An effective relativistic continuum massive Proca Lagrangian action is used to account for the Lorentz vector condensation effects on the equation of state of the strongly interacting fermions system. The interior quantum fluctuation effects are incorporated as an external field approximation indirectly through a fictive generalized Thomson Problem counterterm background. The general analytical formulas for the d -dimensional thermodynamics are given near the unitary limit region. In the non-relativistic limit for d = 3, the universal dimensionless coefficient ξ = 4/9 and energy gap Δ/ε f = 5/18 are reasonably consistent with the existing theoretical and experimental results. In the unitary limit for d = 2 and T = 0, the universal coefficient can even approach the extreme occasion ξ = 0 corresponding to the infinite effective fermion mass m * = ∞, which can be mapped to the strongly coupled two-dimensional electrons and is quite similar to the three-dimensional Bose–Einstein condensation of ideal boson gas. Instead, for d = 1, the universal coefficient ξ is negative, implying the non-existence of phase transition from superfluidity to normal state. The solutions manifest the quantum Ising universal class characteristic of the strongly coupled unitary fermions gas.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
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 .
The dimensionless universal coefficient ξ defines the ratio of the unitary fermions energy density to that for the ideal non-interacting ones in the non-relativistic limit with T = 0. The classical Thomson problem is taken as a nonperturbative quantum many-body arm to address the ground state energy including the low energy nonlinear quantum fluctuation/correlation effects. With the relativistic Dirac continuum field theory formalism, the concise expression for the energy density functional of the strongly interacting limit fermions at both finite temperature and density is obtained. Analytically, the universal factor is calculated to be ξ = 4/9. The energy gap is Δ = (5/18) k f 2 /(2 m ).
We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to those of the ideal quantum gas with integer ratios through a general term formula. Counterintuitively, contrary to those of the ideal bosons ( ) or fermions ( ), the second virial coefficient a 2 of Fermi gas at unitarity is found to be equal to zero. With the vanishing leading order quantum correction, the BCS–BEC crossover thermodynamics manifests the famous pure classical Boyle’s law in the Boltzmann regime. The non-Gaussian correlation phenomena can be validated by studying the Joule–Thomson effect.
We investigate here the BCS-BEC crossover in relativistic systems using a variational construct for the ground state and the minimization of the thermodynamic potential. This is first studied in a four-fermion point interaction model and with a BCS type ansatz for the ground state with fermion pairs. It is shown that the antiparticle degrees of freedom play an important role in the BCS-BEC crossover physics, even when the ratio of Fermi momentum to the mass of the fermion is small. We also consider the phase structure for the case of fermion pairing with imbalanced populations. Within the ansatz, thermodynamically stable gapless modes for both fermions and antifermions are seen for strong coupling in the Bose-Einstein condensation (BEC) regime. We further investigate the effect of fluctuations of the condensate field by treating it as a dynamical field and generalize the BCS ansatz to include quanta of the condensate field also in a boson-fermion model with quartic self-interaction of the condensate field. It is seen that the critical temperature decreases with inclusion of fluctuations.
Strongly interacting Fermi gas systems play an important role in many areas of physics. Their systematic analytical study has been a difficult theoretical endeavour as it is outside the reach of ordinary perturbation theory. Much progress has been made with the development of the large-N expansion, where a fermion is generalized to have N (or 2N) components and the Lagrangian’s global symmetry is extended to the unitary group U(N) (or the compact symplectic group USp(2N)). The chosen symmetry group dictates the system’s properties as U(N) enhances the fermion-hole interaction channel and suppresses the fermion-fermion channel, whereas USp(2N) treats both channels equally by allowing two independent interaction terms in the Lagrangian density, the density-density interaction which enhances the fermion-hole channel and the pairing interaction which enhances the fermion-fermion channel. Literature has been either only focusing on the pairing interaction term or including both but restricting their respective coupling constants to be the same. The most general renormalizable quantum field theory (QFT) compatible with a global USp(2N)-symmetry contains both interaction terms with independent, dimensionless coupling constants. By studying such a system at zero temperature, we want to understand how the two interaction terms influence each other by independently tuning the coupling constants from the attractive to the repulsive regime. To this end, we calculate the ground state energy density to leading order in the large-N expansion in four different cases which are classified by the coupling constant regime and map out the phase diagram. We find that the density-density interaction renders the theory unstable if its coupling constant g₁ is tuned above 2π and that the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover extends to 𝜇̃ /(ϵB) = −g₁/(2π) for g₁ < 2π, where ϵB is the Cooper pair binding energy in vacuum and 𝜇̃ the renormalized chemical potential. Furthermore, we find an exotic phase at the phase boundary (g₁= 2π, 𝜇̃ = −ϵB ) resulting from a flat direction in the leading-order effective potential, where the degenerate ground states span the entire BCS–BEC crossover.