Computing Exact Self-Energies with Polynomial Expansion

TL;DR

This paper presents a method using Chebyshev polynomial expansion to accurately compute spectral and Green's functions for finite quantum systems, including impurity models and systems with leads, capturing key physical phenomena.

Contribution

It introduces a detailed approach for calculating self-energies and spectral functions with polynomial expansion, exploiting symmetries to reduce computational complexity, and extends to systems connected to leads.

Findings

Successfully captures aspects of Kondo physics in the Anderson model

Demonstrates the effectiveness of polynomial expansion for spectral calculations

Provides a methodology for systems with infinite leads using finite-system self-energies

Abstract

We give details on how to calculate spectral functions and Green's functions for finite systems using the Chebyshev polynomial expansion method. We apply the method to a finite Anderson impurity system, and furthermore give details on how to exploit its symmetry to transform the system into smaller orthogonal subsystems. We also consider systems connected to infinite leads, which we study by approximating the unknown self-energy with an exact self-energy for a finite system. As our test case, we consider the single-impurity Anderson model, where we find that we can capture some aspects of Kondo physics.