Density-functional theory for fermions close to the unitary regime

Abstract

We consider interacting Fermi systems close to the unitary regime and compute the corrections to the energy density that are due to a large scattering length and a small effective range. Our approach exploits the universality of the density functional and determines the corrections from the analyical results for the harmonically trapped two-body system. The corrections due to the finite scattering length compare well with the result of Monte Carlo simulations. We also apply our results to symmetric neutron matter.PACS numbers: 03.75. Ss,03.75.Hh,05.30.Fk,21.65.+f Ultracold fermionic atom gases have attracted a lot of interest since Fermi degeneracy was achieved by DeMarco and Jin [1]. These systems are in the metastable gas phase, as three-body recombinations are rare. Most interestingly, the effective two-body interaction itself can be controlled via external magnetic fields. This makes it possible to study the system as it evolves from a dilute Fermi gas with weak attractive interactions to a bosonic gas of diatomic molecules. This transition from a superfluid BCS state to Bose Einstein condensation (BEC) has been the subject of many experimental [2,3,4,5,6,7,8,9,10,11,12] and theoretical works [13,14,15,16,17,18,19,20,21,22,23].At the midpoint of this transition, the two-body system has a zero-energy bound state, and the scattering length diverges. If other parameters as the effective range of the interaction can be neglected, the interparticle spacing becomes the only relevant length scale. This defines the unitary limit. In this limit, the energy density is proportional of that of a free Fermi gas, the proportionality constant denoted by ξ. Close to the unitary limit, corrections are due to a finite, large scattering length a and a small effective range r 0 of the potential. Within the local density approximation (LDA), the energy density is given asHere,is the energy density of the free Fermi gas. . We are not aware of any estimate for the constant c 2 in Eq. (1) that concerns the correction due to a small effective range. It is the purpose of this work to fill this gap. This is particularly interesting as experiments also have control over the effective range. Note that the regime of a large effective range has recently been discussed by Schwenk and Pethick [20].In this work, we determine the coefficients c 1 , and c 2 via density functional theory. Recall that the density functional is supposed to be universal, i.e. it can be used to solve the N -fermion system for any particle number N , and for any external potential. Exploiting the universality of the density functional, the parameters c 1 and c 2 can be obtained from a fit to an analytically known solution, i.e. the harmonically trapped two-fermion system [33]. This simple approach has recently been applied [27] to determine the universal constant ξ, and will be followed and extended below.Let us briefly turn to the harmonically trapped twofermion system. The wave function u(r) in the relative coordinate r = r 1 − r 2 of the spin-singlet state is ...

Universal properties of infrared oscillator basis extrapolations

“…Bang et al [12] used the method of harmonic oscillator representation of scattering equations (HORSE) for this purpose, and more recent works [13,14] computed phase shifts to develop an EFT for nuclear interactions directly in the oscillator basis [10]. References [13,14] build on the results by Busch et al [15] and their generalization [16] to finite range corrections, and extract scattering information from the energy shifts of bound states in a harmonic oscillator potential. The resulting EFTs are quite efficient for contact interactions and systems such as ultracold trapped fermions, but nuclear potentials with a finite range require an extrapolation of Ω → 0 [13].…”

Ground State Energy of Unitary Fermion Gas with the Thomson Problem Approach

“…The corresponding energy scale is the Fermi kinetic energy ε f = k 2 f /(2m), while m is the fermion mass. According to dimensional analysis, the system details do not contribute to the thermodynamics properties, i.e., the thermodynamics properties are universal [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The energy density should be proportional to that of a free Fermi gas E/V = ξ (E/V ) f ree = ξ How to approach the exact value of ξ analytically is a bewitching topic in the Fermi-Dirac statistical physics.…”

Occupation-number-based energy functional for nuclear masses

“…In Eq. (27), n * (c), z * (c) denote the optimal occupation numbers (we suppress that they depend on the label i) and are taken from the domain…”

“…However, the form of the nonlinear lower-level minimization in Eq. (27) does not satisfy standard regularity conditions that would ensure existence and continuity of the derivatives ∂n * (c) ∂cj and ∂z * (c) ∂cj (see, e.g., [50]). Thus, unavailability of the residual derivatives in our case comes from the dependence of the optimal occupation numbers n * , z * on the coefficients c.…”

“…Furnstahl and colleagues [22][23][24] derived energy-density functionals for dilute Fermi gases with short-ranged interactions. For Fermi gases in the unitary regime, simple scaling arguments suggest the form of the energy-density functional [25][26][27][28]. We similarly use scaling arguments in the development of our functional.…”