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.
Highlights
The forward-scattering approximation (FSA) for disordered quantum systems is the simplest approximation for describing the Anderson and many-body localization at strong disorder [1–5], which for some situations [6,7] is well corroborated by numerics
In the first part we show, using the Efetov’s supersymmetry method [23], that the moments m = 2β < 1 of the real Green’s function Gi j(E) on a Cayley tree are exactly expressed in terms of the largest eigenvalue εβ of the linearized transfer-matrix (TM) equation [24] in the large ri j = r limit:
In this work we present a formal derivation of Eq (3) by the supersymmetry method but without the constraint of the nonlinear sigma-model (NLSM) which significantly simplifies the TM equation
Summary
The forward-scattering approximation (FSA) for disordered quantum systems is the simplest approximation for describing the Anderson and many-body localization at strong disorder [1–. It is proven in this paper that it is the same function εβ that controls both the dynamics of the kink solution of TM equation and the moments of Green’s functions on a Cayley tree at any disorder. If this non-trivial point is taken for granted, one may guess [26, 27] Eq (3) from the results of Ref. An alternative way of justifying Eq (3) is presented in Ref. [29]
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