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East European Journal of Physics

CRITICAL TEMPERATURE OF THE SUPERFLUIDTRANSITION IN FERMI-SYSTEM AT AN ARBITRARY PAIR POTENTIAL

Yurii M. Poluektov, A. A. Soroka

Published Nov 1, 2014• 0 citations
Publisher

Abstract

A method for calculation of the critical temperature of transition of a many-particle Fermi system into a superfluid or a superconducting state at an arbitrary pair potential of the interparticle interaction is proposed. An original homogeneous integral equation, that determines the critical temperature, is transformed into a homogeneous integral equation with a symmetric kernel that enables application of a general theory of integral equations for calculation. Examples are given of calculation of the superconducting transition critical temperature for the BCS potential with the Coulomb repulsion and the critical temperature of the superfluid transition of liquid helium-3 into p -wave pairing state for the Morse potential.

DOI
10.26565/2312-4334-2014-4-03

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A method for calculation of the critical temperature of transition of a many-particle Fermi system into a superfluid or a superconducting state at an arbitrary pair potential of the interparticle interaction is proposed. An original homogeneous integral equation, that determines the critical temperature, is transformed into a homogeneous integral equation with a symmetric kernel that enables application of a general theory of integral equations for calculation. Examples are given of calculation of the superconducting transition critical temperature for the BCS potential with the Coulomb repulsion and the critical temperature of the superfluid transition of liquid helium-3 into p -wave pairing state for the Morse potential.
A method for calculation of the critical temperature of transition of a many-particle Fermi system into a superfluid or a superconducting state at an arbitrary pair potential of the interparticle interaction is proposed. An original homogeneous integral equation, that determines the critical temperature, is transformed into a homogeneous integral equation with a symmetric kernel that enables application of a general theory of integral equations for calculation. Examples are given of calculation of the superconducting transition critical temperature for the BCS potential with the Coulomb repulsion and the critical temperature of the superfluid transition of liquid helium-3 into ppp-wave pairing state for the Morse potential.
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{\mathrm{He}}^{3}He3 or to nuclear matter, it is found for a wide class of trial functions that the system does not become a superfluid.
Fermion pairing in the problem of superconductivity and superfluidity, caused by direct interaction between particles, is studied. A self-consistent equation is obtained for the transition temperature in a system of fermions of different species. It is shown that the results in this case may differ qualitatively from those obtained from the BCS theory. Pairing between electrons and positrons caused by screened Coulomb interaction is investigated. The temperature dependence is found to be asymptotic both for high and low densities of the electron-positron plasma.
The self-consistent equations, which have been derived recently as a microscopic model for the crossover between BCS superconductivity and Bose-Einstein condensation in a three-dimensional interacting Fermi system [R. Haussmann, Z. Phys. B 91, 291 (1993)], are solved numerically by repeated Fourier transformation. We find a superfluid transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ which increases monotonically with increasing attractive coupling strength. Furthermore, we determine the chemical potential \ensuremath{\mu}, the fermion distribution function n(k), and the complex effective mass 2m∗{\mathit{m}}^{\mathrm{*}}m∗ of the fermion pairs at T=Tc{\mathit{T}}_{\mathit{c}}Tc​. The bound fermion pairs cause a power-law tail \ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}4}$ in n(k) for large k and behave as short-living quasiparticles in the crossover region, which is indicated by a large imaginary part of 2m∗{\mathit{m}}^{\mathrm{*}}m∗.
The Fermi-liquid approach generalized to superfluid systems is used to derive general formulas for the normal and abnormal distribution functions (DF) for quasiparticles of a superfluid Fermi liquid (SFL) consisting of electrically neutral fermions with spin-triplet pairing (the spin of a pair s=1, the orbital angular momentum l of the pair being an arbitrary odd number) in a static uniform magnetic field, taking into account the Landau spin-exchange Fermi-liquid interaction at temperatures 0<T<Tc (Tc is the temperature of phase transition from the normal to the superfluid state). In deriving the DF, the explicit form of the energy functional (EF) for SFL is not used. A system of coupled equations is obtained for the order parameter, energy of quasiparticles, and effective magnetic field for the SFL under investigation in the case of an EF quadratic in DF. The obtained DF and the system of equations are applicable for describing both unitary and nonunitary phases of superfluid He3 in a magnetic field, e.g., for determining the magnetization and magnetic susceptibility of these phases.
A Fermi-liquid approach to the theory of superfluidity is developed, based on a generalization of the combinatoric expression for the entropy to the case in which the state of the Fermi liquid is described not only by a normal but also by an anomalous distribution function. The energy of the superfluid Fermi liquid is given in the form of a functional of these distribution functions. In the weak-coupling approximation the Fermi-liquid approach is equivalent to the BCS theory. The interaction between the Fermi liquid and an electromagnetic field is gauge invariant. In addition to the singlet-pairing theory the triplet-pairing theory is also developed. A unified variational principle that gives rise to the theory of superfluidity (superconductivity) is used to obtain the London equations and the Ginzburg–Landau equation. Applications of this theory are made to the derivation of an equation for ideal two-component hydrodynamics.