The Kernel Polynomial Method

TL;DR

This paper reviews the Kernel Polynomial Method and Chebyshev expansion algorithms, highlighting their efficiency, stability, and broad applicability in computational condensed matter physics and related fields.

Contribution

It provides a comprehensive overview of the properties, recent developments, and applications of the Kernel Polynomial Method and Chebyshev expansion algorithms.

Findings

Linear resource scaling with problem size

Successful application in disordered systems and correlated electrons

Integration with techniques like Monte Carlo and Cluster Perturbation Theory

Abstract

Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of Chebyshev expansion based algorithms and the Kernel Polynomial Method. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems we discuss in detail. In addition, we illustrate how the Kernel Polynomial Method is successfully embedded into other numerical techniques, such as Cluster Perturbation Theory or Monte Carlo simulation.