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Physical Review A

Ground-State Energy of a Mixture of Fermions and Bosons in One Dimension with a Repulsive δ -Function Interaction

Cheng Kee Lai, Chen Ning Yang

Published Jan 1, 1971• 64 citations
Publisher

Abstract

The ground-state energy of a mixture of spin-\textonehalf{} fermions and bosons in one dimension, interacting with a repulsive $\ensuremath{\delta}$-function potential, is analyzed. The wave function is given by repeated use of a generalized Bethe hypothesis. The "momenta" in the hypothesis are determined by coupled Fredholm integral equations. Numerical solutions are given. No phase separation is found.

DOI
10.1103/physreva.3.393

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The ground-state energy of a mixture of spin-\textonehalf{} fermions and bosons in one dimension, interacting with a repulsive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential, is analyzed. The wave function is given by repeated use of a generalized Bethe hypothesis. The "momenta" in the hypothesis are determined by coupled Fredholm integral equations. Numerical solutions are given. No phase separation is found.
The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
The repulsive $\ensuremath{\delta}$ interaction problem in one dimension for NNN particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations RRR of the permutation group SN{S}_{N}SN​. For some R′sR'\mathrm{s}R′s this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-\textonehalf{} fermions is reduced to a generalized Fredholm equation.
We consider a gas mixture consisting of spinless fermions and bosons in one dimension interacting via a repulsive [Formula: see text]-function potential. Bosons and fermions are assumed to have equal masses and the interaction strength between bosons and among bosons and fermions is the same. Using the Bethe ansatz solution of the model, we study the ground state properties, the dressed energy potentials for the two bands of rapidities, the elementary particle and hole excitations, the thermodynamics, the finite size corrections to the ground state energy leading to the conformal towers, and the asymptotic behavior at large distances of some relevant correlation functions. The low-energy excitations of the system form a two-component Luttinger liquid. In an elongated optical trap the gas phase separates as a function of the distance from the center of the trap.
The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.