Shell-Model Monte Carlo Simulations of BCS-BEC Crossover in Few-Fermion Systems

The physics of ultracold atomic gases has been intensively pursued experimentally and theoretically in the last decade. Recently there has been great interest in strongly-interacting Fermi gases where Feshbach resonances allow the tuning of the two-body interaction, and studies of the transition from a dilute gas of fermionic atoms to a Bose condensate of molecules are now possible in the laboratory [1,2,3,4,5,6]. While studies of degenerate Fermi gases have mostly dealt with large atom numbers and wide traps, efforts have begun to trap only a few atoms (1-100) in tighter traps [7]. Also, with the implementation of three-dimensional optical lattices, a low-tunneling regime can be reached with essentially isolated harmonic oscillators containing only a few fermions at each site [8]. This means that one can now explore few-body fermionic effects in trapped systems with scattering lengths that are comparable to the inter-particle distance and the trap width.

In this paper we report on a theoretical study of harmonically trapped fermions using the shell-model Monte Carlo (SMMC) approach. This method has been extensively used in nuclear physics to determine nuclear properties at finite temperature in larger model spaces than can be handled by normal nuclear shell-model diagonalization [9,10]. In the SMMC, the many-body problem is described by a canonical ensemble at temperature k B T = β -1 and the Hubbard-Stratonovich transformation is used to linearize the imaginary-time many- * Electronic address: zinner@phys.au.dk body propagator e -βH . Observables are then expressed as path integrals of one-body propagators in fluctuating auxiliary fields. The method is in principle exact and subject only to statistical uncertainties. For equal mixtures of two hyperfine states at low density, the interaction can be modeled with an s-wave zero-range potential. Importantly, this interaction is free of sign problems [11] that are otherwise known to plague quantum Monte Carlo simulations with fermions. We present here the first application of this many-body method to ultracold gas physics.

Previous works have considered few-fermion systems using advanced many-body methods. The Green’s function Monte Carlo methods were applied to homogeneous [12], as well as trapped systems [13]. No-core [14], and traditional shell-models [15], using effective interactions have also recently been applied to these systems, particularly for very low particle numbers where exact results are available [16,17]. Finite-temperature, non-perturbative lattice methods have also been applied to the homogeneous case [18,19]. These works mostly focus on the unitary |a| → ∞ limit and the crossover regime around it. Most previous Monte Carlo approaches have used fixed nodes in the many-body wave function in order to alleviate the sign problem, making the methods variational. As we will now demonstrate the present method has no sign problem [11] and can be used on the BCS side, in the crossover region, and also into the BEC regime.

The model Hamiltonian used is

where we sum over all particles i and [ij] denotes a sum over fermion pairs with opposite internal (hyperfine) states. The trap frequency is ω, and V 0 denotes the interaction strength. The SMMC method was originally set up to handle nucleons where the Hamiltonian above appears in extensively studied pairing problems in nuclei [20]. The two-component Fermi gas can now be mapped onto a single spin 1/2 nucleon species [11,21]. Dimensional arguments reveal that the matrix elements needed in a shell-model approach scale as 1/b 3 , where b = h/mω is the oscillator length. It is therefore natural to redefine the interaction strength in terms of V 0 = -ghωb 3 , where g is a dimensionless strength measure. In order to relate to physical quantities the strength of the zero-range interaction must be regularized [22]. The shell-model works in finite model spaces and this naturally introduces a cut-off in energy E c = α 2 hω, where α 2 = N max + 3/2. Regularization in finite model spaces with discrete energies is notoriously difficult and several prescriptions have been adopted in the literature. Here we will use the simplest strategy and renormalize the coupling g through low-energy scattering parameters defined in the continuum [23]. This defines a relation between g, the s-wave scattering length a, and E c , which is 4πa/b = g/(αg/2π 2 -1). This yields the effective interaction strength for a model space with N max + 1 major harmonic oscillator shells and gives a prescription for varying the interaction strength with model space size in order to keep a fixed.

The relevant parameter regime is expected to be where the natural energy scale, given by the level spacing hω, is comparable to typical two-body matrix elements between the trap states at T = 0 (as a quantitative measure we use 1s1s|V |1s1s ). This turns out to be around g ∼ 10 in our setup at N max = 3, which corresponds to a = 11b in the continuum re

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