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.
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.
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 .
The crossover from weak-coupling Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of tightly bound pairs, as a function of the attractive interaction in Fermi systems, has long been of interest to theoretical physicists. The past decade has seen a series of remarkable experimental developments in ultracold Fermi gases that have realized the BCS-BEC crossover in the laboratory, bringing with it fresh new insights into the very strongly interacting unitary regime in the middle of this crossover. In this review, we start with a pedagogical introduction to the crossover and then focus on recent progress in the strongly interacting regime. Although our focus is on new theoretical developments, we also describe three key experiments that probe the thermodynamics, transport, and spectroscopy of the unitary Fermi gas. We discuss connections between the unitary regime and other areas of physics—quark-gluon plasmas, gauge-gravity duality, and high-temperature superconductivity—and conclude with open questions about strongly interacting Fermi gases.
The crossover from weak coupling Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of tightly bound pairs, as a function of the attractive interaction in Fermi systems, has long been of interest to theoretical physicists. The past decade has seen a series of remarkable experimental developments in ultracold Fermi gases that has realized the BCS-BEC crossover in the laboratory, bringing with it fresh new insights into the very strongly interacting unitary regime in the middle of this crossover. In this review, we start with a pedagogical introduction to the crossover and then focus on recent progress in the strongly interacting regime. While our focus is on new theoretical developments, we also describe three key experiments that probe the thermodynamics, transport and spectroscopy of the unitary Fermi gas. We discuss connections between the unitary regime and other areas of physics -- quark-gluon plasmas, gauge-gravity duality and high temperature superconductivity -- and conclude with open questions about strongly interacting Fermi gases.
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.
This work studies zero-temperature Fermi gases with short-ranged and attractive interactions with focus on the dimensionality aspects by confining the system from three to two dimensions. Interacting Fermi gases are superfluids characterized as being strongly correlated if the scattering length is much larger than the other scales present. In the many-body context, this category of systems requires a careful treatment since mean-field approaches are not expected to hold under the conditions which define the strong pairing. For this reason, we employ quantum Monte Carlo methods to study the dimensionality effects in two-component Fermi gases in the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover. To be able to investigate such aspects, we consider a three-dimensional gas with periodic boundary conditions in the xy plane and a confining potential in the z direction capable of reducing the dimensionality of the system, which defines a dimensional BCS-BEC crossover. We calculate the equation of state as the energy per particle as a function of the dimensionality and as a function of the interaction parameter, which are characterized by the oscillator length in the z direction and the scattering length respectively. The greater confinement in the system lowers the interparticle distances, which increases the effects of the attractive interactions between antiparallel-spin fermions.We also calculate the density profile in the z direction over the crossover for different confinements and different interaction parameters to quantify their influence on the spatial distribution of the particles. This study connects the perspectives between theoretical and experimental works in low-dimensional ultracold Fermi gases by utilizing an approach that is similar to what it is realized in modern experiments.