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Journal of Mathematical Physics

Superfluidity of a Fermi liquid from the viewpoint of a hierarchy of equations for reduced density matrices

В. А. Головко

Published Mar 17, 2004• 2 citations
Publisher

Abstract

The hierarchy of equations for reduced density matrices relevant to thermodynamic equilibrium with account taken of the spin obtained earlier is modified in order to describe the state of a Fermi system with a condensate. Although the procedure is to some extent analogous with the one carried out by the author earlier for a Bose liquid peculiarities relevant to Fermi statistics complicate considerably the treatment. As in the case of the Bose liquid the condensate phase can be superfluid as well as nonsuperfluid, the physical causes of superfluidity being identical. A new mechanism of fermion pairing that acts even in the case of a purely repulsive Hamiltonian is pointed out. Special attention is given to the thermodynamics of a superfluid Fermi system. The example of a hard-sphere system is used to find out the form of phase diagrams, the character of the phase transition to a condensate phase and the properties of the last. Noticeable dissimilarities from a Bose system with the same Hamiltonian are revealed. Application of the present approach to superconductivity is discussed as well.

DOI
10.1063/1.1644903

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The hierarchy of equations for reduced density matrices relevant to thermodynamic equilibrium with account taken of the spin obtained earlier is modified in order to describe the state of a Fermi system with a condensate. Although the procedure is to some extent analogous with the one carried out by the author earlier for a Bose liquid peculiarities relevant to Fermi statistics complicate considerably the treatment. As in the case of the Bose liquid the condensate phase can be superfluid as well as nonsuperfluid, the physical causes of superfluidity being identical. A new mechanism of fermion pairing that acts even in the case of a purely repulsive Hamiltonian is pointed out. Special attention is given to the thermodynamics of a superfluid Fermi system. The example of a hard-sphere system is used to find out the form of phase diagrams, the character of the phase transition to a condensate phase and the properties of the last. Noticeable dissimilarities from a Bose system with the same Hamiltonian are revealed. Application of the present approach to superconductivity is discussed as well.
This paper represents the full version of a paper published earlier in Physica A [246 (1997), 275]. The present paper includes argumentation, proofs and details omitted in the shortened version. The papers are a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose-Einstein condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the papers superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The equations obtained are solved in two simple cases. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well.
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≪1n_0/n\ll 1n0​/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(n-n_0)/n_0\ll 1(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 4^44He. The question of applicability of the Landau criterion in the absence of quantum vortices is discussed.
The hierarchy of equations for reduced density matrices that describes a thermodynamically equilibrium quantum system obtained earlier by the author is investigated in the momentum representation. In the paper it is shown that the use of the momentum representation opens up new opportunities in studies of macroscopic quantum systems both nonsuperfluid and superfluid. It is found that the distribution over momenta in a quantum fluid is not a Bose or Fermi distribution even in the limit of practically noninteracting particles. The distribution looks like a Maxwellian one although, strictly speaking, it is not Maxwellian. The momentum distribution in a quantum crystal depends upon the interaction potential and the crystalline structure. The momentum distribution in a superfluid contains a delta function. The momentum distribution for the condensate in a superfluid crystal consists of delta peaks that are arranged periodically in momentum space. The periodical structure remains if the condensate crystal is not superfluid.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.
The spectrum of elementary excitations in the system of interacting Bose- and Fermi-particles was obtained at a temperature of absolute zero (Dose- condensation being taken into account by means of the Bogolyabov canonic transformations. In terms of a model of permeable spheres'' with finite repulsion it(is shown that the boson spectrum omega (p) in a definite region of parameter has the minimum at p = 0. By selecting parameters one succeeds in obtaining a quite satisfactory quantitative agreement of the theoretical and experimental spectra for superfluid helium. In the case when the Fermi energy is considerably higher than the bosorins maximum energy, a gap characterizing bond energy of the copper pairs can appear in the fermion speetrum due to exchange of virtual supercondensate'' bosons. If in this case the interaction between fermions at infinity is of attraction character, then under certain conditions sustained longitudinal collective excitations of Pose-type with a finite gap'' in the energy spectrum may exist in the superfluid fermion subsystem. Possibility of appearance of similar excitations in superfluid nuclear matter of neutron stars (pulsars) due to gravitational interaction between nucleons is discussed. (auth)
Superfluidity in liquid 3 He, discovered in 1972, is due to pairing, as is the case for superconductivity in ordinary metals. However, the pair is in the 3 P state so that a new aspect of superfluidity appears with internal degrees of freedom. It offers quite an interesting real example of ‘spontaneous symmetry breaking’. This aspect will be emphasised in the following discussion. Fermi liquid 3 He A 3 He atom is a fermion with nuclear spin ½. Although 3 He gas liquefies at low temperatures due to the weak van der Waals force, it does not solidify even at T = 0 unless the pressure exceeds 33.5 atm (3.4 × 10 6 Pa) as shown in the phase diagram (Fig. 6.1 (a)). While Bose liquid 4 He becomes superfluid at about 2K, 3 He is Fermi-degenerate at less than T F ∼ 0.1 K and shows the properties of a normal Fermi gas. For example, the specific heat and the susceptibility are proportional to T at T < T F . It is only at an ultra-low temperature in the mK-region that the system undergoes a transition to the superfluid state. Figure 6.1(b) is the phase diagram showing the superfluid phases with variables T,P and external magnetic field H in the mK-region. It should be noted that there are two thermodynamically distinct phases called the A phase and the B phase even at H = 0, and that for H ≠ 0 the A 1 phase appears.