A pedagogical derivation of dynamical susceptibilities

TL;DR

This paper provides a clear, pedagogical derivation of dynamical susceptibilities in condensed matter physics, making complex response functions more accessible for students and researchers.

Contribution

It introduces a direct derivation method for dynamical susceptibilities in model Hamiltonians and offers an alternative derivation for the irreducible vertex in the Falicov-Kimball model.

Findings

Simplified derivation of dynamical susceptibilities

Alternative derivation of the irreducible vertex in the Falicov-Kimball model

Enhanced understanding of response functions in condensed matter physics

Abstract

Dynamical two-particle susceptibilites are important for a wide range of different experiments in condensed-matter physics and beyond. Nevertheless, most textbooks avoid describing how to derive such response functions, perhaps because they are viewed as too complex. In the literature, most derivations work with generalized susceptibilities, which are more general, but require an even higher layer of complexity. In this work, we show a more direct derivation in the context of model Hamiltonians which can be mapped directly onto an impurity model. We also present an alternative derivation for the irreducible vertex in the context of the Falicov-Kimball model.