7July 2022 Thu 5 pm

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Random matrix theory provides a powerful tool for studying transitions from ergodic to non-ergodic phases in complex quantum systems. A well-established phenomenological model for many-body localization in static (time-independent) systems is provided by the Rosenzweig-Porter random matrix ensemble. This single-parameter ensemble is analytically tractable, and covers both an ergodic, delocalized yet non-ergodic, and localized phase. As such, it serves as one of the simplest models for the level statistics and fractality of eigenstates across the many-body localization transition.

Motivated by the observation of many-body localization in periodically driven (Floquet) systems, we propose a unitary ("circular") analogue of the Rosenzweig-Porter random matrix ensemble. Like Floquet operators, this ensemble consists of unitary matrices with the eigenvalues located on the unit circle in the complex plane. Similar to the role of the Rosenzweig-Porter random matrix ensemble for static systems, this ensemble provides a model for the many-body localization transition in Floquet systems.

We define our ensemble as the outcome of a Dyson Brownian motion process. We provide analytical arguments and show numerical evidence that this ensemble shares key statistical properties with the Rosenzweig-Porter ensemble for both the statistics of the eigenvalues and the eigenstates in each of the phases. Directions for future research include e.g. an extension of the ensemble covering multifractality or the use of the circular Rosenzweig-Porter ensemble as a non-maximally random building block for random quantum circuits.

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