Abstract
The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.
Highlights
Rosenzweig-Porter ensembleThe Rosenweig-Porter (RP) ensemble [23], which was originally proposed in the context of complex atomic nuclei [25], can be seen as generalization of the Gaussian orthogonal ensemble (GOE) [51, 52] with a preferential basis
A profound example of such a model is the Rosenzweig-Porter random matrix ensemble [25]. This model was suggested as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition [23], which separates a many-body localized from a thermal phase
We propose a circular analogue of the Rosenzweig-Porter ensemble, which we define as the result of a Dyson Brownian motion process [42]
Summary
The Rosenweig-Porter (RP) ensemble [23], which was originally proposed in the context of complex atomic nuclei [25], can be seen as generalization of the Gaussian orthogonal ensemble (GOE) [51, 52] with a preferential basis. In the large-N limit, level spacings of the RP ensemble obey WignerDyson level statistics for γ < 2 [53], which are typically observed for chaotic quantum systems. The ratios of consecutive level spacings acquire the values r ≈ 0.386 for Poissonian and r ≈ 0.530 for Wigner-Dyson (GOE) level statistics [55]. Where the bar denotes an average over V in Eq (1), En is the eigenvalue corresponding to the eigenstate |ψn〉, En(0) = (H0)nn, and the so-called spreading width Γ (En) is obtained through Fermi’s golden rule as. For 1 < γ < 2, the spreading width gives the width of the so-called mini-band [23,31,60] In this energy window, the eigenstate amplitudes |〈ψn|m〉|2 are on average of order N −D2 (where D2 is the fractal dimension Dq for q = 2), which is much larger than the value N −1 obtained when averaging over all basis states
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