# T-matrix approach to the phonon-mediated Casimir interaction

## TL;DR

This paper develops a T-matrix based theory for phonon-mediated Casimir interactions between impurities, revealing power-law decay at zero temperature and exponential decay at finite temperature, with differences depending on scattering regimes.

## Contribution

It introduces a novel T-matrix approach to analyze phonon-mediated Casimir interactions, connecting scattering properties to interaction decay behaviors.

## Key findings

- Casimir interaction decays as a power law at zero temperature.
- Interaction becomes exponential at large distances at finite temperature.
- Decay power varies between weak and strong scattering regimes.

## Abstract

We develop a theory of the phonon mediated Casimir interaction between two point-like impurities, which is based on the single impurity scattering T-matrix approach. Within this, we show that the Casimir interaction at $T = 0$ falls off as a power law with the distance between the impurities. We find that the power in the weak and in the unitary phonon-impurity scattering limits differs, and we relate the power law to the low-energy properties of the single impurity scattering T-matrix. In addition, we consider the Casimir interaction at finite temperature and show that at finite temperatures the Casimir interaction becomes exponential at large distances.