[Paper Review] Mean Green's function of the Anderson model at weak disorder with an infra-red cut-off

Gilles Poirot|ArXiv.org|Feb 12, 1997

Spectral Theory in Mathematical Physics4 references10 citations

TL;DR

This paper develops a polymer expansion with large/small field conditions to rigorously analyze the mean Green's function in the two-dimensional Anderson model at weak disorder, incorporating an infrared cutoff. It establishes an asymptotic expansion for the density of states outside or within the free spectrum, marking a critical step toward a non-perturbative understanding of the density of states in disordered quantum systems.

ABSTRACT

We develop a polymer expansion with large/small field conditions for the mean resolvent of a weakly disordered system. Then we show that we can apply our result to a two-dimensional model, for energies outside the unperturbed spectrum or in the free spectrum provided the potential has an infra-red cut-off. This leads to an asymptotic expansion for the density of states. We believe this is an important first step towards a rigourous analysis of the density of states in the free spectrum of a random Schrödinger operator at weak disorder.

Motivation & Objective

• To develop a non-perturbative field-theoretic method for analyzing the mean Green's function in disordered quantum systems.
• To address the challenge of rigorously computing the density of states in the free spectrum of a random Schrödinger operator at weak disorder.
• To introduce an infrared cutoff to control long-wavelength fluctuations in the two-dimensional case.
• To establish a mathematically sound asymptotic expansion for the density of states using a polymer cluster expansion.
• To lay the foundation for a rigorous, non-perturbative analysis of localization and spectral properties in disordered systems.

Proposed method

• A polymer expansion is formulated with large and small field conditions to control fluctuations in the resolvent of the disordered Hamiltonian.
• The method applies to the mean resolvent (Green's function) of the Anderson model in two dimensions.
• An infrared cutoff is imposed on the disorder potential to suppress long-range correlations and ensure convergence.
• The expansion is analyzed using cluster expansion techniques from statistical field theory.
• The approach allows for the systematic resummation of contributions to the Green's function and density of states.
• The framework is applied to energies outside the unperturbed spectrum or within the free spectrum, under the infrared cutoff condition.

Experimental results

Research questions

• RQ1 Can a rigorous asymptotic expansion for the density of states be derived in the two-dimensional Anderson model at weak disorder?
• RQ2 How does the inclusion of an infrared cutoff affect the convergence and analyticity of the mean Green's function?
• RQ3 What is the role of polymer cluster expansion techniques in controlling the resolvent of a random Schrödinger operator?
• RQ4 Can the method handle both the spectral gap region and the free spectrum regime?
• RQ5 What are the mathematical conditions under which the mean Green's function admits a convergent expansion with controlled error bounds?

Key findings

• The polymer expansion with large/small field conditions successfully controls the mean resolvent in the two-dimensional Anderson model at weak disorder.
• An asymptotic expansion for the density of states is rigorously established for energies outside the unperturbed spectrum.
• The expansion remains valid for energies within the free spectrum when the potential includes an infrared cutoff.
• The method provides a non-perturbative, mathematically controlled framework for studying spectral properties in disordered systems.
• The results represent a foundational step toward a rigorous analysis of the density of states in the free spectrum of random Schrödinger operators.
• The infrared cutoff is essential for convergence and enables the treatment of extended states in two dimensions.