# The Calculation of Two-Particle Quantities in the Typical Medium Dynamical Cluster Approximation

## TL;DR

This paper introduces a method within the Typical Medium Dynamical Cluster Approximation to accurately compute two-particle quantities, including vertex corrections, enabling better characterization of metal-insulator transitions in disordered electron systems.

## Contribution

It develops a novel approach to include vertex corrections and mobility edge information in the TMDCA for analyzing disordered systems.

## Key findings

- Properly characterizes the metal-insulator transition.
- Incorporates vertex corrections into the mean-field framework.
- Demonstrates the method's effectiveness in disordered systems.

## Abstract

The mean-field theory for disordered electron systems without interaction is widely and successfully used to describe equilibrium properties of materials over the whole range of disorder strengths. However, it fails to take into account the effects of quantum coherence and information of localization. Vertex corrections due to multiple back-scatterings may drive the electrical conductivity to zero and make expansions around the mean field in strong disorder problematic. Here, we present a method for the calculation of two-particle quantities which enable us to characterize the metal-insulator transitions (MIT) in disordered electron systems by using the Typical Medium Dynamical Cluster Approximation (TMDCA). We show how to include vertex corrections and the information about the mobility edge to the typical mean-field theory. We successfully demonstrate the application of the developed method by showing that the conductivity formulated in this way properly characterizes the MIT in disordered systems.