## Survival probability in Generalized Rosenzweig-Porter random matrix ensemble

Giuseppe De Tomasi, Mohsen Amini, Soumya Bera, Ivan M. Khaymovich, Vladimir E. Kravtsov

SciPost Phys. 6, 014 (2019) · published 30 January 2019

- doi: 10.21468/SciPostPhys.6.1.014
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### Abstract

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 in different phases. The ergodic phase is characterized by the standard power-law decay of $R(t)$ with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of $R(t\to\infty)=k$, finite in the thermodynamic limit $N\rightarrow\infty$, which approaches $k=R(t\to 0)$ in this limit.

### Cited by 2

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^{1 }^{2 }Giuseppe De Tomasi, -
^{3 }Mohsen Amini, -
^{4 }Soumya Bera, -
^{1 }Ivan Khaymovich, -
^{5 }^{6 }^{7 }Vladimir Kravtsov

^{1}Max-Planck-Institut für Physik komplexer Systeme / Max Planck Institute for the Physics of Complex Systems^{2}Technische Universität München / Technical University of Munich [TUM]^{3}دانشگاه اصفهان / University of Isfahan [UI]^{4}भारतीय प्रौद्योगिकी संस्थान मुम्बई / Indian Institute of Technology Bombay [IITB]^{5}University of California, Santa Barbara [UCSB]^{6}Институт теоретической физики им. Л. Д. Ландау / Landau Institute for Theoretical Physics^{7}Centro Internazionale di Fisica Teorica Abdus Salam / Abdus Salam International Centre for Theoretical Physics [ICTP]

- Department of Science and Technology, Ministry of Science and Technology (through Organization: विज्ञान एवं प्रौद्योगिकी विभाग / Department of Science and Technology [DST])
- Deutsche Forschungsgemeinschaft / German Research FoundationDeutsche Forschungsgemeinschaft [DFG]
- National Science Foundation [NSF]
- Российский фонд фундаментальных исследований / Russian Foundation for Basic Research [RFBR]