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.
We study the system of a dilute gas of fermions in 3-dimensions, with attractive interactions tuned to the unitarity point, using the non-perturbative Restricted Path Integral Monte Carlo (R-PIMC) method. The pairing and superfluid properties of this system are calculated at finite temperature. The total energy at very low temperature from our results agrees closely with that of previous ground-state Quantum Monte Carlo calculations. We identify the temperature T � � 0.70ǫF below which pairing correlations develop, and estimate the critical temperature for the superfluid transition Tc � 0.25ǫF from a finite size scaling analysis of the superfluid density. PACS numbers: The recent experiments, starting with generating a degenerate gas of cold atoms[3], the use of Feshbach resonance to tune the effective interaction between the fermions[4, 5], the measurements of the gap to identify a pairing scale[6] and the measurements of vortices[7] to identify the superfluid state, have all given an impetus to the study of superfluidity in the BCS-BEC crossover regime. The nature of superconductivity or superfluidity in a many-particle system with an increasing pairing attraction was shown [1, 2] to interpolate smoothly between the BCS regime for weak attraction between the fermions, and the Bose-Einstein condensation (BEC) regime for strong coupling. In the weak coupling regime, the Cooper pair size is much larger than the interparticle spacing, and the simultaneous pairing and condensation of fermions is well described by the BCS theory. In the strong coupling regime, fermions form strongly bound bosonic molecules at a pairing temperature scale T ∗ that is considerably higher than the BEC temperature Tc. In a two species system of fermions with an attractive interaction between them, the unitary point is defined by the divergence of the zero energy s-channel scattering length, as, for two particles in free space. The unitary point is interesting since it describes a strongly correlated system with universal properties[8]. In general, there are two length scales that define the system: the interparticle spacing ∝ n −1/3 , and the scattering length as which contains information about the interaction potential. At the unitary point however, since as diverges, all the properties of the system are described by a single length scale k −1
We theoretically study the thermodynamic properties of a strongly interacting Fermi gas at the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC), by applying a recently outlined strong-coupling theory that includes pair fluctuations beyond the commonly-used many-body
T -matrix or ladder approximation at the Gaussian level. The beyond Gaussian pair fluctuation (GPF) theory always respects the exact thermodynamic relations and recovers the Bogoliubov theory of molecules in the BEC limit with a nearly correct molecule-molecule scattering length. We show that the beyond-GPF theory predicts quantitatively accurate ground-state properties at the BEC-BCS crossover, in good agreement with the recent measurement by Horikoshi et al. in Phys. Rev. X 7, 041004 (2017). In the unitary limit with infinitely large
s -wave scattering length, the beyond-GPF theory predicts a reliable universal energy equation of state up to 0.6
Tc , where
Tc is the superfluid transition temperature at unitarity. The theory predicts a Bertsch parameter
ξ≃0.365 at zero temperature, in good agreement with the latest quantum Monte Carlo result
ξ=0.367(7) and the latest experimental measurement
ξ=0.367(9) . We attribute the excellent and wide applicability of the beyond-GPF theory in the broken-symmetry phase at sufficiently low temperatures to the empirically useful partial re-summation of Feynman diagrams following a dimensional
ϵ -expansion analysis near four dimensions (
d=4−ϵ ), which gives rise to accurate predictions at the second order
O(ϵ2) . Our work indicates the possibility of further improving the strong-coupling theory of strongly interacting fermions based on the systematic inclusion of large-loop Feynman diagrams at higher orders
O(ϵn) with
n≥3 .
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.
The Feshbach resonance provides precise control over the scattering length and effective range of interactions between ultracold atoms. We propose the ultratransferable pseudopotential to model effective interaction ranges $\ensuremath{-}1.5\ensuremath{\le}{k}_{\mathrm{F}}^{2}{R}_{\mathrm{eff}}^{2}\ensuremath{\le}0$, where
Reff is the effective range and
kF is the Fermi wave vector, describing narrow to broad Feshbach resonances. We develop a mean-field treatment and exploit the pseudopotential to perform a variational and diffusion Monte Carlo study of the ground state of the two-dimensional Fermi gas, reporting on the ground-state energy, contact, condensate fraction, momentum distribution, and pair-correlation functions as a function of the effective interaction range across the BEC-BCS crossover. The limit ${k}_{\mathrm{F}}^{2}{R}_{\mathrm{eff}}^{2}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ is a gas of bosons with zero binding energy, whereas $ln({k}_{\mathrm{F}}a)\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ corresponds to noninteracting bosons with infinite binding energy.
Quantum droplets (QDs) in weakly interacting ultracold quantum gases are typically characterized by mean-field theories incorporating Lee-Huang-Yang (LHY) quantum fluctuations under simplified zero-range interaction assumptions. However, bridging these models to broader physical regimes like superfluid helium requires precise understanding of short-range interatomic interactions. Here, we investigate how finite-range interactions--next-order corrections to zero-range potentials--significantly alter QDs mechanics. Using a consistent effective theory, we derive an analytical equation of state (EOS) for three-dimensional bosonic mixtures under finite-range interactions at zero temperature. Leveraging the Hubbard-Stratonovich transformation, we demonstrate that interspecies attraction facilitates bosonic pairing across components characterized by the non-perturbative parameter of
Δ, leading to nonuniversal LHY terms that encode short-range interaction details while recovering previous universal QDs EOS in the zero-range limit. Extending superfluid hydrodynamic equations for two-component systems, we predict fractional frequency shifts in breathing modes induced by these nonuniversal terms. Experimental observation of these shifts would reveal critical insights into QDs dynamics and interatomic potential characteristics.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also 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. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.