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Universal Properties of the Four-Boson System in Two Dimensions

Platter, L., H.‐W. Hammer, Ulf-G. Meißner

Published Jan 1, 2004• 16 citations
OpenAlex

Abstract

We consider the nonrelativistic four-boson system in two dimensions interacting via a short-range attractive potential. For a weakly attractive potential with one shallow two-body bound state with binding energy B-2, the binding energies B-N of shallow N-body bound states are universal and thus do not depend on the details of the interaction potential. We compute the four-body binding energies in an effective quantum mechanics approach. There are exactly two bound states: the ground state with B-4((0)) = 197.33(1)B-2 and one excited state with B-4((1)) = 25.5 (1)B-2. We compare our results to recent predictions for N-body bound states with large N much greater than 1.

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We consider the nonrelativistic four-boson system in two dimensions interacting via a short-range attractive potential. For a weakly attractive potential with one shallow two-body bound state with binding energy B-2, the binding energies B-N of shallow N-body bound states are universal and thus do not depend on the details of the interaction potential. We compute the four-body binding energies in an effective quantum mechanics approach. There are exactly two bound states: the ground state with B-4((0)) = 197.33(1)B-2 and one excited state with B-4((1)) = 25.5 (1)B-2. We compare our results to recent predictions for N-body bound states with large N much greater than 1.
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.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to 4He^{4}\mathrm{H}\mathrm{e}4He atoms and calculate the binding energy of the 4He^{4}\mathrm{H}\mathrm{e}4He tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to 4He^{4}\mathrm{H}\mathrm{e}4He atoms, the correlation is approximately linear.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
Near two-body unitarity, the three-boson system is characterized by an approximate discrete scale invariance manifest in a geometric tower of bound states (the Efimov effect). In the absence of a strong four-body force, the four-boson system has two states associated with each Efimov state, one very nearly unstable, the other several times deeper. We study correlations between the excited- and ground-state properties, such as binding energies and radii, up to next-to-leading order in an effective field theory for short-range forces. We obtain the parameters in these correlations from similar correlations arising from existing precise calculations based on short-range potentials. We also derive correlations among excited-state properties that emerge from the proximity of the state to the break-up threshold into a boson and a three-boson bound state, using an effective field theory for ``halo'' states.
The presence of few-body bound states is an important issue in the study of interacting bosonic systems due to the instability it introduces into the condensation. The problem is more interesting in two dimensions because the system can host only a finite number of bound states, in contrast to the Efimov effect in three dimensions. The authors considered a two-component bosonic system in two dimensions, with repulsive intra- and attractive inter-species interactions. They found that there is only one three-particle bound state and the binding energies for various scattering lengths fall on an universal curve, where the microscopic details are of no relevance. Their conclusion is pertinent for various systems such as excitons in quantum well structures and bosonic dipoles in bilayer geometries. They developed an efficient renormalization group type method, capable of handling both short- and long- range physics and free of numerical instabilities. This method also makes use of the Hopf coordinates and running basis to reduce the proliferation of hyperspherical harmonics appearing in alternative approaches, yielding significant saving in numerical implementations.