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arXiv:hep-th/9606182 · Physical Review A

δ-function spin-1/2 fermions in a one-dimensional potential well

You‐Quan Li, Jianhui Dai

Published Jun 1, 1996• 4 citations
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Abstract

The quantum-mechanical problem of N fermions with \ensuremath{\delta}-function interaction in a one-dimensional potential well of finite depth is solved. It is shown that there exists exact wave function of Bethe-ansatz form in the case that a single particle tunnels outside of the well. The Bethe-ansatz-like secular equations for the spectrum are obtained. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.3743

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The quantum-mechanical problem of N fermions with \ensuremath{\delta}-function interaction in a one-dimensional potential well of finite depth is solved. It is shown that there exists exact wave function of Bethe-ansatz form in the case that a single particle tunnels outside of the well. The Bethe-ansatz-like secular equations for the spectrum are obtained. \textcopyright{} 1996 The American Physical Society.
The quantum-mechanical problem of bosons, in a square well in one dimension and interacting with a \ensuremath{\delta} function, is analyzed and solved. The resulting Bethe-ansatz wave function is a sum, over the Weyl group of Lie algebra BN{\mathit{B}}_{\mathit{N}}BN​, of nondiffractive waves. The secular equation of the spectrum is obtained and its thermodynamic limit is discussed.
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 ground state of a one-dimensional δ-function attractive spin-1/2 fermions is studied by the nested Bethe ansatz method. Explicitly, the Gaudin integral equation for the system is solved in the form of power series with arbitrary spin polarization. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities, such as the ground state energy and the chemical potentials, are expressed in terms of the dimensionless coupling constant and the polarization , where c is the coupling constant and are the number densities of the spin-up (down) particles. While in the limiting case P = 1, the system consists of spinless free fermions, the other limit P = 0 describes the BCS–BEC crossover in one dimension. As a function of P , the evolution is continuous and smooth.
We investigate the ground-state properties of a one-dimensional spin-1/2 fermionic atoms interacting through the attractive δ-function potential in an external magnetic field. By the thermal Bethe ansatz method, the integral equations for the dressed energies are formulated. Those integral equations at zero-temperature are solved in the power series forms of rapidity and momentum for both strong and weak coupling cases. The magnetization as a function of the coupling constant and the external field is also obtained explicitly. Based on the analytic results, quantum phase transitions are identified among three phases, an unpolarized fully paired state, a fully polarized normal ferromagnetic state and a mixed state of paired and unpaired atoms.
The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.