Exact solutions of two complementary 1D quantum many-body systems on the half-line

TL;DR

This paper presents exact Bethe Ansatz solutions for two complementary 1D quantum many-body models on the half-line, revealing dualities and physical interpretations, with one model related to the massive Thirring model.

Contribution

It provides the first exact solutions for these models with boundary conditions, including a duality relation and physical insights, expanding the understanding of integrable systems with boundaries.

Findings

Exact Bethe Ansatz solutions for both models.

Duality between bosonic and fermionic models.

Physical interpretation related to the massive Thirring model.

Abstract

We consider two particular 1D quantum many-body systems with local interactions related to the root system $C_N$. Both models describe identical particles moving on the half-line with non-trivial boundary conditions at the origin, and they are in many ways complementary to each other. We discuss the Bethe Ansatz solution for the first model where the interaction potentials are delta-functions, and we find that this provides an exact solution not only in the boson case but even for the generalized model where the particles are distinguishable. In the second model the particles have particular momentum dependent interactions, and we find that it is non-trivial and exactly solvable by Bethe Ansatz only in case the particles are fermions. This latter model has a natural physical interpretation as the non-relativistic limit of the massive Thirring model on the half-line. We establish a duality relation between the bosonic delta-interaction model and the fermionic model with local momentum dependent interactions. We also elaborate on the physical interpretation of these models. In our discussion the Yang-Baxter relations and the Reflection equation play a central role.