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arXiv:hep-th/9905120

A Condensation of Interacting Bosons in Two Dimensional Space

S. G. Rajeev

Published May 17, 1999• 0 citations
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Abstract

We develop a theory of non-relativistic bosons in two spatial dimensions with a weak short range attractive interaction. In the limit as the range of the interaction becomes small, there is an ultra-violet divergence in the problem. We devise a scheme to remove this divergence and produce a completely finite formulation of the theory. This involves reformulating the dynamics in terms of a new operator whose eigenvalues give the {\it logarithm} of the energy levels. Then, a mean field theory is developed which allows us to describe the limit of a large number of bosons. The ground state is a new kind of condensate (soliton) of bosons that breaks translation invariance spontaneously. The ground state energy is negative and its magnitude grows {\it exponentially} with the number of particles, rather than like a power law as for conventional many body systems.

DOI
10.48550/arxiv.hep-th/9905120

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We develop a theory of non-relativistic bosons in two spatial dimensions with a weak short range attractive interaction. In the limit as the range of the interaction becomes small, there is an ultra-violet divergence in the problem. We devise a scheme to remove this divergence and produce a completely finite formulation of the theory. This involves reformulating the dynamics in terms of a new operator whose eigenvalues give the {\it logarithm} of the energy levels. Then, a mean field theory is developed which allows us to describe the limit of a large number of bosons. The ground state is a new kind of condensate (soliton) of bosons that breaks translation invariance spontaneously. The ground state energy is negative and its magnitude grows {\it exponentially} with the number of particles, rather than like a power law as for conventional many body systems.
We investigate the ground-state properties of N bosons with attractive zero-range interactions characterized by the scattering length a > 0 and confined to the surface of a sphere of radius R. We present the analytic solution of the problem for N = 2, mean-field analysis for N→∞, and exact diffusion Monte Carlo results for intermediate N. For finite N, we observe a smooth crossover from the uniform state in the limit a/R≫1 (weak attraction) to a localized state at small a/R (strong attraction). With increasing N, this crossover narrows down to a discontinuous transition from the uniform state to a soliton of size ∼R/N. The two states are separated by an energy barrier, tunneling under which is exponentially suppressed at large N. The system behavior is marked by a peculiar competition between space-curvature effects and beyond-mean-field terms, both breaking the scaling invariance of a two-dimensional mean-field theory.
The isotropic scattering phase shift is calculated for nonrelativistic bosons interacting at low energies via an arbitrary finite-range potential in ddd space-time dimensions. Scattering on a $(d\ensuremath{-}1)$-dimensional torus is then considered, and the eigenvalue equation relating the energy levels on the torus to the scattering phase shift is derived. With this technology in hand, and focusing on the case of two spatial dimensions, a perturbative expansion is developed for the ground-state energy of NNN identical bosons which interact via an arbitrary finite-range potential in a finite area. The leading nonuniversal effects due to range corrections and three-body forces are included. It is then shown that the thermodynamic limit of the ground-state energy in a finite area can be taken in closed form to obtain the energy per particle in the low-density expansion by explicitly summing the parts of the finite-area energy that diverge with powers of NNN. The leading and subleading finite-size corrections to the thermodynamic limit equation of state are also computed. Closed-form results---some well known, others perhaps not---for two-dimensional lattice sums are included in an Appendix.
We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)
A non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two-dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean-field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan–Symanzik equation) for the principal operator of the model is derived and the β function is exactly calculated for the general case, which includes all particle numbers.
We consider a system of N nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for N large, but below some critical value, the properties of the N-boson bound state are universal. In particular, the ratio of the binding energies of (N+1)- and N-boson systems, B(N+1)/B(N), approaches a finite limit, approximately 8.567, at large N. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state B(0)(3)=16.522 688(1)B2 and for the excited state B(1)(3)=1.270 409 1(1)B2.