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arXiv:1306.4706 · Physical Review A

Superfluidity of a spin-imbalanced Fermi gas in a three-dimensional optical lattice

Rafael Mendoza, Mauricio Fortes, M. A. Solís, Zlatko G. Koinov

Published Sep 6, 2013• 12 citations
arXivHTMLPDF

Abstract

We study fermion pairing in a population-imbalanced mixture of ${}^{6}$Li 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.

DOI
10.1103/physreva.88.033606

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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.
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 present an extensive study of two-dimensional Larkin-Ovchinnikov (LO) superfluidity in a spin-imbalanced two-component atomic Fermi gas. In the context of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, we explore a wide and generic class of pairing gap functions with explicit spatial dependency. The mean-field theory of such phases is applied through the Bogoliubov-de Gennes equations in which the pairing gap can be determined self-consistently. To systematically explore the configuration space we consider both the canonical and grand canonical ensembles where we control the polarization or chemical potentials of the system, respectively. The mean-field calculations enable us to understand the nature of the phase transitions in the fully paired Bardeen-Cooper-Schrieffer (BCS) state, exotic LO phase, and partially polarized free Fermi gas (NPP{N}_{\text{PP}}NPP​). The order of the phase transitions has been examined and, in particular, we find a weak first-order phase transition between the exotic inhomogeneous LO phase and the BCS phase. In comparison to the three-dimensional case, where the phase diagram is dominated by a generic separation phase, we predict a broader parameter space for the spatially inhomogeneous LO phase. By computing the superfluid density of the LO phase at different polarization, we show how the superfluidity of the system is suppressed with increasing spin polarization.
We consider pairing in a two-component atomic Fermi gas, in a three-dimensional optical lattice, when the components have unequal densities, i.e. the gas is polarized. We show that a superfluid where the translational symmetry is broken by a finite Cooper pair momentum, namely a Fulde–Ferrel–Larkin–Ovchinnikov (FFLO)-type state, minimizes the Helmholtz free energy of the system. We demonstrate that such a state is clearly visible in the observable momentum distribution of the atoms, and analyse the dependence of the order parameter and the momentum distribution on the filling fraction and the interaction strength.
The zero-temperature phase diagrams of imbalanced two-species Fermi gases are investigated in asymmetric optical lattices with arbitrary potential depths, based on the exact spectrum instead of the Fermi-Hubbard model. We study the effect of lattice potentials and atomic densities to the fully paired Bardeen-Cooper-Schrieffer (BCS) state and particularly the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. It is found that the increasing lattice potential favors BCS at low densities because of the enhanced effective coupling, whereas FFLO is favored at intermediate densities when the system undergoes a dimensional crossover. Finally, using the local density approximation, we study the evolution of the phase profile in the presence of external harmonic traps by merely tuning the lattice potentials.
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.
Pairing in a population-imbalanced Fermi system in a two-dimensional optical lattice is studied using determinant quantum Monte Carlo simulations and mean-field calculations. The approximation-free numerical results show a wide range of stability of the Fulde-Ferrell-Larkin-Ovchinnikov phase. Contrary to claims of fragility with increased dimensionality, we find that this phase is stable across a wide range of values for the polarization, temperature, and interaction strength. Both homogeneous and harmonically trapped systems display pairing with finite center-of-mass momentum, with clear signatures either in momentum space or real space, which could be observed in cold-atomic gases loaded in an optical lattice. We also use the harmonic level basis in the confined system and find that pairs can form between particles occupying different levels, which can be seen as the analog of the finite center-of-mass momentum pairing in the translationally invariant case. Finally, we perform mean-field calculations for the uniform and confined systems and show the results to be in good agreement with quantum Monte Carlo. This leads to a simple picture of the different pairing mechanisms, depending on the filling and confining potential.