Remarkably the one-dimensional (1D) many-fermion fluid with pairwise-attractive \ensuremath{\delta}-function interactions is exactly solvable in that one can determine the exact many-body ground-state energy and chemical potential for all values of the coupling strength and/or density. Bardeen-Cooper-Schrieffer (BCS) theory is tested in this model by numerically determining the BCS total ground-state energy and chemical potential as a function of the coupling strength and/or density, and comparing with the exact results. As is the case for 2D and 3D theories, two regimes are apparent: (a) a BCS-proper regime of weakly coupled, overlapping Cooper pairs and (b) a Bose-gas regime of strongly interacting fermions which pair to form an ideal Bose gas at low density. In the two extremes the BCS energy and chemical potential are identical to the exact values and are moderately close for intermediate coupling and/or density.
Remarkably the one-dimensional (1D) many-fermion fluid with pairwise-attractive \ensuremath{\delta}-function interactions is exactly solvable in that one can determine the exact many-body ground-state energy and chemical potential for all values of the coupling strength and/or density. Bardeen-Cooper-Schrieffer (BCS) theory is tested in this model by numerically determining the BCS total ground-state energy and chemical potential as a function of the coupling strength and/or density, and comparing with the exact results. As is the case for 2D and 3D theories, two regimes are apparent: (a) a BCS-proper regime of weakly coupled, overlapping Cooper pairs and (b) a Bose-gas regime of strongly interacting fermions which pair to form an ideal Bose gas at low density. In the two extremes the BCS energy and chemical potential are identical to the exact values and are moderately close for intermediate coupling and/or density.
The dynamical similarities between the three-dimensional (3D) electron fluid ``jellium'' model and the exactly soluble 1D fermion fluid with attractive \ensuremath{\delta}-function pairwise potentials motivates solving the Cooper-pair problem in this model for all couplings and/or densities. This is accomplished exactly in graphical form. For weak coupling, the essential singularity characteristic of standard 3D low-temperature superconductivity reemerges, and for infinite coupling, BCS theory reproduces the exact ground-state system energy.
We theoretically investigate a quartet superfluid state in fermionic matter by using the quartet Bardeen-Cooper-Schrieffer (BCS) variational theory and the Green's function method. We demonstrate that the quartet BCS theory with the multiple-infinite-product ansatz successfully reproduces an exact four-body result in a one-dimensional four-component Fermi gas at the dilute limit, in contrast to the single-infinite-product ansatz. To see the validity of the quartet BCS state, we derive the self-consistent equation for the quartet superfluid order parameter within the generalized imaginary-time Nambu-Gor'kov formalism, which is found to be consistent with the quartet BCS variational equation. Moreover, by numerically computing the momentum-resolved single-particle spectral function in a one-dimensional system, we discuss how the single-particle spectra evolve with increasing the strength of the four-body cluster formation. We show that a coherent BCS-like quasiparticle branch on the weak-coupling side evolves into a strongly damped, continuum-dominated spectrum in the strong-coupling side, while nonzero quartet superfluid order parameter persists throughout the crossover regime. Our results would be useful for understanding beyond-BCS pairing effects and four-body cluster formations in fermionic systems in an interdisciplinary way.
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.
Abstract The superfluid density ns(T) n s ( T ) of a superconductor is calculated based on the generalized Bose–Einstein condensation (GBEC) theory that addresses a fully-interacting ternary boson-fermion gas mixture of free electrons as fermions, plus two-electron Cooper pairs (2eCPs) and also, explicitly, two-hole Cooper pairs (2hCPs), both as bosons. Here we consider two special cases (i) 100%–0% (i.e., with no condensed 2hCPs) and (ii) 0%–100% (i.e., with no condensed 2eCPs). Subsumed in GBEC are the Bardeen–Cooper–Schrieffer (BCS) and Bose–Einstein condensation (BEC) theories along with the BCS-BEC crossover theory extended with 2hCPs. We find that in the weak-coupling regime ns(0) n s ( 0 ) agrees with data from the Uemura et al. (2004) graph for several elemental SCs by taking in 3D with a quadratic energy-dispersion relation while in 2D with a linear relation are much too far below the data. In the strong-coupling regime the linear behavior of critical temperature Tc T c vs ns(0) n s ( 0 ) obtained here is just as Božović et al. (2016) found. However, in 2D with a linear relation accounting for 0%–100%, ns(T)/ns(0) n s ( T ) / n s ( 0 ) compares well with some high- Tc T c -cuprate SC data between the two coupling regimes.
Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length as. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}kFas, where kF is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.
We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the s-wave scattering length and the Fermi wave vector, akF. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG. When a>0, the interaction is strong enough to form bound molecules with energy Emol. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.