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arXiv:cond-mat/0508139 · Physical Review Letters

Effective Theory of Excitations in a Feshbach-Resonant Superfluid

W. Vincent Liu

Published Feb 28, 2006• 8 citations
arXivHTMLPDF

Abstract

A strongly interacting Fermi gas, such as that of cold atoms operative near a Feshbach resonance, is difficult to study by perturbative many-body theory to go beyond mean-field approximation. Here I develop an effective field theory for the resonant superfluid based on broken symmetry. The theory retains both fermionic quasiparticles and superfluid phonons, the interaction between them being derived nonperturbatively. The theory converges and can be improved order by order, in a manner governed by a low energy expansion rather than by a coupling constant. I apply the effective theory to calculate the specific heat and discuss the theory with a recent heat capacity experiment.

DOI
10.1103/physrevlett.96.080401

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•Sep 28, 2011•0 citations
A strongly interacting Fermi gas, such as that of cold atoms operative near a Feshbach resonance, is difficult to study by perturbative many-body theory to go beyond mean-field approximation. Here I develop an effective field theory for the resonant superfluid based on broken symmetry. The theory retains both fermionic quasiparticles and superfluid phonons, the interaction between them being derived nonperturbatively. The theory converges and can be improved order by order, in a manner governed by a low energy expansion rather than by a coupling constant. I apply the effective theory to calculate the specific heat and discuss the theory with a recent heat capacity experiment.
For calculating low-energy properties of a dilute gas of atoms interacting via a Feshbach resonance, we develop an effective theory in which the parameters that enter are an atom-molecule coupling strength and the magnetic moment of the molecular resonance. We demonstrate that, for resonances in the fermionic systems 6Li and 40K that are under experimental investigation, the coupling is so strong that many-body effects are appreciable even when the resonance lies at an energy large compared with the Fermi energy. We calculate a number of many-body effects, including the effective mass and the lifetime of atomic quasiparticles in the gas.
Summary form only given. For non-resonant atomic interactions the properties of ultra-cold scattering can be accurately represented by the scattering length. When the interactions are resonant, for instance when the atoms are colliding via a Feshbach resonance, the scattering properties become strongly dependent on energy, and the scattering length approximation is no longer valid. We show that there is a limited set of parameters that fully characterizes the scattering in the ultra-cold regime, and that this description can be easily incorporated into a theory of many-body physics. In effect, it can be shown that with a Hamiltonian where the resonant bound-state is explicitly treated, the resulting many-body mean-field equations have built in the microscopic two-body coupled-channels scattering equations. We have used this many-body theory to derive a theory of resonance superfluidity, and applied it to both fermionic /sup 40/K and /sup 6/Li. Since the scattering length a does not appear as an expansion parameter of the theory it can be shown that this field theory does not break down as a goes to infinity. In fact, the theory describes the high-T/sub c/ behavior of the system and predicts a critical temperature which can be as high as half the Fermi temperature.
A mechanism is proposed to explain the puzzling damping of collective excitations, which was recently observed in the experiments of strongly interacting Fermi gases below the superfluid critical temperature on the fermionic (BCS) side of Feshbach resonance. Sound velocity, superfluid density, and damping rate are calculated with effective field theory. We find that a dominant damping process is due to the interaction between superfluid phonons and thermally excited fermionic quasiparticles, in contrast to the previously proposed pair-breaking mechanism. Results from our effective model are compared quantitatively with recent experimental findings, showing a good agreement.
We develop an effective field theory to describe the superfluid pairing in strongly interacting fermions with arbitrary short-range attractions by extending Kaplan's idea of coupling fermions to a fictitious boson state [presented in Nucl. Phys. B 494, 471 (1997)]. This boson field is assigned with an unconventional kinetic term to recover the exact scattering phase shift obtained either from scattering data or model calculations. The theory works even if the explicit form of the interaction potential has not been constructed from scattering data. The contact boson-fermion coupling allows us to go beyond mean-field to include Gaussian pair fluctuations, yielding reliable predictions on equations of state. As an application, we use our theory to address the nonuniversal ground-state energy of strongly paired fermions due to the nontrivial momentum dependence of the phase shift characterized, for example, by effective range. We find a good agreement between our predictions and recent quantum Monte Carlo simulations on the effective-range dependence in both three and two spatial dimensions. We propose that in cold-atom experiments, the nonuniversal dependence in thermodynamics can be probed using dark-state optical control of Feshbach resonances.
A self-consistent theory is derived to describe the BCS--Bose-Einstein-condensate crossover for a strongly interacting Fermi gas with a Feshbach resonance. In the theory the fluctuation of the dressed molecules, consisting of both preformed Cooper pairs and ``bare'' Feshbach molecules, has been included within a self-consistent TTT-matrix approximation, beyond the Nozi\`eres and Schmitt-Rink strategy considered by Ohashi and Griffin. The resulting self-consistent equations are solved numerically to investigate the normal-state properties of the crossover at various resonance widths. It is found that the superfluid transition temperature Tc{T}_{c}Tc​ increases monotonically at all widths as the effective interaction between atoms becomes more attractive. Furthermore, a residue factor Zm{Z}_{m}Zm​ of the molecule's Green function and a complex effective mass have been determined to characterize the fraction and lifetime of Feshbach molecules at Tc{T}_{c}Tc​. Our many-body calculations of Zm{Z}_{m}Zm​ agree qualitatively well with recent measurments of the gas of 6Li^{6}\mathrm{Li}6Li atoms near the broad resonance at 834G834\phantom{\rule{0.3em}{0ex}}\mathrm{G}834G. The crossover from narrow to broad resonances has also been studied.
In the first part of this thesis, we present an inhomogeneous theory for the low-temperature properties of a resonantly interacting Fermi mixture in a trap that goes beyond the local-density approximation. We compare a Bogoliubov-de Gennes and a Landau-Ginzburg approach that both include mean-field-like self-energy effects to account for the strong interactions at unitarity. We conclude that the Landau-Ginzburg approach is more appropriate when dealing with a first-order phase transition. We show diagrammatically how these self-energy effects arise from fluctuations in the superfluid order parameter. Gradient terms of the order parameter are included to account for inhomogeneities. This approach incorporates the state-of-the-art knowledge of the homogeneous mixture with a population imbalance exactly and gives good agreement with the experimental density profiles of Shin et al. [Nature 451, 689 (2008)]. We calculate the universal surface tension due to the observed interface between the equal-density superfluid and the partially polarized normal state of the mixture. We find that the exotic and gapless superfluid Sarma phase can be stabilized at this interface, even when this phase is unstable in the bulk of the gas. We also discuss the possibility of a metastable state to explain the deformation of the superfluid core that is seen in the experiment of Partridge et al. [Science 311, 503 (2006)]. In the second part, we show that a two-channel mean-field theory for a Bose gas near a Feshbach resonance allows for an analytic computation of the chemical potential, and therefore the universal constant β, at unitarity. To improve on this mean-field theory, which physically neglects condensate depletion, we study a variational Jastrow ansatz for the ground-state wavefunction and use the hypernetted-chain (HNC/0) approximation to minimize the energy for all positive values of the scattering length. We also show that other important physical quantities such as Tan's contact and the condensate fraction can be directly obtained from this approach.