Pavel A. Nosov, Ivan M. Khaymovich, Andrey Kudlis, Vladimir E. Kravtsov
SciPost Phys. 12, 048 (2022) ·
published 1 February 2022

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The accuracy of the forward scattering approximation for twopoint Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfermatrix equation is proved in the framework of the supersymmetric functionalintegral method. The new largedisorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the twopoint Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a twopoint Green's function.
SciPost Phys. 11, 045 (2021) ·
published 31 August 2021

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We consider the static and the dynamic phases in a RosenzweigPorter (RP) random matrix ensemble with a distribution of offdiagonal matrix elements of the form of the largedeviation ansatz. We present a general theory of survival probability in such a randommatrix model and show that the {\it averaged} survival probability may decay with time as a simple exponent, as a stretchexponent and as a powerlaw or slower. Correspondingly, we identify the exponential, the stretchexponential and the frozendynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretchexponent $\kappa$ in the thermodynamic limit. As another example we consider the logarithmicallynormal RP random matrix ensemble and find analytically its phase diagram and the exponent $\kappa$. Our theory allows to describe analytically the finitesize multifractality and to compute the critical length with the exponent $\nu_{MF}=1$ associated with it.
Giuseppe De Tomasi, Mohsen Amini, Soumya Bera, Ivan M. Khaymovich, Vladimir E. Kravtsov
SciPost Phys. 6, 014 (2019) ·
published 30 January 2019

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We study analytically and numerically the dynamics of the generalized RosenzweigPorter 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 highlyexcited nonstationary 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 powerlaw 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 nontrivial 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.
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