The two-boundary sine-Gordon model

TL;DR

This paper analyzes the ground state energy of a free bosonic system on a finite interval with boundary interactions of sine-Gordon or Kondo type, addressing technical challenges in applying Bethe ansatz techniques.

Contribution

It provides detailed methods for handling complex analytic continuations in Bethe ansatz solutions for boundary sine-Gordon and Kondo models.

Findings

Successful application of Bethe ansatz with complex continuations

Detailed analysis of boundary interaction effects

Potential implications for condensed matter and string theory

Abstract

We study in this paper the ground state energy of a free bosonic theory on a finite interval of length $R$ with either a pair of sine-Gordon type or a pair of Kondo type interactions at each boundary. This problem has potential applications in condensed matter (current through superconductor-Luttinger liquid-superconductor junctions) as well as in open string theory (tachyon condensation). While the application of Bethe ansatz techniques to this problem is in principle well known, considerable technical difficulties are encountered. These difficulties arise mainly from the way the bare couplings are encoded in the reflection matrices, and require complex analytic continuations, which we carry out in detail in a few cases.