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Physical review. B, Condensed matter

Twisting in superfluid A3 and consequences for hydrodynamics atT=0

R. Combescot, T. Dombre

Published Jan 1, 1986• 32 citations
Publisher

Abstract

We study the effects of twisting on the low-energy excitations of superfluid $^{3}\mathit{A}$. They are small compared to the effects of bending, but they have a different symmetry which makes them important. The structure of the excitation wave functions has strong analogies with the eigenstates of a charged particle in a magnetic field. The group velocity is parallel to curl l^. The density of states at zero energy has a contribution proportional to twisting which is odd with respect to k^. Because of this term a density fluctuation produces a fluctuation of the current carried by the excitations. This current fluctuation turns out to be exactly equal to the fluctuation of the ${C}_{0}$ term in the current. This shows that the ${C}_{0}$ term fluctuation is carried by the normal liquid and leads us to believe that the ${C}_{0}$ term as a whole is linked to the normal liquid. Since calculations making use of the Bardeen-Cooper-Schrieffer wave function consider only the condensate, our results offer a possible explanation for the ``angular-momentum paradox.'' We physically interpret the ${\mathrm{v}}_{n}$ term in the Josephson equation at T=0 and show that it is readily obtained from the microscopic description of the normal liquid. In the same way we deduce from the momentum conservation law an equation of motion for ${\mathrm{v}}_{n}$ and we show that it can be entirely understood from the microscopic framework.

DOI
10.1103/physrevb.33.79

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We present a new calculation of the current $\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$ flowing in a ground state of the Bardeen-Cooper-Schrieffer (BCS) form for a weakly inhomogeneous superfluid with the symmetry of $^{3}\mathrm{He}$-$A$. When the structure of the order parameter not determined by symmetry is appropriate to $^{3}\mathrm{He}$-$A$ and when the mass density $\ensuremath{\rho}$ of the helium is essentially uniform, our current reduces to that calculated by Cross. If the mass density is allowed to vary, we find a generalization of the Cross current which shows that when ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}_{s}=0$ and the anisotropy axis $\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$ is uniform, then the current is simply $(\frac{\ensuremath{\hbar}}{4M}){\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}}_{\ensuremath{\rho}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$. We show that this property of the BCS ground state, which taken with the Cross definition leads to an "intrinsic angular momentum density" of $\frac{\ensuremath{\rho}\ensuremath{\hbar}}{2M}$ at zero temperature, also follows directly from the Gor'kov equations. If the range of the order parameter is taken to be small compared with the interatomic separation, then the ground state does not describe $^{3}\mathrm{He}$-$A$, but a Bose-Einstein condensate of tightly bound diatomic molecules. In this limit our current reduces to the form calculated by Ishikawa et al. We indicate why their analysis is only valid in this limit, and offer some rather more general remarks on the differences between Cooper pairing and the Bose-Einstein condensation of diatomic molecules.

Physical review. B, Condensed matter•Feb 1, 1980•84 citations
We study the effects of twisting on the low-energy excitations of superfluid 3A^{3}\mathit{A}3A. They are small compared to the effects of bending, but they have a different symmetry which makes them important. The structure of the excitation wave functions has strong analogies with the eigenstates of a charged particle in a magnetic field. The group velocity is parallel to curl l^. The density of states at zero energy has a contribution proportional to twisting which is odd with respect to k^. Because of this term a density fluctuation produces a fluctuation of the current carried by the excitations. This current fluctuation turns out to be exactly equal to the fluctuation of the C0{C}_{0}C0​ term in the current. This shows that the C0{C}_{0}C0​ term fluctuation is carried by the normal liquid and leads us to believe that the C0{C}_{0}C0​ term as a whole is linked to the normal liquid. Since calculations making use of the Bardeen-Cooper-Schrieffer wave function consider only the condensate, our results offer a possible explanation for the ``angular-momentum paradox.'' We physically interpret the vn{\mathrm{v}}_{n}vn​ term in the Josephson equation at T=0 and show that it is readily obtained from the microscopic description of the normal liquid. In the same way we deduce from the momentum conservation law an equation of motion for vn{\mathrm{v}}_{n}vn​ and we show that it can be entirely understood from the microscopic framework.
Particle interactions in a Fermi gas may be such as to attract pairs near the Fermi surface more strongly in l=1,2,3l=1, 2, 3l=1,2,3 or higher states than in the simple spherically symmetrical sss 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=1and2l=1 \mathrm{and} 2l=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=2l=2l=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{\mathrm{He}}^{3}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.
A system of hydrodynamic equations has been obtained for superfluid Fermi systems with singlet pairing and nonzero momentum which are formed, e.g., in the case of small polarization of the spin system. As a result, the energy gap is a periodic function of the coordinates and a spin-density wave is formed. For such systems the complete set of dynamic variables includes both purely hydrodynamic (superfluid and normal) velocities and the displacement vector corresponding to the violations of the translational invariance. All of the tensors of normal and superfluid densities that appear in the system as well as the elastic moduli have been calculated. A dispersion relation has been obtained for thermal vibrations (second sound) and elastic vibrations, and this relation has turned out to be substantially anisotropic. Observation of the predicted effects is most probable in 3He–He II solutions in magnetic fields of the order of several kilooersteds at low temperatures.
The effects of possible superfluidity on the cranking moment of a large many-fermion system moving under periodic boundary conditions are investigated within the framework of the theory of superconductivity recently formulated by Bogolyubov. The Hamiltonian is initially subjected to Bogolyubov's general unitary transformation. The collective excitations of the fermions are then considered in the usual pair approximation; the appropriate cranking terms are linear in the boson pair operators. On performing a unitary transformation which transforms away these linear terms, one obtains an expression for the moment of inertia of the system which includes both the effects of possible superfluidity and collective excitation. This expression, by virtue of its being stationary with respect to arbitrary variations in the amplitude associated with the latter unitary transformation, is then utilized as a variational principle for the moment of inertia. For the normal state, the result previously obtained by the author, that the moment of inertia has the rigid value, is rederived in more compact form. For the superfluid state, one finds that collective excitations effect a marked increase in the superfluid moment at intermediate coupling strengths although the resulting moment is still quite small compared to the rigid value. In contrast to the normal state case, where particle-hole pairs play a major role, this increase is almost entirely due to excitations consisting of particle-pairs or hole-pairs. The precise magnitude of the apparent resonance in the moment produced by the ddd-wave part of the cranking interaction is dependent to some extent on the features of the particle-particle potential which leads to the superfluid state. Variational expressions for the moment are exhibited for both Yukawa and delta-function shell potentials. These results are identical in charged and neutral Fermi systems. A calculation of the cranking moment at finite temperatures is presented in an Appendix along with an interpretation of it in terms of Bardeen's two-fluid model of superconductivity.
A theory of a Fermi superfluid is formulated based on the condensate wave function of the configuration space. First, two basic ingredients are extracted from the wave function, i.e. the effective condensate wave function v ( x 1 , x 2 ) and the quasiparticle field γ( x ) with x denoting the space and spin coordinates r α. After transforming a Hamiltonian into a normal-ordered product of γ, a perturbation expansion with respect to the quasiparticle interaction is formulated for both the thermodynamic potential Ω and the one-particle density matrix ρ. The thermodynamic potential thus obtained includes the normal-state expression of Luttinger and Ward as the limit of v → 0. The density matrix is written as a sum of the coherent part ρ ( c) and the quasiparticle part ρ ( q) , where ρ ( c) is expressed only with respect to v ( x 1 , x 2 ). It is shown that v and γ are responsible for the coherent and thermodynamic properties respectively. Thus the present formalism gives a microscopic foundation for the two-fluid model. Connections with the Gor'kov equations and the Bogoliubov-de Gennes equations are also clarified. It is found that the one-particle density matrices of those formalisms are not equivalent to ρ of the present formalism. The origin of the angular momentum paradox of 3 He - A is traced to this difference.
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
We present a new calculation of the current $\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$ flowing in a ground state of the Bardeen-Cooper-Schrieffer (BCS) form for a weakly inhomogeneous superfluid with the symmetry of 3He^{3}\mathrm{He}3He-AAA. When the structure of the order parameter not determined by symmetry is appropriate to 3He^{3}\mathrm{He}3He-AAA and when the mass density $\ensuremath{\rho}$ of the helium is essentially uniform, our current reduces to that calculated by Cross. If the mass density is allowed to vary, we find a generalization of the Cross current which shows that when ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}_{s}=0$ and the anisotropy axis $\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$ is uniform, then the current is simply $(\frac{\ensuremath{\hbar}}{4M}){\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}}_{\ensuremath{\rho}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$. We show that this property of the BCS ground state, which taken with the Cross definition leads to an "intrinsic angular momentum density" of $\frac{\ensuremath{\rho}\ensuremath{\hbar}}{2M}$ at zero temperature, also follows directly from the Gor'kov equations. If the range of the order parameter is taken to be small compared with the interatomic separation, then the ground state does not describe 3He^{3}\mathrm{He}3He-AAA, but a Bose-Einstein condensate of tightly bound diatomic molecules. In this limit our current reduces to the form calculated by Ishikawa et al. We indicate why their analysis is only valid in this limit, and offer some rather more general remarks on the differences between Cooper pairing and the Bose-Einstein condensation of diatomic molecules.