (V ≡ |Vµ,ν |) becomes Gaussian. In general, there are both contributions The condition of stability of the multifractal phase against hybridization is derived similar to the Anderson criteria of stability, Eq. (4), of the localized. The difference is that now we have to replace the matrix element between the resonant sites U by the matrix element V between the resonant non-ergodic states and take into account that on each of M = N 1−D 1 different support sets there are m = N D 1 wave functions which belong to the same mini-band and thus are already in resonance with each other. Therefore the total number of independent states-candidates for hybridization with a given state should be smaller than the total number of states M m = N and larger than the number of support sets M . This number is in fact equal to their geometric mean With this comment, the criterion of stability of the multifractal phase reads in the limit N → ∞ as The contribution of the Gaussian part P Gauss in Eq. (17) to Eq. (18) is: where and for stability it must be finite as N → ∞. The contribution of P LN in Eq. (17) to the stability criterion Eq. (18) is [21] Thus the multifractal phase is stable against hybridization if the following inequalities are both fulfilled D1 + ∆(D1) ≥ 2.
The functions γ eff (D1) and ∆(D1) are computed in Appendix C and discussed in the next Section. A particular case D1 = 0 of Eq. (22), Eq. (23) describes the stability criterion of the localized phase. If the localized phase is not stable, then hybridization produces an avalanche of multifractal states living on fractal support which dimensionality grows until inequalities Eq. (22), Eq. (23) are both fulfilled for the first time at some 0 < D min 1 < 1. If this is possible in some parameter region then the multifractal state is stable, otherwise the only stable extended phase is ergodic.

Fractal dimension of the NEE support set
In this section we re-consider the phase diagram Fig. 1 from the viewpoint of stability criteria given in the previous section by Eq. (22), Eq. (23) and derive the expression for the fractal dimension D1(γ) of the support set of multifractal wave functions.
To this end in Fig. 3 we plot and as functions of α = D1/γ. Here γAT (p) ≥ 2 and γET (p) ≥ 1 are given by Eq.  (25) should be compared to 2/γ, see Fig. 3. First, we note that the localized phase which formally corresponds to D1 = 0, is stable if the lowest of the blue and orange curves in Fig. 3 is higher than 2/γ at α = 0 and it is unstable otherwise. One can see that at α = 0 for all values of p the log-normal contribution to Eq. (17) (orange curve) is lower than the Gaussian one (blue curve). This means that the stability of the localized phase is always determined by the log-normal part of P (V ). Moreover, since at α = 0 Eq. (24), Eq. (25) reduce to α + γ eff (α)/γ = 1 and α + ∆(α)/γ = 2/γAT , respectively, the stability of the localized phase implies that γ > γAT (p) ≥ 2 in agreement with Eq. (11).
If the localized phase is unstable then different localized states hybridize and form a multifractal state with D1 > 0. Those states are, however, unstable until their support set reaches the fractal dimension D min 1 > 0 where Eq. (22), Eq. (23) are both fulfilled for the first time.
As the parameter γ decreases below the critical value γAT , the stable fractal dimension D1(γ) increases from D min 1 being always determined by the intersection of the horizontal line y = 2/γ > 2/γAT (p) (red line in Fig. 3) with the blue line. Thus the stable fractal dimension D1(γ) is always determined by the Gaussian part of P (V ) and according to the second line of Eq. (24) and Fig. 3 is equal to: At γ = γET the fractal dimension D1(γ) reaches unity, and at this point a continuous ergodic transition happens. Thus the critical point of ergodic transition coincides with that determined by Eq. (12). Note that, unlike the ergodic transition, the Anderson transition is discontinuous: the stable fractal dimension D1(γ) is separated by a finite gap D min 1 = D1(γAT ) from the localized state D1 = 0:  This gap is shown by the gray dotted arrow in Fig. 3. As we show in the next section this jump reveals itself in the slope of the Kullback-Leibler divergence, see Fig. 5 and Eq. (32). The right panel of Fig. 3 demonstrates that for p ≥ 1 the minimal fractal dimension D min 1 = 1, so that the multifractal phase is no longer possible in LN-RP model Eq. (1). However, it is restored if the LN distribution is truncated at |U | ∼ N −γ tr with γtr > 0 (see Appendix A for details).

Kullback-Leibler (KL) measure
The numerical verification of Eq. (11), Eq. (12) and determination of the critical exponents at the Anderson localization and ergodic transitions is done in this paper using the Kullback-Leibler divergence (KL) (29-31, 42) (for more detailed multifractal analysis of this model see (43)).
The Kullback-Leibler correlation functions KL1 and KL2 are defined as follows (31,42). The first one is defined in terms of wave functions of two neighboring in energy states ψµ(i) and ψµ+1(i) at the same disorder realization: The second one is similar but the states ψ andψ correspond to different (and totally uncorrelated) disorder realizations: The idea to define such two measures is the following. In the ergodic phases each of the states has an amplitude |ψ(i)| 2 ∼ N −1 of the same order of magnitude. Then the logarithm of their ratio is of order O(1), and for the normalized states For fully-ergodic states the eigenfunction coefficients are fully uncorrelated, even for the neighboring in energy states. Thus there is no difference between KL1 and KL2. Using the Porter-Thomas distribution one finds: For weakly-ergodic states KL2 is still O(1) but is larger than the Porter-Thomas value due to the fact that there are 'population holes' where N |ψ| 2 is N -independent but small. Deeply in the localized phase ln |ψµ(i)| 2 ∼ −|i − iµ|/ξ, where iµ is the position of the localization center. Since the positions of localization centers iµ are not correlated even for the states neighboring in the energy, the logarithm of the ratio of the two wave function coefficients in Eq. (28), Eq. (29) is divergent in the thermodynamic limit. For Anderson localized states on finite-dimensional lattices this divergence is linear in the system size L. However, localization on graphs such as RRG and RP models is not a conventional localization (2,20). In this case there is a power-law in 1/N background with most probable (typical) value of |ψ| 2 min ∼ N −α 0 far from the localization center and therefore: with α0 = (γAT /2)(γ − γAT ) + 2 for LN-RP model. A qualitative difference between KL1 and KL2 is in the multifractal phase. In this phase the neighboring in energy states |ψµ(i)| 2 and |ψµ+1(i)| 2 are most probably belonging to the same support set (44) and hence they are strongly overlapping: |ψµ(i)| 2 ∼ |ψµ+1(i)| 2 . Furthermore, eigenfunctions on the same fractal support set can be represented as: ψµ(i) = Ψ(i) φµ(i), where Ψ(i) is the multifractal envelope on the support set and φµ(i) is the fast oscillating function with the Porter-Thomas statistics (2). Thus the ratio |ψµ(i)|/|ψµ+1(i)| and hence KL1 in MF phase has the same statistics as in the ergodic one. We conclude that KL1 is not sensitive to the ergodic transition but is very sensitive to the localization one, Fig. 4.
In contrast, the eigenfunctions ψ(i) andψ(i) in KL2 corresponding to different realizations of a random Hamiltonian, overlap very poorly in MF phase. This is because the fractal support sets which contain a vanishing fraction of all the sites, do not typically overlap when taken at random. Therefore at the Anderson transition, γ = γAT , originates from the jump in D1, Eq. (27). Numerically it is clearly seen in the derivative of KL2 over ln N versus γ shown in Fig. 5. We also show in Fig. 6 that KL2 is sensitive to the FWE transition and can be operative in identifying it.
A more detailed theory of KL1 and KL2 in the multifractal phase is given in Appendix D. The main conclusion of this analysis is that the curves for KL1(γ, N ) for different N have an intersection point at the critical point γ = γAT of the Anderson localization transition. At the same time, the intersection point for curves for KL2(γ, N ) coincides with the ergodic transition (31), provided that it is continuous and well separated from the Anderson localization transition. If the localization and ergodic transition merge together and the multifractal state exists only at the transition point, then intersection of KL2 curves is smeared out and may disappear whatsoever (as in 3D Anderson model). However, the intersection of KL1 curves remains sharp in this case too (see Fig. 4).
The intersection of finite-size curves for KL1 and KL2 helps to locate numerically the critical points γAT and γET . More  precise determination of the critical points and the corresponding critical exponents ν1 and ν2 is done by the finite-size scaling (FSS) data collapse (see insets in Fig. 4 and Appendix E). The results are shown in the Table 1. On the basis of these numerical results we conclude that our expressions Eq. (11), Eq. (12) for the Anderson and ergodic transition points are accurate and conjecture on the p-dependence of the critical exponents ν1 and ν2 of AT and ET obtained from KL1 and KL2. (see right panel of Fig. 4).

Numerical location of the FWE transition
For numerical verification of Eq. (13) for FWE transition point we make use of the ratio of the typical ρtyp and mean ρav average ln ρtyp = ln ρ(x, E + iη) , ρav = ρ(x, E + iη) , [34] of local density of states (LDOS) ρ(x, E + iη) = Im µ |ψµ(x)| 2 /(E + iη − Eµ) . [35] As is shown in Ref. (3), at small bare level width η EBW /N , where EBW = max(Γ, W ) is the level bandwidth, this ratio ρtyp/ρav ∼ η N D 1 /EBW grows linearly with η but then saturates at ρtyp/ρav ∼ N −1+D 1 . In the ergodic phase D1 = 1 and the plateau in ρtyp/ρav tends to a finite limit as N → ∞. This behavior is well seen in the inset of Fig. 6. We used the properly defined † plateau value of φ = 1 − ρtyp/ρav as the order parameter for the FWE transition. For γ < γF W E this parameter φ = 0, signaling of the fully-ergodic phase. For γ > γF W E the order parameter is non-zero. This behavior is seen in Fig. 6, where the black curve represents φ = φ∞(γ) extrapolated to N = ∞ from the finite N values φN (γ) obtained by exact diagonalization (see also an inset in the lower-left panel of Fig. 1 for p = 1/2, where Eq. (13) predicts γF W E = 2/3). In spite of imperfect extrapolation that does not allow to get a true singularity at γ = γF W E , the dashed gray lines of continuation of the black curve intersect exactly at γ = 1/2 which is the predicted value of γF W E at † at the maximum of the second derivative of this ratio vs. η, see Appendix B for details   Fig. 6.They all suggest that the FWE transition does exist and is described by Eq. (13).

Conclusion
In this paper we introduce a log-normal Rosenzweig-Porter (LN-RP) random matrix ensemble characterized by a longtailed distribution of off-diagonal matrix elements with the variance controlled by the parameter p. We calculate analytically the phase diagram of LN-RP using the recently suggested Anderson localization and Mott ergodicity criteria for random matrices and complement it by the new criterion for the transition between the fully-and weakly-ergodic phases. We give arguments that the two ergodic phases are indeed connected by the new FWE transition and found an analytical expression for the FWE transition point. An alternative approach to localization and ergodic transitions based on the analysis of stability with respect to hybridization of multifractal wave functions developed in this paper gives results identical to those obtained from the above criteria and consistent with numerical calculations. Using this approach we computed analytically the dimension D1 of the eigenfunction fractal support set and show that the Anderson localization transition is discontinuous with D min 1 > 0 at all p > 0.
Our results show how the rare off-diagonal matrix elements which are much larger than the typical ones, change the character of the eigenfunction statistics giving rise to a new fragile ergodic phase and a new phase transition.

A. Truncated LN-RP and fragility of ergodic phase.
The phase diagram shown in Fig. 1 of the main text confirmed numerically by calculations of the KL-divergence and by the ratio of typical and mean local density of states (LDOS) demonstrates the collapse of the multifractal phase at p ≥ 1 and existence of the tricritical points in LN-RP model at p = 0 and p = 1.
In this section we show that the weakly-ergodic (WE) phase that emerges at p > 0 and replaces fully the multifractal (MF) phase and partly the fully-ergodic (FE) one at p ≥ 1 is unstable with respect to a deformation of LN-RP model such that P (U ) is cut from above at: O(1), (γtr > 0). [36] As the result of this truncation the multifractal phase re-appears by substituting a part of the ergodic phase in a non-truncated LN-RP model (see right bottom panel in Fig. 1) (48). To this end we use the expression that generalizes Eq. (10): and apply the same criteria Eq. (4), Eq. (7), Eq. (9) to find the critical points of the localization and both ergodic transitions.
Then we obtain that the critical point γ AT of the Anderson localization transition is affected as follows only if γtr > γ AT (1 − p), 0. In the opposite case truncation does not affect γ AT . For the critical point γ ET of the ergodic transition in the same way we find the effect only for γtr > γ ET (1 − 2p), 0 given by The criterion for the fully-weakly ergodic (FWE) transition does not have any truncation of U 2 at U ∼ W , thus it is affected by the truncation at all γtr > γ F W E (1 − 2p) (even negative ones if p > 1/2). As a result FWE transition occurs for γtr > γ F W E (1−2p) at [40] Note that Eq. (38) and Eq. (39) give real solutions for γtr < γ AT (0) = 2 and γtr < γ ET (0) = 1, respectively, and both these solutions increase with the tail weight p. At the same time FWE transition replaces ET one for all γtr > 1 as γ F W E (γtr = 1) = γ ET (γtr = 1) = 1 for all p. Similar thing happens for γtr > 2, when FWE transition replaces ALT as well, with γ AT (γtr = 2) = 2, but in this case γ F W E (γtr = 2) = 2 only for p → 0.
One can see that at any positive non-zero γtr the multifractal NEE phase emerges at p ≥ 1 in between of the localized and ergodic ones. Indeed, at small γtr 1 the line of localization transition is almost insensitive to truncation close to p = 1 (p > 1 − γtr/(4p)) while the line of ergodic transition is pushed to smaller values of γ at 2p > 1 − γtr/(4p) corresponding to larger typical transition matrix elements U (smaller effective disorder). Thus, the width of the MF phase increases linearly with γtr 1 This proves the fact that the weakly ergodic phase in LN-RP with p ≥ 1 is very fragile and exists only due to atypically large transition matrix elements. It is substituted by the multifractal NEE phase as soon as such matrix elements are made improbable by truncation.
In the limit γtr 1 the width of the WE phase can be approximated at 2p > 1 − γtr/(4p) as , [44] showing linear decrease with γtr and giving a reasonable approximation of the value of γtr 1 where this phase disappears. Here we use [45]

B. Ratio of typical and mean LDOS
In this section we consider in more details the technical issue with the determination of the order parameter for the FWE transition [46] being the ratio of the typical, ρtyp, and the mean, ρav, LDOS given by the expressions ln ρtyp = ln ρ(x, E + iη) , ρav = ρ(x, E + iη) , [47] with the LDOS before averaging written as [48] The averaging in Eq. (47) is taken over the disorder realizations, over all coordinates x (which are statistically equivalent in LN-RP) and over 100 energy values in the middle half of the spectrum. As mentioned in the main text the ratio ρtyp/ρav develops the plateau ∼ N −1+D 1 in some range of bare level width parameter η δ large compared to the typical level spacing δ. However, at any finite sizes this plateau has a finite slope, especially for the WE phase where N D 1 = f N with a N -independent constant f < 1 and, thus, the plateau is also N -independent which is zero in the FE phase, f = 1, and finite in the WE one, f < 1.
In order to find the FWE transition accurately we develop the procedure of the automatic selection of η in the middle of the underdeveloped plateau. For this purpose we take the second derivative of the ratio ρtyp/ρav versus η after the smoothening it with the 5-degree spline and find the maximal point of this derivative lying in between two local minima (see the lower panels in Fig. 7). Figure 7 shows several examples for p = 0.01 and p = 1 where the positions of the maxima of the second derivative are shown by crosses of the corresponding color for all system sizes N .

C. Analysis of stability
In this section we calculate the contributions to P (V ) from the log-normal P LN (V ) and Gaussian P Gauss (V ) parts to Eq. (17).
One can easily compute the variance of the Gaussian part of P Gauss (V ) leaving in it only the bi-diagonal terms with i = i and j = j : The maximum in Eq. (50) at g belonging to region II in Fig. 8 can be reached (i) inside the region II at g = g * 1 , (ii) at the border of this region at g = g * 2 , and (iii) at the cut-off of P (g) at g * = 0 (see Fig. 8 and Fig. 9(left)).
The expression for γ eff (D 1) takes the form: [51] Next we compute the function [52] in Eq. (21). The details of the calculation which is similar to calculation of γ eff (D 1 ) in Eq. (50) are illustrated in Fig. 9(right). The resulting expression for ∆(D i ) is: In the end of this section we consider the question of the distribution of g ij = − ln |H ij − H ij |/ ln N with log-normal distributed H = N −g 1 and H = N −g 2 . Neglecting rare events of very small differences |H ij − H ij | |H ij | we approximate and, thus, the distribution P (g) is given by P (g) = P (g 1 = g) ∞ g P (g 2 )dg 2 P (g 1 = g) g < γ/2 P 2 (g 1 = g) g > γ/2 [55] As one can see from Fig. 9 the latter region g > γ/2 is actual only for the upper branch g * 2 of ∆(D 1 ) + D 1 for D 1 > γ/(8p) which never contributes to the phase diagram.

D. Kullback-Leibler measures in the multifractal phase
In this section we give a more detailed quantitative description of KL1 and KL2 measures.
We begin by considering the simpler correlation function, KL2. For that we employ the ansatz for the wavefunction moments: [56] where Dq is the fractal dimension in the corresponding phase and fq(x) is the crossover scaling function: const. N (q−1)Dq localized phase [57] that tends to a constant as L → ∞.
Note that graphs with the local tree structure and for LN-RP matrices the length scale L ∝ ln N , so that the scaling function is in general a function of two arguments ln N/ξq and N/e ξq representing the length-and volume scaling (10,11). On the finite-dimensional lattices N ∝ L d , and the volume scaling can be represented as the length scaling in the modified scaling function. In this case a single argument L/ξq is sufficient.
When L ∝ ln N the volume scaling is the leading one for L ξq, and it is this scaling that provides the asymptotic behavior Eq. (57). The length scaling is important in the crossover region L ξq. Below for brevity we will use the short-hand notation L/ξq in all the cases.
There are two trivial cases: M 0 = N and M 1 = 1 (which follows from the normalization of wave function). As a consequence we have D 0 = 1 and f 0 (x) = f 1 (x) ≡ 1. [58] Next using the statistical independence of ψ andψ in Eq. (29) and normalization of wave functions we represent Now we express both terms in Eq. (59) in terms of Mq using the identity: ln The first term is equal to: [61] The second term can be expressed as: Now expanding M 1+ and M in the vicinity of q = 0, 1 and defining we obtain: where KL2c is logarithmically divergent, as in Eq. (33): [66] Here we used the identity for α 0 describing the typical value of the wave function amplitude: Note that, generally speaking, the characteristic lengths ξ 0 ∼ |γ − γc| −ν (0) and ξ 1 ∼ |γ − γc| −ν (1) in φ 0 and φ 1 may have different critical exponents ν (0) and ν (1) . If this is the case, the smallest one will dominate the finite-size corrections near the critical point: Eq. (69) is employed in this paper for the numerical characterization of the phases by finite size scaling (FSS). One can see from  52) is reached at the edge of the right segment of region I, g = g * 2 (not to be confused with the edge of the left segment g = g * 2 , see Fig. 8 ). It leads to a higher branch of the orange curve ∆(α)/γ + α in Fig. 3 (not shown in Fig. 3) which is separated by a gap from the blue curve in Fig. 3 and thus is irrelevant for our analysis. Eq. (66) that KL2 is logarithmically divergent in the multifractal phase, as α 0 > 1 and D 1 < 1 and the scaling functions ϕ 0 (x) and ϕ 1 (x) tend to a finite N -independent limit. It is also logarithmically divergent in the localized phase, as in Eq. (32), where one can formally set D 1 = 0 in Eq. (66): However, in the ergodic phase the logarithmic divergence of KL2 is gone, since in this case α 0 = D 1 = 1 in Eq. (66). One can easily show using the Porter-Thomas distribution: that KL2 = 2 in the fully-ergodic phase. At the continuous ergodic transition, where the correlation length ξ = ∞ and α 0 = D 1 = 1, the critical value KL2c(N ) of KL2 is independent of N . This results in crossing at γ = γ ET of all the curves for KL2 at different values of N which helps to identify the ergodic transition (31).
However, if the ergodic transition coincides with the Anderson localization transition and is discontinuous, (i.e. α 0 and D 1 are not equal to 1 at the transition), the critical value KL2c(N ) is no longer N -independent. In this case the crossing is smeared out and can disappear whatsoever. Nonetheless, by subtracting KL2c from KL2 one can still locate the transition point from the best collapse of KL2 vs. γ curves by choosing an optimal γc and ν 2 in Eq. (69). However, it is safer to use KL1 in this case.
[78] We obtain: KL1 = Φ 1 (L|γ − γc| ν 1 ). [79] where ν 1 = ν (1) ≥ ν 2 and the crossover scaling function Φ 1 (x) is: [80] As it is seen from Eq. (79), KL1 is independent of N at the Anderson transition point γ = γ AT . Thus all curves for KL1 at different values of N intersect at γ = γ AT . This gives us a powerful instrument to identify the Anderson localization transition point. Note that the coefficient in front of ln N in KL2 may help to detect discontinuity of the Anderson transition. Indeed, one can use the Mirlin-Fyodorov symmetry of fractal dimensions to establish the relation, see Eq. (33): [81] This tells us immediately that for continuous Anderson transition which is characterized by vanishing D 1 both just below and just above the transition, the coefficient in front of ln N in KL2 is equal to 2. In particular, we conclude that α 0 on the localized side of the transition is equal to 2. It appears that in LN-RP this value α 0 = 2, (γ = γ AT + 0). [82] in the localized phase just above the transition remains equal to 2 also in the case where the transition is discontinuous. This is in contrast to the corresponding coefficient 2(1 − D 1 ) in front of ln N in KL2 just below the transition which is smaller than 2 if the transition is discontinuous. Such a jump in the coefficient in front of ln N in KL2 is a signature of the discontinuity of the transition which is the most easily detectable numerically, see Fig. 5.

E. Finite-size scaling collapse for KL1 and KL2.
The next step is to analyze the finite-size scaling (FSS) by a collapse of the data for KL1 and KL2 at different N in the vicinity of the localization and ergodic transition, respectively. To this end we use the form of FSS derived in IS D. [83b] The input data for the collapse is KL1 and KL2 versus γ for 7 values of N is shown in Fig. 4. The fitting parameters extracted from the best collapse are ν 1 (ν 2 ) and the critical points γ AT (γ ET ). The critical value of KL2c(N ) = KL2(γ ET , N ) is determined by the best fitting for γ ET . For the localization transition where the critical point γ AT is well defined by the intersection in KL1, one may look for the best collapse by fitting only ν 1 . The plots of Fig. 10 demonstrate the quality of the collapse for several representative cases. In the insets of the figures we show the ln N -dependence of the critical values of KL1, KL2 which were obtained numerically from KL1(γ = γ AT , N ) and KL2(γ = γ ET , N ), respectively, with γ AT and γ ET found from the best collapse. It is demonstrated that the critical value of KL1 is almost N -independent, as well as the critical value of KL2 at p = 1/2 when the continuous ergodic transition is well separated from the Anderson localization one. However, at p = 1 when ET and AT merge together the critical value of KL2 increases linearly with ln N , signaling of the critical multifractal state at the Anderson transition point, very similar to the case of 3D Anderson transition. This ln N -dependence of KL2c is the reason of smearing out of the intersection point in KL2 shown in Fig. 4.