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Physical Review C

Spectral functions in an exactly solvable self-boundA-body system

Dimitri Van Neck, S. Rombouts, S. Verdonck

Published Nov 29, 2005• 2 citations
Publisher

Abstract

We consider an exactly solvable Hamiltonian for bosons in one-dimension interacting through zero-range attractive forces, and construct a complete basis of its A-particle eigenstates. The structure of the single-particle spectral function in the removal domain is investigated, by taking the overlap of the A-particle ground state with the various excited states of the $(A\ensuremath{-}1)$ system. In particular we study the contribution to the spectral function of the different break-up channels in the $A\ensuremath{-}1$ continuum, and compare the results to general statements available in the literature. It is shown that the asymptotic behavior in coordinate space does not agree with conventional assumptions. The relation to recent $(e,{e}^{'}p)$ experiments at large values of missing energy and momentum is pointed out.

DOI
10.1103/physrevc.72.054318

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We consider an exactly solvable Hamiltonian for bosons in one-dimension interacting through zero-range attractive forces, and construct a complete basis of its A-particle eigenstates. The structure of the single-particle spectral function in the removal domain is investigated, by taking the overlap of the A-particle ground state with the various excited states of the (A\ensuremath−1)(A\ensuremath{-}1)(A\ensuremath−1) system. In particular we study the contribution to the spectral function of the different break-up channels in the A\ensuremath−1A\ensuremath{-}1A\ensuremath−1 continuum, and compare the results to general statements available in the literature. It is shown that the asymptotic behavior in coordinate space does not agree with conventional assumptions. The relation to recent (e,e′p)(e,{e}^{'}p)(e,e′p) experiments at large values of missing energy and momentum is pointed out.
We consider a simple but nontrivial many-body system interacting through short-range forces, and confirm a property of such systems that was recently proposed on general grounds, namely, that single-particle overlap functions, spectroscopic factors, and separation energies of the bound (A-1)-particle eigenstates can be derived from the one-body density matrix of the A-particle system in its ground state. The basis of natural orbitals for the system is constructed and its properties are discussed. We also investigate the high-momentum content of the bound-state overlap functions and momentum distribution. It is found that the mean field provides a good approximation for the bound-state overlap functions even in the region of large momenta, where the total momentum distribution is already enhanced by several orders of magnitude over the mean-field result. \textcopyright{} 1996 The American Physical Society.
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.
A one-parameter solvable model for three bosons subject to $\ensuremath{\delta}$-function attractive interactions in one dimension with periodic boundary conditions is studied. The energy levels and wave functions are classified and given explicitly in terms of three momenta. In particular, eigenstates and eigenvalues are described as functions of the model parameter ccc. Some of the states are given in terms of complex momenta and represent dimer or trimer configurations for large negative ccc. The asymptotic behavior for small and large values of the parameter, and at thresholds between real and complex momenta, is provided. The properties of the potential energy are also discussed.
We investigate the single-particle spectral density of interacting bosons within the nonperturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where the perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation, which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasifree bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozi\`eres and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.
We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes à la Kreĭn–Višik–Birman and à la von Neumann. We produce a class of canonical models for which we also analyze the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e. sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularization that prevents such spectral instability while retaining an effective short-scale pattern. Besides the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.
The exactly solvable Lieb-Liniger model of interacting bosons in one dimension has attracted renewed interest as current experiments with ultracold atoms begin to probe this regime. Here we numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to 20 particles for attractive interactions. We discuss the features of the solutions, and how they deviate from the well-known string solutions [Thacker, Rev. Mod. Phys. 53, 253 (1981)] at finite densities. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Finally we compare our results to those of exact diagonalization of the many-body Hamiltonian in a truncated basis. We also present excited state solutions and the excitation spectrum for the repulsive one-dimensional Bose gas on a ring.