Session U14: Focus Session: Exotic Phases in Ultracold Fermi Gases

FFLO states in resonant Fermi gases

We discuss the possible phases of two-component Fermi gas with population imbalance. In particular, we consider the various states proposed by Fulde-Ferrell and Larkin-Ovchinnikov. We distinguish between the plane-wave state Delta $\sim $ e$^{iqr}$, where the magnitude of the order parameter is uniform in space but the phase varies continuously in space, from those where the order parameters are real but change sign from one spatial region to the other. The later states, considered first by Larkin and Ovchinnikov, occupy a much larger region in the uniform phase diagram than previously suggested by other authors. If time permits, we shall discuss also the situation in a harmonic trap.   

Is There an FFLO Region in a Polarized Trapped Unitary Fermi Gas?

We have studied strongly interacting polarized gases in a harmonic trap beyond the local density (LDA) approximation using the Bogoliubov-deGennes equations. In particular, we are interested in the region separating an unpolarized superfluid core in the center and a fully polarized majority gas in the outer edge. Several authors have found that in this region the order parameter oscillates in a way similar to an FFLO phase. We will present the results of a detailed analysis of the properties of this system as a function of polarization, system size, and high energy cutoff used in the calculations. We find that the order parameter oscillations are an artifact of a finite (not sufficiently large) cutoff. Moreover, we find that the intermediate region shows a scaling with number of particles which makes it consistent with an interface. Our BdG calculation thus gives us a microscopic theory of the interface in a trapped unitary fermi gas.   

Polarized Fermi superfluids between one and three dimensions

We theoretically explore the phase diagram of a polarized two-component Fermi gas divided into an array of tubes by a two-dimensional optical lattice. By increasing the intensity of the optical lattice one suppresses the inter-tube hopping, and one can drive a crossover from three to one-dimensional behavior. We show that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superfluid phase is stabilized by the lattice, and we argue that the most promising parameters for observing the FFLO phase are in the intermediate lattice limit, where the anisotropy in the atomic motion enhances Fermi surface `nesting', but there is still a small amount of tunneling between the tubes to provide long-range order. Finally we discuss the spatial arrangement of phases in a trap: in 3D the homogenous superfluid phase sits in the center of the trap, while in 1D it lies on the edge. We explain how this pattern evolves as one changes the lattice intensity.   

Dimensional crossover from Quasi-1D to 3D in spin-polarized Fermi superfluids

We use a zero temperature Bogoliubov de Gennes mean field theory to study the evolution of the exotic FFLO superfluid in a spin-imbalanced Fermi gas as one progresses from quasi-1D to 3D by changing the coupling between an array of tubes. The boundary between the uniform BCS superfluid and the FFLO state is determined by examining the energetics of a single $\pi$-domain wall in the superfluid order parameter. In the quasi-1D limit, each tube contains a single excess particle at the center of each domain wall and the spectrum of single particle excitations is gapped. As one approaches 3D there is a phase transition where this commensurability condition is relaxed and gapless single particle excitations can be found. [1] M. M. Parish, S. K. Baur, E. J. Mueller, and D. A. Huse, arXiv:0709.1120   

Effect of the surface tension on the shape of the superfluid region in population imbalanced unitary Fermi gases.

We use a variational approach to determine the shape of the central superfluid shell of apopulation imbalanced unitary Fermi gases. We find that the surface tension between superfluid and normal regions significantly distorts the superfluid shell from an ellipsoidal shape to a cylindrical shape as experimentally seen in highly anisotropic traps. Comparing with experimental data, we find that the surface tension has strong temperature dependence and this allows us to compare the temperatures of various available experiments.   

Shape of the Normal-Superfluid Boundary in Polarized Fermi Gases

We model the normal-superfluid boundary in a trapped polarized Fermi gas as an elastic membrane and calculate the density profile. For weak trapping anisotropy, the normal-superfluid boundary remains elliptical, in agreement with the LDA. However, for strong anisotropy, the boundary becomes distorted into a capsule-like shape. As one moves axially from the edge of the trap to the center, the radius of the boundary almost discontinuously jumps. In addition to full numerical calculations, we present a simple model that predicts the density profile in the limit of large trapping anisotropy.   

Imbalanced Fermi superfluids beyond mean-field.

Cold atomic Fermi gases undergo pairing transitions leading to superfluidity, as has been demonstrated in recent experiments. In superconducting metals, the number of spin-up and spin-down fermions forming the Cooper pairs is always equal, but in the experiments with atomic superfluids, the number of each pairing partner can be tuned individually. This allows to probe the effect of population imbalance on the pairing properties, and has rekindled much theoretical interest in these systems. Here, we describe how to tackle fluctuations beyond mean field, and at nonzero temperature through an extension of the path-integral scheme developed by Randeria and co-workers. The results are discussed in the context of the recent (sometimes conflicting) experimental observations of imbalanced Fermi superfluids.   

Evolution from weak to strong coupling pairing of Dirac Fermions

We study the pairing of Dirac Fermions with attractive interaction from weak to strong coupling regime, highlighting the differences and resemblances with that of the BCS-BEC crossover in the systems with extended Fermi surface. Dirac Fermions model at low doping limit is solved by mean field approximation. Exact Quantum Monte Carlo method, auxiliary field Quantum Monte Carlo, is used to simulate the single band attractive Hubbard model on a honeycomb lattice. Quantities for probing the crossover, double occupancy, spin susceptibility, on-site pair correlation, and kinetic energy are obtained impartially. We find that these quantities indicate the BCS-BEC crossover of the model. This can be interpreted as a competition between Fermionic modes and Bosonic modes which coexist in the single band Hubbard model on a honeycomb lattice with attractive interaction.   

P-wave Pairing in Two-Component Fermi System with Unequal Population near Feshbach Resonance

We explore $p$-wave pairing in a single-channel two-component Fermi system with unequal population near Feshbach resonance. Our analytical and numerical study reveal a rich superfluid (SF) ground state structure as a function of imbalance. In addition to the state $\Delta_{\pm 1} \propto Y_{1\pm 1}$, a multitude of ``mixed'' SF states formed of linear combinations of $Y_{1m}$'s give global energy minimum under a phase stability condition; these states exhibit variation in energy with the relative phase between the constituent gap amplitudes. States with local energy minimum are also obtained. We provide a geometric representation of the states. A $T$=0 polarization vs. p-wave coupling phase diagram is constructed across the BEC-BCS regimes. With increased polarization, the global minimum SF state may undergo a quantum phase transition to the local minimum SF state.   

P-wave Paired Ground state Structure of Two-component Population Imbalanced Fermi Systems with Unequal Mass

Effects of unequal mass on p-wave pairing in population imbalanced two-component Fermi systems is studied using a single-channel fermionic Hamiltonian. Provided certain phase criteria are satisfied among the orbital gap parameters, a rich structure of the ground state is obtained. The global and local minima ground state energies are determined analytically. By numerically solving the gap and number equations, we construct T=0 polarization versus p-wave coupling phase diagrams across the BEC-BCS regime for different mass ratios. This shows the existence of two superfluid phases, SF1 and SF2, corrsponding to the global and local energy minima respectively; phase separation occurs at higher polarizations. For small mass ratios, SF1 is enhanced; SF2 is not significantly affected. Competition between mass ratio and polarization is studied; the superfluid transition temperature shows interesting behavior versus the mass ratio at high polarizations. At large polarizations, a stable p-wave superfluid would survive only for small mass ratios, and for mass ratios close to unity.   

Orbital ordering in an atomic Mott insulator of p-band fermions

We derive the low energy effective model describing the orbital degrees of freedom of strongly interacting spinless p-orbital fermionic atoms in 2D optical lattices. Virtual hopping processes of $p_x$ and $p_y$ fermions give rise to direct and multi-particle orbital exchanges in the strong coupling regime. For the square lattice, we show that the effective orbital Hamiltonian is equivalent to a quantum spin-1/2 XXZ model. In the limit where the transverse hopping is much smaller than the longitudinal hopping, the XXZ model reduces to an antiferromagnetic Ising model. Thus the atomic Mott insulator is antiferro-orbitally ordered. We also present results for other simple 2D lattices and discuss the experimental signatures of various orbital ordering.   

Density waves and supersolidity in rapidly rotating atomic Fermi gases

We study theoretically the low-temperature phases of a two-component atomic Fermi gas with attractive $s$-wave interactions under conditions of rapid rotation [1], a problem related to the discussion of high-field superconductivity in the solid state [2]. The regime of interest for atomic gases differs substantially from solid state conditions: the rotation does not lead to any Zeeman splitting which might suppress high-field SC order; the short-range interactions allow density wave order to develop (contrary to long-range Coulomb interactions). We show that the low-temperature phases of an atomic Fermi gas with attractive interactions involve an interesting interplay between CDW and superconducting phases. In the extreme quantum limit, when only the lowest Landau level is occupied, we employ a renormalization group approach [3] to show that the system is unstable to CDW order along the rotation axis. At lower rotation rates, we show how CDW and SC can coexist, leading to supersolid behaviour.\\[0pt] [1] G. M\"oller and N. R. Cooper, Phys. Rev. Lett {\bf 99}, 190409 (2007).\\[0pt] [2] Z. Tesanovic, M. Rasolt and L. Xing, Phys. Rev. Lett. {\bf 63}, 2425 (1989).\\[0pt] [3] V.~M. Yakovenko, Phys.~Rev.~B {\bf 47}, 8851 (1993).   

Quantum Hall Transition near a Feshbach Resonance in Fast Rotating Fermi Gases

We consider two-species of fermions in a rotating trap that interact via an s-wave Feshbach resonance, at total Landau level filling factor two (or one for each species). We show that the system undergoes a quantum phase transition from a fermion integer quantum Hall state to a boson fractional quantum Hall state as the pairing interaction strength increases, with the transition occurring near the resonance. The effective field theory for the transition is shown to be that of a (emergent) massless relativistic bosonic field coupled to a Chern-Simons gauge field, with the coupling giving rise to semionic statistics to the emergent particles.