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Physical Review Letters

Rotonlike Fulde-Ferrell Collective Excitations of an Imbalanced Fermi Gas in a Two-Dimensional Optical Lattice

Zlatko G. Koinov, Rafael Mendoza, Mauricio Fortes

Published Mar 7, 2011• 13 citations
Publisher

Abstract

We address the question of whether superfluidity can survive in the case of fermion pairing between different species with mismatched Fermi surfaces using as an example a population-imbalanced mixture of 6Li atomic Fermi gas loaded in a two-dimensional optical lattice at nonzero temperatures. The collective mode is calculated from the Bethe-Salpeter equations in the general random phase approximation assuming a Fulde-Ferrell order parameter. The numerical solution shows that, in addition to low-energy (Goldstone) mode, two rotonlike minima exist, and therefore, the superfluidity can survive in this imbalanced system.

DOI
10.1103/physrevlett.106.100402

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Bulletin of the American Physical Society•Jun 12, 2015•2 citations
We address the question of whether superfluidity can survive in the case of fermion pairing between different species with mismatched Fermi surfaces using as an example a population-imbalanced mixture of 6Li atomic Fermi gas loaded in a two-dimensional optical lattice at nonzero temperatures. The collective mode is calculated from the Bethe-Salpeter equations in the general random phase approximation assuming a Fulde-Ferrell order parameter. The numerical solution shows that, in addition to low-energy (Goldstone) mode, two rotonlike minima exist, and therefore, the superfluidity can survive in this imbalanced system.
We study fermion pairing in a population-imbalanced mixture of 6{}^{6}6Li atomic gas loaded in a three-dimensional lattice at very low temperatures. Using the number equation for each population, the gap equation, and the equation for the Helmholtz free energy, we determine the gap, chemical potentials, and pair-momentum as functions of polarization. These parameters define the stability regions for a Fulde-Ferrell-Larkin-Ovchinnikov phase, a phase separation region where Bardeen-Cooper-Schrieffer and normal phases coexist, a Sarma phase when the pair-momentum vanishes, and the transition to the normal phase when the gap disappears. The collective-mode energies are then calculated using a Bethe-Salpeter approach in the generalized random-phase approximation assuming that the system is well described by the single-band Hubbard model. An interesting result is that this fermionic gas has a superfluid phase revealed by rotonlike minima in the asymmetric collective-mode energy spectrum.
A functional integral technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green's function and the Bethe-Salpeter equation for the two-particle Green's function and the associated collective modes of a population-imbalanced spin-orbit-coupled atomic Fermi gas loaded in a two-dimensional optical lattice at zero temperature. The collective-mode excitation energy is calculated within the Gaussian approximation, and from the Bethe-Salpeter equation in the generalized random phase approximation assuming the existence of a Sarma superfluid state. It is found that the Gaussian approximation overestimates the speed of sound of the Goldstone mode. More interestingly, the Gaussian approximation fails to reproduced the rotonlike structure of the collective-mode dispersion which appears after the linear part of the dispersion in the Bethe-Salpeter approach. We use the Gaussian approximation and the Bethe-Salpeter approach to investigate the speed of sound of a balanced spin-orbit-coupled atomic Fermi gas near the boundary of the topological phase transition driven by an out-of-plane Zeeman field. It is shown that within both approaches, the minimum of the speed of sound is located at the topological phase transition boundary, and this fact can be used to confirm the existence of a topological phase transition.
We present theoretical calculations of collective modes of the one-band attractive Hubbard model which is widely used to study the s-wave superfluid phases of atomic Fermi gases of two-hyperfine states loaded in a deep optical lattice. To make our theory applicable for both superconductivity and superfluidity, we assume the more general t−U−Jt-U-Jt−U−J Hamiltonian. Using the functional differentiation we derive Schwinger-Dyson equations for the single-particle Green's functions. The method of Legendre transform is used to give a systematic derivation of the Bethe-Salpeter (BS) equation for the two-particle Green's function and the associated collective modes. The numerical solution of the BS equation in the limit J→0J\rightarrow 0J→0 shows the existence of two rotonlike collective modes with different low-energy Goldstone dispersions and different positions of the rotonlike minima. The two rotonlike modes lie outside of the region determined by the lower boundary of the particle-hole continuum, and therefore, the two modes are not damped and they should be experimentally observable. In the presence of superfluid flow at a certain critical flow momentum, the minimum of the first rotonlike mode reaches zero energy, but this occurs before the minimum of the second mode and the lower boundary of the particle-hole continuum do, i.e. there are two critical flow momenta related to the existence of two rotonlike excitations.
A low-energy theory of the Nambu-Goldstone excitation spectrum and the corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and Potassium-40 atoms in a two-dimensional optical lattice at finite temperatures with the Fulde-Ferrell order parameter has been formulated. It is assumed that the two-species interacting Fermi gas is described by the one-band Hubbard Hamiltonian with an attractive on-site interaction. The discussion is restricted to the BCS side of the Feshbach resonance where the Fermi atoms exhibit superfluidity. The quartic on-site interaction is decoupled via a Hubbard-Stratonovich transformation by introducing a four-component boson field which mediates the Hubbard interaction. A functional integral technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green’s function and the Bethe-Salpeter equation for the two-particle Green’s function and the associated collective modes. The numerical solution of the Bethe-Salpeter equation in the generalized random phase approximation shows that there exist two distinct sound velocities in the long-wavelength limit. In addition to low-energy (Goldstone) mode, the two-species Fermi gas has a superfluid phase revealed by two roton-like minima in the asymmetric collective-mode energy.
Spin-imbalanced ultracold Fermi gases have been widely studied recently as a platform for exploring the long-sought Fulde-Ferrell-Larkin-Ovchinnikov superfluid phases, but so far conclusive evidence has not been found. Here we propose to realize an Fulde-Ferrell (FF) superfluid without spin imbalance in a three-dimensional fermionic cold atom optical lattice, where s- and p-orbital bands of the lattice are coupled by another weak moving optical lattice. Such coupling leads to a spin-independent asymmetric Fermi surface, which, together with the s-wave scattering interaction between two spins, yields an FF type of superfluid pairing. Unlike traditional schemes, our proposal does not rely on the spin imbalance (or an equivalent Zeeman field) to induce the Fermi surface mismatch and provides a completely new route for realizing FF superfluids.
Spin-imbalanced ultra-cold Fermi gases have been widely studied recently as a platform for exploring the long-sought Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phases, but so far conclusive evidence has not been found. Here we propose to realize an FF superfluid without spin imbalance in a three-dimensional (3D) fermionic cold atom optical lattice, where sand p-orbital bands of the lattice are coupled by another weak moving optical lattice. Such coupling leads to a spin-independent asymmetric Fermi surface, which, together with the s-wave scattering interaction between two spins, yields an FF type of superfluid pairing. Unlike traditional schemes, our proposal does not rely on the spin imbalance (or an equivalent Zeeman field) to induce the Fermi surface mismatch and may provide a completely new route for realizing FF superfluids.