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arXiv:0705.1502 · Journal of Physics A Mathematical and Theoretical

Three fermions in a box at the unitary limit: universality in a lattice model

Ludovic Pricoupenko, Yvan Castin

Published Oct 9, 2007• 31 citations
arXivHTMLPDF

Abstract

We consider three fermions with two spin components interacting on a lattice model with an infinite scattering length. Low-lying eigenenergies in a cubic box with periodic boundary conditions, and for a zero total momentum, are calculated numerically for decreasing values of the lattice period. The results are compared to the predictions of the zero-range Bethe–Peierls model in continuous space, where the interaction is replaced by contact conditions. The numerical computation, combined with analytical arguments, shows the absence of negative energy solution, and a rapid convergence of the lattice model towards the Bethe–Peierls model for a vanishing lattice period. This establishes for this system the universality of the zero-interaction range limit.

DOI
10.1088/1751-8113/40/43/003

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We consider three fermions with two spin components interacting on a lattice model with an infinite scattering length. Low-lying eigenenergies in a cubic box with periodic boundary conditions, and for a zero total momentum, are calculated numerically for decreasing values of the lattice period. The results are compared to the predictions of the zero-range Bethe–Peierls model in continuous space, where the interaction is replaced by contact conditions. The numerical computation, combined with analytical arguments, shows the absence of negative energy solution, and a rapid convergence of the lattice model towards the Bethe–Peierls model for a vanishing lattice period. This establishes for this system the universality of the zero-interaction range limit.
The properties of two-component Fermi gases become universal if the interspecies sss-wave scattering length as{a}_{s}as​ and the average interparticle spacing are much larger than the range of the underlying two-body potential. Using an explicitly correlated Gaussian basis set expansion approach, we determine the eigenenergies of two-component Fermi gases in a cubic box with periodic boundary conditions as functions of the interspecies sss-wave scattering length and the effective range of the two-body potential. The universal properties of systems consisting of up to four particles are determined by extrapolating the finite-range energies to the zero-range limit. We determine the eigenenergies of states with vanishing and finite momenta. In the weakly attractive BCS regime, we analyze the energy spectra and degeneracies using first-order degenerate perturbation theory. Excellent agreement between the perturbative energy shifts and the numerically determined energies is obtained. For the infinitely large scattering length case, we compare our results---where available---with those presented in the literature.
We investigate the e↵ect of a spherical cuto↵ in the single particle momentum on the lattice calculations of unitary fermions, and use this to explore the ground state configuration. The unitarity limit, corresponding to an infinite s -wave scattering length and zero interaction range, is achieved by tuning the lattice coupling of four-fermi interactions to reproduce the scattering phase shift of σ = π/2 in the continuum. We particularly compute the ground state energies of three unitary fermions at zero and non-zero net momenta. We show that in the continuum limit the spherical cuto↵ does not a↵ect the results, while it violates the unitarity for a boosted pair of fermions. Our findings suggest that the pair of fermions in the ground state of three unitary fermions has zero momentum, while the total net momentum of the system is carried by the unpaired fermion.
We investigate the e↵ect of a spherical cuto↵ in the single particle momentum on the lattice calculations of unitary fermions, and use this to explore the ground state configuration. The unitarity limit, corresponding to an infinite s-wave scattering length and zero interaction range, is achieved by tuning the lattice coupling of four-fermi interactions to reproduce the scattering phase shift of σ = π/2 in the continuum. We particularly compute the ground state energies of three unitary fermions at zero and non-zero net momenta. We show that in the continuum limit the spherical cuto↵ does not a↵ect the results, while it violates the unitarity for a boosted pair of fermions. Our findings suggest that the pair of fermions in the ground state of three unitary fermions has zero momentum, while the total net momentum of the system is carried by the unpaired fermion.
We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
Abstract We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the celebrated (and ubiquitous in the literature of this field) Bethe–Peierls contact condition with given two-body scattering length within the Krein–Višik–Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system.