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arXiv:cond-mat/0605169 · New Journal of Physics

Fermion pairing with spin-density imbalance in an optical lattice

T Koponen, J. J. Kinnunen, J.-P. Martikainen, L M Jensen, Päivi Törmä

Published Sep 5, 2006• 40 citations
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Abstract

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.

DOI
10.1088/1367-2630/8/9/179

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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.
We theoretically investigate a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type superfluid phase transition in a driven-dissipative two-component Fermi gas. The system is assumed to be in the nonequilibrium steady state, which is tuned by adjusting the chemical potential difference between two reservoirs that are coupled with the system. Including pairing fluctuations by extending the strong-coupling theory developed in the thermal-equilibrium state by Nozi\`eres and Schmitt-Rink to this nonequilibrium case, we show that a nonequilibrium FFLO (NFFLO) phase transition can be realized without spin imbalance, under the conditions that (1) the two reservoirs imprint a two-edge structure on the momentum distribution of Fermi atoms and (2) the system is loaded on a three-dimensional cubic optical lattice. While the two edges work like two Fermi surfaces with different sizes, the role of the optical lattice is to prevent the NFFLO long-range order from destruction by NFFLO pairing fluctuations. We also draw the nonequilibrium mean-field phase diagram in terms of the chemical potential difference between the two reservoirs, a fictitious magnetic field to tune the spin imbalance of the system, and the environmental temperature of the reservoirs to clarify the relation between the NFFLO state and the ordinary thermal-equilibrium FFLO state discussed in spin-imbalanced Fermi gases.
We study the properties of a one-dimensional (1D) gas of fermions trapped in a lattice by means of the density matrix renormalization group method, focusing on the case of unequal spin populations, and strong attractive interaction. In the low-density regime, the system phase separates into a well-defined superconducting core and a fully polarized metallic cloud surrounding it. We argue that the superconducting phase corresponds to a 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, with a quasicondensate of tightly bound bosonic pairs with a finite center-of-mass momentum that scales linearly with the magnetization. In the large density limit, the system allows for four phases: in the core, we either find a Fock state of localized pairs or a metallic shell with free spin-down fermions moving in a fully filled background of spin-up fermions. As the magnetization increases, the Fock state disappears to give room for a metallic phase, with a partially polarized superconducting FFLO shell and a fully polarized metallic cloud surrounding the core.
In three-dimensional Fermi gases with spin imbalance, a competition exists between Cooper pairing with zero and finite momentum. The latter gives rise to the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superfluid phase, which only exists in a restricted area of the phase diagram as a function of chemical potential imbalance and interaction strength. Applying an optical potential along one direction enhances the FFLO region in this phase diagram. In this paper, we construct the phase diagram as a function of polarization and interaction strength in order to study the competition between the FFLO phase and the spin balanced Bardeen–Cooper–Schrieffer (BCS) phase. This allows us to take into account the region of phase separation, and provides a more direct connection with experiment. Subsequently, we investigate the effects of the wavelength and the depth of the optical potential, which is applied along one direction, on the FFLO state. It is shown that the FFLO state can exist up to a higher level of spin imbalance if the wavelength of the optical potential becomes smaller. Our results give rise to an interesting effect: the maximal polarization at which the FFLO state can exist decreases when the interaction strength exceeds a certain critical value. This counterintuitive phenomenon is discussed and the connection to the optical potential is explained.
We study the interplay between the long- and short-range interaction of a one-dimensional optical lattice system of two-component dipolar fermions by using the density matrix renormalization group method. The atomic density profile, pairing-pairing correlation function, and compressibility are calculated in the ground state, from which we identify the parameter region of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state, half-metal (HM) state, FFLO-HM state, and the normal polarized state, and thus the phase diagram in the coordinates of the long- and short-range interaction strength. The effect of the long-range dipolar interaction on the FFLO state is discussed in detail. We find that the long-range part of the dipole-dipole interaction does not sweep away the FFLO superconducting region that is driven by the short-range interaction in the Hubbard model, and thus the FFLO state survives in the wide parameter space of the long-range interaction, polarization, and filling.
We consider a two-component gas of fermions in optical lattices in the presence of a population imbalance within a mean-field theory. We study phase transitions from a normal gas of unpaired fermions to a superfluid phase of Bose-condensed Cooper pairs. The possibility of Cooper pairs with a nonzero centre-of-mass momentum is included, which corresponds to a so-called Fulde–Ferrel–Larkin–Ovchinnikov (FFLO) state. We find that for population-imbalanced systems such states can form the ground state. The FF and LO state are compared and it is shown that actually the LO state is energetically more favourable. We complete the mean-field phase diagram for the LO phase and show that it is qualitatively in excellent agreement with recent diagrammatic Monte Carlo calculations. Subsequently, we calculate the atomic density modulations in the LO phase.
The question whether a spin-imbalanced Fermi gas can accommodate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state has been the subject of intense study. This state, in which Cooper pairs obtain a nonzero momentum, has hitherto eluded experimental observation. Recently, we demonstrated that the FFLO state can be stabilized in a 3D Fermi gas, by adding a 1D periodic potential. Until now it was assumed that the FFLO wave vector always lies parallel to this periodic potential (FFLO-P). In this contribution we show that, surprisingly, the FFLO wave vector can also lie skewed with respect to the potential (FFLO-S). Starting from the partition sum, the saddle-point free energy of the system is derived within the path-integral formalism. Minimizing this free energy allows us to study the different competing ground states of the system. To qualitatively understand the underlying pairing mechanism, we visualize the Fermi surfaces of the spin-up and spin-down particles. From this visualization, we find that tilting the FFLO wave vector with respect to the direction of the periodic potential can result in a larger overlap between the pairing bands of both spin species. This skewed FFLO state can provide an additional experimental signature for observing FFLO superfluidity in a 3D Fermi gas.