In this paper we consider an extension of the Rosenzweig-Porter (RP) model, the Lévy-RP (L-RP) model, in which the off-diagonal matrix elements are broadly distributed, providing a more realistic benchmark to develop an effective description of non-ergodic extended (NEE) states in interacting many-body disordered systems. We put forward a simple, general, and intuitive argument that allows one to unveil the multifractal structure of the mini-bands in the local spectrum when hybridization is due to anomalously large transition amplitudes in the tails of the distribution. The idea is that the energy spreading of the mini-bands can be determined self-consistently by requiring that the maximum of the matrix elements between a site $i$ and the other $N^{D_1}$ sites of the support set is of the same order of the Thouless energy itself $N^{D_1 - 1}$. This argument yields the fractal dimensions that characterize the statistics of the multifractal wave-functions in the NEE phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions are confirmed both analytically, by a thorough investigation of the self-consistent equation for the local density of states obtained using the cavity approach, and numerically, via extensive exact diagonalizations.
The Rosenzweig-Porter model is a one-parameter family of random matrices with three different phases: ergodic, extended non-ergodic and localized. We characterize numerically each of these phases and the transitions between them. We focus on several quantities that exhibit non-analytical behaviour and show that they obey the scaling hypothesis. Based on this, we argue that non-ergodic chaotic and ergodic regimes are separated by a continuous phase transition, similarly to the transition between non-ergodic chaotic and localized phases.
We consider the Rosenzweig-Porter model $H = V + \sqrt{T}\, Φ$, where $V$ is a $N \times N$ diagonal matrix, $Φ$ is drawn from the $N \times N$ Gaussian Orthogonal Ensemble, and $N^{-1} \ll T \ll 1$. We prove that the eigenfunctions of $H$ are typically supported in a set of approximately $NT$ sites, thereby confirming the existence of a previously conjectured non-ergodic delocalized phase. Our proof is based on martingale estimates along the characteristic curves of the stochastic advection equation satisfied by the local resolvent of the Brownian motion representation of $H$.
Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robust multifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phase in GRP model is oversimplified: the spectrum of fractal dimensions is degenerate and the mini-band in the local spectrum is not multifractal. In this paper we suggest an extension of the GRP model by adopting a logarithmically-normal (LN) distribution of off-diagonal matrix elements. A family of such LN-RP models is parametrized by a symmetry parameter $p$ and it interpolates between the GRP at $p\rightarr…
Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach
Nanocenter CENN（纳米中心CENN）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Jozef Stefan Institute（约瑟夫·斯特凡研究所）；Abdus Salam International Center for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）
The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a non-ergodic extended phase between the ergodic extended metallic phase and the localized phase. Such a phase is relevant to many physical models from the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity, to the interacting many-body localization in condensed matter physics and quantum computing. This phase is characterized by fractal behavior of the wavefunctions, and a postulated correlated mini-band structure of the energy spectrum. Here we will seek evidence for the latter in the spectrum. Since…
We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability $R(t)$, the probability of finding the initial state after time $t$. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that $R(t)$ can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified …
Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach
Nanocenter CENN, Ljubljana, Slovenia（卢布尔雅那纳米中心）；Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia（卢布尔雅那大学数学物理学院）；Jozef Stefan Institute, Ljubljana, Slovenia（约瑟夫·斯蒂芬研究所）
In this paper we suggest an extension of the Rosenzweig-Porter (RP) model, the LN-RP model, in which the off-diagonal matrix elements have a wide, log-normal distribution. We argue that this model is more suitable to describe a generic many body localization problem. In contrast to RP model, in LN-RP model a fragile weakly ergodic phase appears that is characterized by broken basis-rotation symmetry which the fully-ergodic phase, also present in this model, strictly respects in the thermodynamic limit. Therefore, in addition to the localization and ergodic transitions in LN-RP model there exis…
We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with the tailed distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the {\it averaged} survival probability may decay with time as the simple exponent, as the stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson model on Ran…
We consider eigenvectors of the Hamiltonian $H_0$ perturbed by a generic perturbation $V$ modelled by a random matrix from the Gaussian Unitary Ensemble (GUE). Using the supersymmetry approach we derive analytical results for the statistics of the eigenvectors, which are non-perturbative in $V$ and valid for an arbitrary deterministic $H_0$. Further we generalise them to the case of a random $H_0$, focusing, in particular, on the Rosenzweig-Porter model. Our analytical predictions are confirmed by numerical simulations.
We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter $N\times N$ random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies $\{a_i\}$ and a structurally disordered hopping, we found that each eigenstate is delocalised over $N^{2-γ}$ sites close in energy $|a_j-a_i|\leq N^{1-γ}$ in agreement with Kravtsov \emph{et al}, arXiv:1508.01714. Our other main result, obtained combining a recurrence relation for the resolvent mat…
Motivated by the problem of Many-Body Localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two localization transitions as the parameter $γ$ of the model varies from 0 to $\infty$. One of them is the Anderson transition from the localized to the extended states that happens at $γ=2$. The other one at $γ=1$ is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the G…
We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson's model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emerge…
We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are…
Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(τ) = 1 + b K_{1}(τ) + b^{2} K_{2}(τ) + ... $ in powers of $b \ll 1$ with the coefficients $K_{m}(τ)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(τ)$, we develop a diagrammatic technique which is based on the Trotter formula and on th…
Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models
Budapest University of Technology and Economics（布达佩斯技术与经济大学）
