Spread of wave packets in disordered hierarchical lattices

We consider the spreading of the wave packet in the generalized Rosenzweig-Porter random matrix ensemble in the region of non-ergodic extended states 1 < γ < 2. We show that despite non-trivial fractal dimensions 0 < Dq = 2− γ < 1 characterize wave function statistics in this region, the wave packet spreading 〈r〉 ∝ t is governed by the “diffusion” exponent β = 1 outside the ballistic regime t > τ ∼ 1 and 〈r〉 ∝ t in the ballistic regime for t < τ ∼ 1. This demonstrates that the multifractality exhibits itself only in local quantities like the wave packet survival probability but not in the large-distance spreading of the wave packet. Introduction. – The idea of the absence of diffusion in disordered non-interacting systems which leads to Anderson Localization(AL) was first studied more than half a century ago [1]. Modeling the disordered environment, Anderson considered hopping of a single particle from atom to atom in presence of an external random potential which can localized the particle exponentially in real space. This matter wave localization phenomena, which is essentially a quantum mechanical effect, means that an initial spreading of an evolving wave-packet eventually stops in presence of sufficient amount of static disorder and leads to an Anderson transition [2] in dimension d > 2. On the other hand, during the last recent years, the problem of Many-body localization(MBL) [3] has attracted significant interests and can be understood as localization in the Fock space of Slater determinants which has hierarchical structure due to two-body interaction effect [4]. So MBL can be studied in terms of disordered hierarchical lattices which shows an intermediate nonergodic extended phase with multifractal eigenfunctions [5, 6]. Furthermore, it is also observed that the propagation of dipolar excitations among immobile molecules randomly spaced in a lattice due to exchange via dipole shows the same behavior [7]. Moreover, Random Matrix Theory (RMT) which is introduced by Wigner [8] describes the statistical properties of ensembles of random matrices. The theory was then expanded significantly and became a major theoretical tool to calculate the relevant statistical properties of complex quantum systems in the field of quantum chaos [9, 10]. However, the conventional RMT is unable to describe important phenomena such as eigenstate localization. For this reason, much attention has been paid to the so-called Band Random Matrices as an approach to study the AL problem theoretically [11]. Very recently [12], the existence of non-ergodic extended phase has been proven for a generalization of Rosenzweig-Porter(RP) RMT suggested in 1960 [13]. This model also possesses the multifractality of eigenvectors in the non-ergodic extended phase in a certain range of parameter 1 < γ < 2. Furthermore, it is shown that the properties of the non-ergodic delocalized phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit [14]. So the generalized RP model seems to be the proper RMT to study the hierarchical lattices as well as statistical properties of MBL. However, much less is clear about the spreading of wave packets in hierarchical lattices described by generalized RP model. In this paper, we studied the spreading of initially δ-like wave packets in the generalized orthogonal RP model of RMT, with particular attention to their behavior on different time scales. Our calculation which is confirmed by numeric shows that in this model even in the non-ergodic extended phase, where the eigenstates are multifractal, there is either ballistic transport, or the diffusive one. In the following sections, we will first introduce the model which we are interested in and then we will describe the method which we used to monitor the wave packet spreading and finally we will report the results.