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arXiv:1511.01701 · Journal of Modern Optics

Larkin-Ovchinnikov phases in two-dimensional square lattices

J. E. Baarsma, Päivi Törmä

Published Jan 18, 2016• 12 citations
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Abstract

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.

DOI
10.1080/09500340.2015.1128009

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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.
We study attractively interacting spin-12\frac{1}{2}21​ fermions on the square lattice subject to a spin population imbalance. Using unbiased diagrammatic Monte Carlo simulations we find an extended region in the parameter space where the Fermi liquid is unstable towards formation of Cooper pairs with nonzero center-of-mass momentum, known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. In contrast to earlier mean-field and quasi-classical studies we provide quantitative and well-controlled predictions on the existence and location of the relevant Fermi-liquid instabilities. The highest temperature where the FFLO instability can be observed is about half of the superfluid transition temperature in the unpolarized 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 study the pairing states in a largely imbalanced two-component Fermi gas loaded in an anisotropic two-dimensional optical lattice, where the spin-up and spin-down fermions are filled to the s- and p x -orbital bands, respectively. We show that owing to the relative inversion of the band structures of the s and p x orbitals, the system favors pairing between two fermions on the same side of the Brillouin zone, leading to a large stable regime for states with a finite center-of-mass momentum, i.e., the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. In particular, when two Fermi surfaces are close in momentum space, a nesting effect stabilizes a special type of π-FFLO phase with a spatial modulation of π along the easily tunneled x direction. We map out the zero-temperature phase diagrams within the mean-field approach for various aspect ratios within the two-dimensional plane and calculate the Berezinskii–Kosterlitz–Thouless (BKT) transition temperatures T BKT for different phases.
The attractive Fermi-Hubbard Hamiltonian is solved via the Bogoliubov-de Gennes formalism to analyze the ground state phases of population imbalanced two-component fermion mixtures in harmonically trapped optical lattices. In the low density limit, the superfluid order parameter has Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type oscillations in the radial direction towards the trap edges to accommodate the unpaired fermions that are pushed away from the trap center with a single peak in their density. However, in the high density limit, while the order parameter has oscillations in the radial direction towards the trap center for low imbalances, it also has oscillations towards the trap edges with increasing imbalance until the superfluid to normal phase transition occurs beyond a critical imbalance. This leads to a single peak in the density of unpaired fermions for low and high imbalances, but leads to double peaks for intermediate imbalances.
We study the phase diagram of an imbalanced two-component Fermi gas in optical lattices of 1–3 dimensions (1D–3D), considering the possibilities of the Fulde–Ferrel–Larkin–Ovchinnikov (FFLO), Sarma/breached pair, BCS and normal states as well as phase separation, at finite and zero temperatures. In particular, phase diagrams with respect to average chemical potential and the chemical potential difference of the two components are considered, because this gives the essential information about the shell structures of phases that will occur in the presence of an additional (harmonic) confinement. These phase diagrams in 1D, 2D and 3D show in a striking way the effect of Van Hove singularities on the FFLO state. Although we focus on population imbalanced gases, the results are relevant also for the (effective) mass imbalanced case. We demonstrate by LDA calculations that various shell structures such as normal–FFLO–BCS–FFLO–normal, or FFLO–normal, are possible in presence of a background harmonic trap. The phases are reflected in noise correlations: especially in 1D the unpaired atoms leave a clear signature of the FFLO state as a zero-correlation area ('breach') within the Fermi sea. This strong signature occurs both for a 1D lattice as well as for a 1D continuum. We also discuss the effect of Hartree energies and the Gorkov correction on the phase diagrams.
The recent experimental realization of spin-orbit (SO) coupling for ultracold atoms opens a completely new avenue for exploring new quantum matters. In experiments, the SO coupling is implemented simultaneously with a Zeeman field. Such SO-coupled Fermi gases are predicted to support Majorana fermions with non-Abelian exchange statistics in one dimension (1D). However, as shown in recent theory and experiments for 1D spin-imbalanced Fermi gases, the Zeeman field can lead to the long-sought Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids with nonzero center-of-mass momentum Cooper pairings, in contrast to the zero center-of-mass momentum pairings in Majorana superfluids. Therefore a natural question to ask is which phase, FFLO or Majorana superfluids, will survive in SO-coupled Fermi gases in the presence of a large out-of-plane Zeeman field. In this paper, we address this question by studying the mean-field quantum phases of 1D (quasi-1D) SO-coupled fermionic cold-atom optical lattices.