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arXiv:cond-mat/0312386 · Journal of Physics A Mathematical and General

Ground state of 1D bosons with delta interaction: link to the BCS model

Murray T. Batchelor, Xi-Wen Guan, James McGuire

Published Oct 7, 2004• 36 citations
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Abstract

The Bethe roots describing the ground-state energy of the integrable one-dimensional (1D) model of interacting bosons with weakly repulsive two-body delta interactions are seen to satisfy the set of Richardson equations appearing in the strong coupling limit of an integrable BCS pairing model. The BCS model describes boson–boson interactions with zero centre of mass momentum of pairs. It follows that the Bethe roots of the weakly interacting boson model are given by the zeros of Laguerre polynomials. The ground-state energy and the lowest excitation are obtained explicitly via the Bethe roots. A direct link has thus been established, in the context of integrable 1D models, between bosons interacting via weakly repulsive two-body delta interactions and strongly interacting BCS boson pairs.

DOI
10.1088/0305-4470/37/42/l01

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The Bethe roots describing the ground-state energy of the integrable one-dimensional (1D) model of interacting bosons with weakly repulsive two-body delta interactions are seen to satisfy the set of Richardson equations appearing in the strong coupling limit of an integrable BCS pairing model. The BCS model describes boson–boson interactions with zero centre of mass momentum of pairs. It follows that the Bethe roots of the weakly interacting boson model are given by the zeros of Laguerre polynomials. The ground-state energy and the lowest excitation are obtained explicitly via the Bethe roots. A direct link has thus been established, in the context of integrable 1D models, between bosons interacting via weakly repulsive two-body delta interactions and strongly interacting BCS boson pairs.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb–Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem.
Analytic expressions are derived for the root-mean-square (rms) radius of a pair of fermions in a BCS many-fermion state in one, two, and three dimensions, in terms of the BCS gap energy and the associated chemical potential. These expressions are valid for any coupling strength of any pair interaction model implying a momentum-independent gap energy. The latter holds, e.g., for an attractive \ensuremath{\delta} pair potential examined in the one-dimensional (1D) case (whose N-fermion ground state can be determined exactly) or for the BCS (electron-phonon) model interaction in any dimension. Weak-coupling and/or high-density limits for the rms radius are identical in 1D, 2D, and 3D, and reduce to the familiar well-known Pippard result to within a factor of order unity. In contrast, strong-coupling and/or low-density limits coincide in 1D and 3D, but differ by a factor of order unity in the 2D limit, and in each case are essentially the size of a single, isolated pair. The 1D \ensuremath{\delta} interaction McGuire-Yang-Gaudin many-fermion model is studied in detail. The interaction renormalization scheme of Miyake and of Randeria, Duan, and Shieh, and the BCS interaction model, both in 2D, are employed to analyze cuprate superconductor empirical results. Reasonable agreement between theoretical rms radii with experimental coherence lengths suggests that cuprates can be described moderately well as weakly coupled superconductors within the BCS-Bose formalism.