The possible superfluidity of a system of strongly interacting fermions is investigated on the assumption that an adequate description of the system in its "normal" state is given by independent fermions in a momentum-dependent potential. On the basis of this assumption we have investigated whether a correlated wave function of the form used by Bardeen, Cooper, and Schrieffer minimizes the ground-state energy. The nonzero terms in the expectation value of the Hamiltonian contain the modified kinetic energy and the full two-body potential between the fermion pairs. An integral equation is obtained in configuration space for the correlation function between pairs. This integral equation is meaningful even for potentials with hard cores, and a nonzero solution implies the existence of a superfluid state. A variational method is devised which provides a criterion for superfluidity and a lower bound for the transition temperature into the superfluid state. We find that a repulsive hard core does not in principle forbid the existence of a superfluid state, but whereas in the absence of a hard core an attractive two-body potential always leads to a superfluid state at sufficiently low temperatures, in the presence of a repulsive core there appears to be a critical strength of attraction needed to form a superfluid state. When the variational principle is applied to liquid ${\mathrm{He}}^{3}$ or to nuclear matter, it is found for a wide class of trial functions that the system does not become a superfluid.
The possible superfluidity of a system of strongly interacting fermions is investigated on the assumption that an adequate description of the system in its "normal" state is given by independent fermions in a momentum-dependent potential. On the basis of this assumption we have investigated whether a correlated wave function of the form used by Bardeen, Cooper, and Schrieffer minimizes the ground-state energy. The nonzero terms in the expectation value of the Hamiltonian contain the modified kinetic energy and the full two-body potential between the fermion pairs. An integral equation is obtained in configuration space for the correlation function between pairs. This integral equation is meaningful even for potentials with hard cores, and a nonzero solution implies the existence of a superfluid state. A variational method is devised which provides a criterion for superfluidity and a lower bound for the transition temperature into the superfluid state. We find that a repulsive hard core does not in principle forbid the existence of a superfluid state, but whereas in the absence of a hard core an attractive two-body potential always leads to a superfluid state at sufficiently low temperatures, in the presence of a repulsive core there appears to be a critical strength of attraction needed to form a superfluid state. When the variational principle is applied to liquid He3 or to nuclear matter, it is found for a wide class of trial functions that the system does not become a superfluid.
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).
One considers the superfluid (SF) state of a Bose liquid with a strong repulsion between bosons, in which at T=0, along with a weak single-particle Bose-Einstein condensate (BEC), there exists an intensive pair coherent condensate (PCC), analogous to the Cooper condensate in a Fermi liquid with an attraction between the fermions. Such a PCC emerges in a system of bosons due to an oscillating sign-changing momentum dependence of the Fourier component of the pair interaction potential, which is characteristic of a certain family of repulsion potentials. In such cases, the Fourier component is negative in some domain of nonzero momentum transfer, which corresponds to an effective attraction. The collective effects of renormalization (``screening'') of the initial interaction lead to a suppression of the repulsion and an enhancement of the effective attraction. It is the ratio of the BEC density to the full density of the liquid n0/n≪1 that is used as a small parameter---unlike in the Bogolyubov theory for a quasi-ideal Bose gas, in which the small parameter is the ratio of the number of supracondensate excitations to the number of particles in an intensive BEC, (n−n0)/n0≪1. A closed system of nonlinear integral equations for the normal and anomalous self-energy parts is obtained, in the framework of a renormalized perturbation theory built on combined hydrodynamic and field variables. In the framework of the hard-spheres model, a spectrum of quasiparticles is obtained, which is in good accordance with the experimental spectrum of elementary excitations in superfluid 4He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.
Particle interactions in a Fermi gas may be such as to attract pairs near the Fermi surface more strongly in l=1,2,3 or higher states than in the simple spherically symmetrical s state. In that case the Bardeen-Cooper-Schrieffer condensed state must be generalized, and the resulting state is an anisotropic superfluid. We have studied the properties of this type of state in considerable detail, especially for l=1and2. We have derived expressions for the energy, the moment of inertia, the magnetic susceptibility and the specific heat. We also derive the density correlation function and the density-current density correlation; in some cases the latter implies that the liquid has net surface currents and a net orbital angular momentum. The ground state for l=2 is different from those previously considered, and has cubic symmetry and no net angular momentum. A general method for replacing the possibly rather complicated potential by a simple scattering matrix is given. A brief discussion of possible collective effects is included. We apply our results to liquid He3; after correction for scattering by a method due to Suhl, it is found that the predicted transition should take place below 0.02\ifmmode^\circ\else\textdegree\fi{}K. Other possible applications are suggested.
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
Quantum degenerate Fermi gases can be created in the laboratories using alkali atoms. These gases can be in different regimes of density and interaction strength and provide an ideal test bed for the basic properties of the quantum and statistical mechanics. Also some astrophysical objects such as neutron stars can be idealized as interacting fermion systems. In this thesis work, the ground state properties of dilute Fermi gases with attractive short range two-body interactions are reported. Main results of this work are produced by application of quantum Monte Carlo methods. We introduce variational nodal structure to the trial wave function in order to deal with the fermion sign problem. We consider the possibility of Cooper pairing between fermions of different species. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the s-wave scattering length a and the Fermi wave vector kF. We also review some of the conventional variational methods such as the Lowest Order Constrained Variational which is suitable for Fermi fluids without pairing correlation. The ground state energy, pairing gap Δ, quasi-particle spectrum, two-body distribution function, and momentum distribution of the superfluid were studied with greater accuracy using ab initio quantum Monte Carlo techniques. In the case of the two component Fermi fluid, we find that in the weak coupling regime, 1/akF < -1, the system is a Bardeen-Cooper-Schrieffer type superfluid and the energy gap Δ is much smaller than the Fermi gas energy EFG. For 1/akF ≥ 0:5 we find that weakly interacting composite bosons or molecules are formed in the superfluid gas with Δ and gas energy per particle approaching half of the molecular energy jEmolj=2. In this region we seem to have Bose-Einstein condensation of molecules. The behavior of the energy, chemical potential and energy gap in the transition region, -0:5 < 1/akF < 0:5, is discussed. The study is extended to the three component Fermi fluid in relation to the spin polarized alkali gases and color superconductivity of the quark matter. Unlike the two component fluids, zero potential range limit cannot be reached without causing collapse of the system. We also study the scaling behavior of the three-body system. We calculate the threshold of the three-body bound state as a function of the potential range and strength. For the many-body systems, we find that ground state can be achieved by broken symmetry pairing. However, the restored symmetry pairing scenario in the strongly interacting regime is not conclusive. We also discuss the pairing gap as well as the structural properties such as two-body distribution functions.
We introduce a new fermionic variational wavefunction, generalizing the Bardeen-Cooper-Schrieffer (BCS) wavefunction, which is suitable for interacting multi-species spinful systems and sustaining superfluidity. Applications range from quark matter to the high temperature superconductors. A wide class of Hamiltonians, comprising interactions and hybridization of arbitrary momentum dependence between different fermion species, can be treated in a comprehensive manner. This is the case, as both the intra-species and the inter-species interactions are treated on equally rigorous footing, which is accomplished via the introduction of a new quantum index attached to the fermions. The index is consistent with known fermionic physics, and allows for heretofore unaccounted for fermion-fermion correlations. We have derived the finite temperature version of the theory, thus obtaining the renormalized quasiparticle dispersion relations, and we discuss the appearance of charge and spin density wave order. We present numerical solutions for two electron species in 2 dimensions. Based on these solutions, we show that, for equivalent spin up and down fermions, the Fermi occupation factor (per spin) equals 1/2 deep in the Fermi sea. This constitutes a unique experimental prediction of the theory, both for the normal and superfluid states. Interestingly, this result, obtained in the thermodynamic limit, is consistent with Fermi occupation factor (in-)equalities for finite systems of electrons, derived (in a different context) by Borland and Dennis, J. Phys. B {\bf 5}, 7 (1972) and by Altunbulak and Klyachko, Commun. Math. Phys. {\bf 282}, 287 (2008).