Bethe-ansatz study of the Bose-Fermi mixture

TL;DR

This paper analytically investigates a one-dimensional Bose-Fermi mixture with contact interactions using Bethe-ansatz, deriving exact results for excitation velocities, Drude weights, and their relations, highlighting the model's integrability and low-energy properties.

Contribution

It provides an exact Bethe-ansatz solution for the Bose-Fermi mixture, deriving analytical expressions for excitation velocities and Drude weights, and explores their physical implications.

Findings

Calculated excitation velocities from compressibility and Drude weight matrices.

Derived exact expressions for the Drude weight matrix elements.

Confirmed the consistency with Galilean invariance and momentum coupling.

Abstract

We consider a one-dimensional mixture of bosons and spinless fermions with contact interactions. In this system, the elementary excitations at low energies are described by four linearly dispersing modes characterized by two excitation velocities. Here we study the velocities in a system with equal interaction strengths and equal masses of bosons and fermions. The resulting model is integrable and admits an exact Bethe-ansatz solution. We analyze it and analytically derive various exact results, which include the Drude weight matrix. We show that the excitation velocities can be calculated from the knowledge of the matrices of compressibility and the Drude weights, as their squares are the eigenvalues of the product of the two matrices. The elements of the Drude weight matrix obey certain sum rules as a consequence of Galilean invariance. Our results are consistent with the presence of a momentum-momentum coupling term between the two subsystems of bosons and fermions in the effective low-energy Hamiltonian. The analytical method developed in the present study can be extended to other models that possess a nested Bethe-ansatz structure.