# Field-theoretical approach to the Casimir-like interaction in a one-dimensional Bose gas

## TL;DR

This paper analyzes the fluctuation-induced interaction between impurities in a one-dimensional Bose gas using field theory, revealing different behaviors at various distances and temperatures, and discussing experimental implications.

## Contribution

It provides analytical expressions for impurity interactions in a 1D Bose gas at zero and finite temperatures, highlighting classical and quantum regimes.

## Key findings

- Interaction is exponential at short distances due to classical effects.
- At long distances, the interaction scales as the inverse cube of the separation.
- Finite temperature suppresses the interaction beyond the thermal length.

## Abstract

We study the fluctuation-induced interaction between two impurities in a weakly-interacting one-dimensional Bose gas using the field theoretical approach. At separations between impurities shorter and of the order of the healing length of the system, the induced interaction has a classical origin and behaves exponentially. At separations longer than the healing length, the interaction is of a quantum origin and scales as the third power of the inverse distance. Finite temperature destroys the quasi-long-range order of the Bose gas and, accordingly, the induced interaction becomes exponentially suppressed beyond the thermal length. We obtain analytical expressions for the induced interaction at zero and finite temperature that are valid at arbitrary distances. We discuss experimental realizations as well as possible formation of bound states of two impurities, known as bipolarons.