Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

Cecile Monthus

doi:10.1088/1751-8121/aa77e1

paper

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

arXiv:1609.01121 · doi:10.1088/1751-8121/aa77e1

Abstract

The delocalized non-ergodic phase existing in some random $N \times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_{0}$ . The main output of this approach is the inverse $Γ_{j_{0}} (N)$ of the characteristic time to leave the state $j_{0}$ that provides some broadening $Γ_{j_{0}} (N)$ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $Γ_{j_{0}} (N)$ is smaller in scaling than the level spacing $Δ_{j_{0}} (N) \propto \frac{1}{N}$ , while the delocalized non-ergodic phase corresponds to the region where the broadening $Γ_{j_{0}} (N)$ decays with $N$ but is bigger in scaling than the level spacing $Δ_{j_{0}} (N)$ . Then the number $\frac{Γ _{j_{0}} ( N )}{Δ _{j_{0}} ( N )}$ of resonances grows only sub-extensively in $N$ . This approach allows to recover the multifractal spectrum of the Generalized-Rosenzweig-Potter (GRP) Matrix model [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. We then consider the Lévy generalization of the GRP Matrix model, where the off-diagonal matrix elements are drawn with an heavy-tailed distribution of Lévy index $1 < μ < 2$ : the dynamics is then governed by a stretched exponential of exponent $β = \frac{2 ( μ - 1 )}{μ}$ and the multifractal properties of eigenstates are explicitly computed.

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Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach

Abstract

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