Large-Nexpansion for unitary superfluid Fermi gases

Abstract

We analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number N of spin-1/2 fermion flavors with Sp(2N ) symmetry. For N → ∞ this problem is exactly solved by the BCS-BEC mean-field theory, with corrections small in the parameter 1/N . The large-N expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of N = 1, among other quantities, we predict in the unitarity regime, the energy of the gas to be ξ = 0.28 times that for the non-interacting gas and the pairing gap to be 0.52 times the Fermi energy.

Radio-frequency spectroscopy of a strongly imbalanced Feshbach-resonant Fermi gas

“…However, we stress that, given that the hyperfine state |3 is empty, the single particles properties of the hyperfine states |1 and |2 do not depend on the values of a 23 and a 13 . In this sense their properties are universal and our numerical result are applicable to other fermionic resonant atoms, e.g., 40 K. On the other hand, the properties of the hyperfine state |3 are not universal in the sense that they depend on the interaction couplings.…”

“…For instance [41] in 40 K, (where |1 , |2 , and |3 are |F = 9/2, m F = −9/2 , |F = 9/2, m F = −7/2 and |F = 9/2, m F = −5/2 ) a collision term such as ψ † 1 ψ † 3 ψ 2 ψ 2 conserves the total quantum number m F is therefore allowed by symmetry. However, effects of such interactions, that do not conserve the atom number in each hyperfine state are energetically suppressed by virtue of 2ω 2 =ω 1 +ω 3 , due to the large quadratic Zeeman splitting between the three levels.…”

“…Our computations of these universal functions is however not exact, and next we describe our computational method. We build on our earlier studies of the superfluid state [39,40]: our theoretical approach to the treatment of this strongly interacting (small parameter-free) normal state is based on the introduction of an artificial small parameter 1/N , with N the number of distinct "spin"-1/2 fermion flavors in the generalized model. The advantage of such a generalization is that, for N → ∞, the problem In free space, the resonance would be at ν = 0.…”

Superfluidity and phase transitions in a resonant Bose gas

“…In the presence of twisted boundary conditions one generalizes the variational Hamiltonian to the form 26) in whichã k is the Fourier transform ofψ. All variational coefficients depend on the twist wavevector k 0 , but N k , P k are even functions of k as before, while M k is an odd function of k. In fact, as will seen shortly, M k = v 0 · k, but this is not assumed at the outset.…”

“…13,15 In such narrow FR systems the crossover to BEC takes place when the FR detuning ν (quasimolecule's rest energy) ranges from twice the Fermi energy 2ǫ F (when it first becomes favorable to convert a finite fraction of the Fermi-sea into molecules stabilized by Pauli-blocking) down to zero energy, where all the fermions have become bound into Bose-condensed diatomic molecules. The complementary broad resonance regime of most experiments, 19 particularly near a universal unitary point 20 has been successfully studied using quantum Monte Carlo 21,22,23 and field theoretic ǫ-expansion 24,25 and 1/N -expansion 25,26 methods borrowed from critical phenomena.…”