# Many-impurity phonon Casimir effect in atomic chains

## TL;DR

This paper introduces a new approach to analyze the long-range phonon Casimir interactions among multiple impurities in atomic chains at various temperatures, extending beyond the two-impurity case.

## Contribution

It develops a method to study many-impurity phonon Casimir effects at zero and finite temperatures, advancing understanding of complex impurity interactions.

## Key findings

- Extended the analysis of phonon Casimir interactions to multiple impurities.
- Demonstrated the temperature dependence of impurity interactions.
- Provided a framework for future studies of impurity effects in condensed matter.

## Abstract

Phonon Casimir effect is the long-range interaction between impurities in condensed matter systems, mediated by vacuum fluctuations of the phonon field. For pairs of impurities, this interaction has been shown to follow a quasi-power law at zero-temperature and evolve into an exponentially decaying form as the temperature is increased. This work introduces an approach to deal with systems of more than two impurities, both at zero and finite temperatures.