Localization and fractality in disordered Russian Doll model

Report 1 of the Referee Report In this manuscript, the authors consider the eigenstate statistics of an infinite-ranged "hopping" model with onsite disorder, inspired in part by the Richardson (reduced BCS) and "Russian Doll" BCS models. As a many-body or reduced (Cooper) problem, the latter was shown to exhibit a novel "periodic RG", associated to the formation of a tower of BCS-type eigenstate solutions. The emphasis here however is on the associated one-body (random-matrix) version of the dirty Russian Doll model (RDM), which does not exhibit the RG recurrence. This is an elaboration of the corresponding one-body Richardson (or "simplex") model studied previously. That model exhibits "localization" of all eigenstates but one. More precisely, in the one-body Richardson model all eigenstates take the form $\psi_i = 1/(E - e_i)$, where $E$ is the eigenenergy and $e_i$ is the potential at site i. Most of the eigenstates are "localized" in the sense that IPR gives a multifractal exponent $\tau(q) = 0$, due to the infrared divergence for $e_i < E < e_{i+1}$. By contrast, the addition of the imaginary, antisymmetric hopping term "h" in the present manuscript appears to allow for ergodic, fractal, and localized regimes, depending on how the hopping terms are scaled with N. The latter is controlled by an exponent $\gamma$. The authors use a version of the RDM RG [LeClair et al] to decimate out the highest-energy states in momentum space. Given the strongly divergent dispersion $\sim cot(k)$, this is a sensible procedure, and produces an effective Hamiltonian that the authors then use to investigate fractality of the model. For $0 < \gamma < 2$, they predict $D = 1 - \gamma/2$, a result that they also check numerically.

We thank the referee for careful reading of our manuscript and for summarizing our results. We would like just to mention a couple of points which seem to be a bit unclear:

(a) the RG recurrence mentioned by the referee does not break down in the single-particle (random matrix) sector of the disorder-free Russian Doll model. It is the diagonal disorder, which breaks down the structure of renormalized Hamiltonian (please see the reply (c) to the question (3) and [34] for more details). (b) The RG used in our manuscript has the origin from [7-8], but is not just a version of it: the main idea of decimating out the highest-energy states is the same in both, but in our case, we have - decimated out the states which are non-ergodic in the momentum space, - not observed any RG recurrence, and therefore - had to introduce the stopping criterion for the RG procedure based on the optimization of the broadening-based fractal dimension. In the revised version of the manuscript we have emphasized both above issues more clearly.

Weaknesses (1) The biggest weakness of this paper is rather unfocused nature of the introduction. The opening paragraphs discuss the many-body Richardson and Russian Doll models, but while these "inspire" the study here, they have no direct relevance. The dirty RDM studied here is a problem in random matrix theory, and the basic conclusions (tunable regimes for ergodic, non-ergodic, and localized states) are more intelligible in that context. This is especially true because the many-body Richardson model and RDM exhibit a host of peculiar features (e.g. integrable quench dynamics, an infinite tower of BCS "ground" states) which have no relevance here. Requested changes (1) I suggest placing the work here in greater context earlier in the introduction, i.e. that the physics studied here is more analogous to the Rosenzweig-Porter random matrix model than the many-body Richardson or RD models. In particular it would be helpful to state that the limit cycle RG is NOT an ingredient here (despite the title).

We thank the Referee for the suggestion, but we kindly disagree with his/her vision on the relevance of many-body RDM and Richardson models to the considered problem. Indeed, the presence of an infinite tower of BCS "ground" states in RDM has immediate similarity to the hierarchy of divergent single-particle levels which are both ergodically delocalized in the coordinate basis and their number scales in the same way with $N$. Bethe ansatz integrability of the many-body Richardson and RDM models is also relevant to the single-particle sector as already in the Richardson model this integrability does lead to localization, but cannot avoid level repulsion similar to the Gaussian random matrix case. Therefore already on these simple examples one can see the relevance of the interplay of Bethe-ansatz integrability, disorder, and time-reversal symmetry breaking which (unlike Richardson model) leads to the non-ergodic delocalization at $\gamma<2$. The question of the many-body sectors of RDM is still an open issue and should be addressed in the further investigations. Having all this in mind, we prefer to keep the structure of the introduction, only slightly modifying the accents there.

(2) The analytical and numerical results for the fractal dimension D appear to be independent of the angle $\theta$ (except for extreme cases, i.e. "Richardson" with $\theta = 0$). It would be helpful if the authors can explain concisely why this is.

We thank the Referee for the question. Indeed, our results do not depend on $\theta$ for any non-zero values of this parameter. Formally from Eqs. (32-33) one can see that for finite-size models the number of RG steps $r_{opt}$ and the corresponding broadening $\Gamma_n(r_{opt})$ have the common crossover $\sin\theta \sim W N^{-1+\gamma/2}\ll 1$ beyond which $r_{opt}<1$ and $\Gamma_n<\delta$. The later returns us back to the Richardson's model as one does not need to make a single RG step and obtains localization. The origin of this finite-size crossover is given by the divergence of the disorder-free spectrum (8) at least at the largest energy scale $E_{2k+1=1}$. As soon as $E_1 \leq W$ the disorder hybridize it with the rest disorder-free levels and the derivation of the RG in the momentum space does not work anymore (see the discussion after Eq. (8) in the end of the section 3). In the revised version of the manuscript, we have slightly reformulate this discussion in order to emphasize the above points.

(3) A perhaps general comment with respect to "localization" in this model and the one-body Richardson one is the following. In the latter, all eigenstates actually take the same form, $\psi_i = 1/(E - e_i)$, where $E$ is the eigenenergy and $e_i$ is the potential at site i. There is no dependence on the hopping in this case, except indirectly through the energy eigenvalue E. If we "lexicographically order" the energies $e_1 < e_2 < .. < e_N$, then these eigenstates are power-law peaked with a center that merely drifts linearly with E. Moreover, these "localized" states show power-law Chalker scaling, since the states with nearby energies sit on nearby $e_i$'s. This is very different from a true Anderson insulator, where the Chalker correlator is $\sim 0$ for energy differences less than the typical localization volume level-spacing. I would therefore prefer to refer to such states as "frozen" rather than localized, where freezing refers to $\tau(q) = 0$ for $q \geq 1$, but the spatial structure can consist of rare peaks that are separated by arbitrarily large distances. Such states also show Chalker scaling. (see e.g. Chou and Foster, PHYSICAL REVIEW B 89, 165136 (2014)). Are the localized states obtained here (for $\gamma > 2$) also "frozen" in this sense?

We thank the Referee for pointing out this issue of Chalker scaling and possible frozen nature of the localized phase. In order to reply to the Referee's question, we need to mention at least 3 points: (a) First of all, if one speaks about the localized phase, $\gamma>2$, the Russian Doll, the Richardson, and the Rosenzweig-Porter models show identical results given by the perturbation theory (see [12] for the discussion of correlations and [16] for the Chalker-like scaling). In this case, all three models show power-law localization in "lexicographically ordered" energies $e_1 < e_2 < .. < e_N$ with the power $a=2$, see, e.g., Fig. 6 of the revised version. Thus, the corresponding Chalker scaling gives in this case for the overlap correlation function $K(\omega)\sim (\delta/\omega)^2$. (b) Second, generally in all long-range models with power-law Anderson localization (see, e.g., the well-known power-law random banded matrices (PLRBM) or less known Burin-Maksimov model in [12, 26, 27]) the critical exponents $\tau_q$ are non-zero for $q<1/(2a)$, where $a>1$ is the power of the localization, $|\psi_n(r)|\sim 1/|n-r|^a$. The same is true for the above-mentioned Rosenzweig-Porter model at $\gamma>2$, where $\tau_q>0$ for $q<1/\gamma$. In all these cases formally the localized states can be called frozen, but non-zero critical exponents appear only somewhere at $q<1/2$. The same is true for the disordered Russian Doll model as in the localized phase it shows the same structure as the Rosenzweig-Porter model.

(c) Another important issue is the difference between Richardson or Rosenzweig-Porter models and, e.g., PLRBM and Russian Doll model in terms of the Chalker's ansatz in the non-ergodic extended phase. Both in the Richardson and Rosenzweig-Porter models the wave-function structure is given by the diagonal disorder in non-ergodic phases, due to the absence of any other spatial structure in the model. In the Richardson's model it appears due to the eigenstate structure $\psi_i = 1/(E - e_i)$, given by the Bethe ansatz solution and mentioned by the Referee. For the Rosenzweig-Porter model the similar structure of the Breit-Wigner form appears due to the self-averaging in the broadening $\Gamma_n$ and its independence of the space index $n$. Unlike this, in the PLRBM model there is the competing structure of the power-law decaying hopping terms (in the usual order of sites) and the one of the diagonal elements (in the lexicographic order of sites mentioned by the Referee). It is this competition of two reordered spaces, which make the critical states multifractal in the PLRBM and introduces a non-trivial power to the Chalker's scaling. In the case of the Russian Doll model, this competition is present, but it is given minimally by the presence of the sign-distance dependence ${\rm sign}(d(m,n))$ in the TRS-breaking term in (2). But in any case the presence of such a competition immediately breaks down the RG considered in [7-8] and leads to the power-law decaying hopping in the effective Hamiltonian, the second case in (34). This changes the wave-function structure of the Richardson's model, $\psi_i = 1/(E - e_i)$ at $\gamma<2$ with respect to the Russian Doll model and therefore should affect the corresponding Chalker scaling. We leave the investigation of the later for the further study.

In the revised manuscript we have added the discussion of the above issues in a brief form.

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