Abstract
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 exists also the transition between the two ergodic phases (FWE transition). We suggest new criteria of stability of the non-ergodic phases which give the points of localization and ergodic transitions and prove that the Anderson localization transition in LN-RP model involves a jump in the fractal dimension of the eigenfunction support set. We also formulate the criterion of FWE transition and obtain the full 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.
Highlights
The structure of many-body wave function is important for a variety of problems that range from many-body localization (MBL) to quantum computation
We suggest new criteria of stability of the nonergodic phases that give the points of localization and ergodic transitions and prove that the Anderson localization transition in log-normal RosenzweigPorter (LN-RP) model involves a jump in the fractal dimension of the egenfunction support set
We introduce a log-normal RosenzweigPorter (LN-RP) random matrix ensemble characterized by a long-tailed distribution of off-diagonal matrix elements
Summary
The structure of many-body wave function is important for a variety of problems that range from many-body localization (MBL) (see Ref. [1] and a recent review [2]) to quantum computation. More importantly, few mini bands in the local spectrum of this model [25,31] are compact and absolutely continuous in the energy space, and not multiple and fractal as in realistic many-body systems [17] [see Fig. 1(a)] This behavior is intimately related to the compactness of distribution of the wave function coefficients on the support set and can be traced back to the property of the moments of the Gaussian distribution |U |q = U 2 q/2. The ergodic states in the GRP model remain ergodic in any basisis [26], like in the classic Wigner-Dyson (WD) random matrix ensemble Another example of extended ergodic wave functions which distribution does not tend to the Porter-Thomas in the thermodynamic limit was earlier given in Ref.
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