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Journal of Mathematical Physics

Boson–fermion model with two-body interactions: Exact results

Y. Wang, Fu‐Cho Pu

Published Nov 1, 1996• 0 citations
Publisher

Abstract

A one-dimensional boson–fermion model with two-body interactions between the two types of chiral fermions is considered. It is shown that the model is exactly soluble and the general Bethe eigenstates are constructed. On the basis of the Bethe ansatz equations, the ground state, the low lying elementary excitations, and the thermodynamics are also given in some closed integral equations.

DOI
10.1063/1.531712

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A one-dimensional boson–fermion model with two-body interactions between the two types of chiral fermions is considered. It is shown that the model is exactly soluble and the general Bethe eigenstates are constructed. On the basis of the Bethe ansatz equations, the ground state, the low lying elementary excitations, and the thermodynamics are also given in some closed integral equations.
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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.
We develop a bosonization approach for one-dimensional models based on Bethe ansatz equations. The operator formalism of the exact soluble models in the low energy limit provides a systematic method to calculate the asymptotic correlation functions. As examples with and without internal degrees of freedom, the Calogero-Sutherland (C-S) model and the repulsive Hubbard model are considered respectively. We verify that the low energy behavior of the C-S model is controlled by two classes of c=1 conformal field theories, depending on whether the C-S interactions are among bosons or among fermions. For the Hubbard model, we show the explicit charge-spin separation at low energy for arbitrary U>0. The low energy behavior of the system is described by the (semi-) direct product of two independent Virasoro algebras with c=1.