Session L32: Focus Session: BEC/BCS Crossover

Large N expansion for superfluid Fermi gases at unitarity

We study an s-wave resonant Fermi gas near the unitarity point. We treat this problem by generalizing the Fermi gas to a model with $2N$ hyperfine states (with Sp($2N$) symmetry). We show that for $N=\infty$, the model can be solved exactly by the BEC-BCS mean field solution. In order to address the physically relevant problem ($N=1$), we perform a systematic $1/N$ loop expansion around the BEC-BCS solution. For $N=1$, we obtain a variety of thermodynamic quantities, including the energy, the pairing gap, and the upper critical polarization. We compare our results to experimental data and other theoretical approaches.   

Four-fermion problem in hyperspherical coordinates

The four-particle system is the simplest few-body system which contains the fundamental physics involved in ultracold fermionic gases. We solve the quantum four-body problem in the adiabatic hyperspherical representation. Our approach yields a set of coupled-channel equations which can in turn be solved for all elastic and inelastic processes. These rates are expected to play an important role in the lifetime of molecules in ultracold fermi gases. This provides insights into the nature of ultracold fermi systems and the physics of the BCS-BEC cross-over.   

Correlation effects in the BCS/BEC crossover.

We use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance and show how to correctly describe the dimer-dimer scattering on the BEC side.   

Pairing and Superfluid Properties of Dilute Fermion Gases at Unitarity

We study the pairing and superfluid properties of a dilute gas of fermions in 3-dimensions with attractive interactions tuned to the unitarity point [1]. The finite temperature, non- perturbative, Restricted Path Integral Monte Carlo (R-PIMC) method is used for our simulations and tested against previous ground-state Quantum Monte Carlo calculations. From the growth of the density correlations for unequal spins, we identify the pseudogap crossover temperature scale T*~0.70 Ef below which pairing correlations develop. We estimate the critical temperature for condensation Tc~0.24 Ef from a finite size scaling analysis of the superfluid density. The pseudogap phase is characterised by the spin susceptibility and compressibility. We will also present results for unequal populations of fermions. \newline \newline [1] V. Akkineni, N. Trivedi, D.M. Ceperley, cond-mat/0608154   

Exact Relations for A Strongly-Correlated Fermi Gas With Large Scattering Length

A 2-component Fermi gas with a large and tunable scattering length $a$ is considered. If the interfermionic forces have a range much shorter than the average interparticle spacing, the characteristic de Broglie wavelength, and $\mid\! a\!\mid$, the system is in a universal regime in which the interaction is described by a single parameter, $a$. We show that the energy, the momentum distribution, the pressure, the change of energy during a real-time ramp of the scattering length, and the energy spectrum of such a Fermi gas satisfy a few simple \emph{exact} relations. The importance of the $C/k^4$ tails of the momentum distributions at large $k$ is stressed. Implications of these results for experiments on ultracold atomic Fermi gases near Feshbach resonances are discussed.   

Dilute Bose and Fermi gases with large generalized scattering lengths

Dilute weakly-interacting Bose and Fermi gases can be described to a very good approximation by a single atomic physics parameter, the $s$-wave scattering length. Utilizing broad Feshbach resonances, strongly-interacting two-component Fermi gases with infinitely large interspecies scattering lengths can now be studied experimentally. In this so-called unitary regime, the only remaining energy scale is the energy $E_{FG}$ of the non-interacting Fermi gas, and it has been shown that the energy of the Fermi gas becomes about $0.44 E_{FG}$. We investigate Bose and Fermi gases with non-vanishing angular momentum using the lowest order constrained variational method. In particular, we focus on the regime where the generalized scattering length becomes infinite. For example, we show that the energy of $d$-wave interacting fermions depends not only on $E_{FG}$ but additionally on an energy scale set by the range of the underlying two-body potential.   

Luttinger theorem and Fermi liquid behavior close to a Feshbach resonance

Based on the results obtained in a previous paper \thanks{cond-mat/0505306}, we derive the thermodynamic properties of a Fermi gas, deep into the quantum degenerate regime and provide a useful test for the validity of Luttinger theorem. We show that, if Luttinger theorem holds, a first order phase transition has to occur in the normal phase as a function of the interaction strength, U, as a consequence of a jump occurring in the compressibility, spin susceptibility and specific heat. The signature of the transition is given by the presence of a non-zero latent heat. We also show that a volume change occurs at finite temperatures from the BEC to the BCS side of the Feshbach resonance, in the normal phase. The transition has an end point close to the BCS critical temperature. Thus, observation of these properties will require suppression of the superfluid phase. Also we demonstrate that a paramagnetic system in equilibrium, close to a diverging scattering length, expels any applied magnetic field and as a consequence, there is no Clogston limit in the in the superfluid phase.   

Collisional hydrodynamic mode frequencies in the BCS-BEC crossover near unitarity

In the collisional region at finite temperatures (produced by the large value of the $s$-wave scattering length), the collective modes of a superfluid Fermi gas are expected to be described by the Landau two-fluid hydrodynamic equations. These equations predict two types of modes: an in-phase oscillation of the normal and superfluid components as well as an out-of-phase oscillation. We prove that at unitarity and at all temperatures, the in- phase breathing mode solution of the two-fluid equations has a frequency identical to that calculated at $T=0$ by Cozzini and Stringari. This temperature-independence has been verified in recent experiments by Thomas and coworkers. For the special case of an isotropic trap, we find the temperature-independent frequency $\omega = 2\omega_0$, a result predicted to be valid under all conditions at unitarity by Castin. We also discuss the more interesting finite-$T$ out-of--phase (the analogue of second sound) breathing mode frequency given by the Landau-two- fluid equations at unitarity.   

Suppression of $T_c$ by Medium Effects from Dilute to Dense Regime: A Crossing-symmetric Approach

We study medium effects on superfluid transition temperature of 1- and 2-component strongly correlated Fermi systems. A crossing-symmetric approach allows us to explore this across dilute and dense regimes within a single framework. We include many-body effects such as density, spin-density, and current fluctuations. Pairing interactions are deduced from scattering amplitudes in the pairing channel. For 2-component systems, we find the known factor-of-2 suppression in $T_c$ to be robust across both regimes, except near the unitarity limit, where the suppresion is more pronounced. For the 1-component case, the suppresion can be greater, and not universal across the regimes. We discuss possible physical causes for the $T_c$ suppression.   

Quantum fluctuations in the superfluid state of the BCS-BEC crossover

While the Leggett-BCS mean-field approach gives a reasonable `zeroth order' description of the superfluid ground state in the BCS-BEC crossover, there are many ways in which it is inadequate. In addition to quantitative discrepancies with quantum Monte Carlo and experimental results at unitarity and with exact results for dimer-dimer scattering in the BEC limit, the mean field theory also misses the qualitative effects of quantum depletion of the condensate in a strongly interacting Fermi system. To address these concerns, we include the effects of zero-point motion of collective excitations and of the pair continuum in calculating various ground state properties. We implement this RPA in a functional integral formalism which ensures that the feedback of the collective modes on the saddle point respects Goldstone's theorem. We will present results on the ground state energy, gap, compressibility and collective mode frequency as a function of $1/k_f a_s$.   

The universal phase diagram of fermionic quantum liquids near the unitarity limit

We consider several models of particles with short-range attractive interactions whose universal properties are controlled by an unstable renormalization-group fixed point at zero density and temperature. The fixed point corresponds to the Feshbach resonance, and relevant perturbations are the detuning of the resonance, and parameters that control the particle densities. Some critical exponents are determined exactly as expansions about two and four spatial dimensions. The existence of a renormalization-group fixed point implies a universal phase diagram as a function of density, temperature, population imbalance, and detuning. We study this phase diagram in the context of BEC-BCS crossover of s-wave paired fermions. We develop a 1/N expansion, based upon models with Sp(2N) symmetry, and use it to systematically analyze the universal properties of interacting fermions near the unitarity limit. This approach overcomes several limitations of the expansions about two and four dimensions, and allows a well controlled exploration of the full phase diagram of imbalanced fermion populations in the experimentally relevant three-dimensional space.   

Properties and dynamics of four particles in a trap in the BCS-BEC crossover.

The Hamiltonian of two spin up and two spin down fermions in a trap is calculated using a correlated gaussian basis in the vicinity of the BCS-BEC crossover. From the spectrum, key properties of the few-body system are deduced as a function of the 2-body scattering length. After a diabatization procedure, the wavefunctions are used to evolve in time an initial configuration, mimicking molecule formation experiments with Fermi gases in the BCS-BEC crossover. The dynamics are successfully modeled as a sequence of Landau-Zener transitions. Finally, atom-molecule coherent quantum beats in this system are studied and a ramping scheme is proposed for experimental investigation.   

Describing the degenerate Fermi in a renormalized hyperspherical treatment

We describe the degenerate Fermi gas with zero-range density-dependent renormalized interactions (eprint cond-mat/0610848) in an isotropic trap using a variational hyperspherical approach. This method reduces the complex many body Hamiltonian to a simple one-dimensional effective potential in a collective coordinate, the hyperradius, which can be thought of as the rms size of the gas. Exploring the behavior of the effective potential in the unitarity region where the two-body scattering length becomes very large and negative produces interesting effects. The low energy collective excitation frequency of a two spin component gas approaches the noninteracting frequency, as has been seen in hydrodynamic treatments. For larger numbers of spin components an interesting dynamical instability develops.