Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

Abstract

The accuracy of the forward scattering approximation for two-point 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 transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point 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 two-point Green's function.

24 pages, 4 figures, 39 references

References in corpus (8)

• Localization of interacting fermions at high temperature
• Many-body localization edge in the random-field Heisenberg chain
• Integrals of motion in the Many-Body localized phase
• Recent progress in many-body localization
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• Critical behavior at the localization transition on random regular graphs
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3

Cited by in corpus (6)

• Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization
• Non-ergodic delocalized phase with Poisson level statistics
• Sub-diffusive Thouless time scaling in the Anderson model on random regular graphs
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach
• Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree