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Physical Review A

Landau-Ginsburg equations for an anisotropic superfluid

Vinay Ambegaokar, Pierre-Gilles DeGennes, Dierk Rainer

Published Jun 1, 1974• 414 citations
Publisher

Abstract

A discussion of an anisotropic ($p$ state) Fermi superfluid with a spatially varying order parameter is presented. We restrict ourself to the Landau-Ginzburg region and consider spatial variations governed by the linear gap equation. The kernel of the linearized gap equation is shown to be related to a current-current correlation function of quasiparticles in the normal state. This relation permits the calculation of the two coherence lengths ${\ensuremath{\xi}}_{L}$ and ${\ensuremath{\xi}}_{T}$ defined earlier in the framework of a phenomenological Landau-Ginzburg theory, and a discussion of boundary conditions for the order parameter at interfaces. A significant anisotropy in the coherence lengths is found (${\ensuremath{\xi}}_{L}=\sqrt{3}{\ensuremath{\xi}}_{T}$), which reflects itself in the nature of the supercurrents. The boundary conditions at specularly and diffusely reflecting surfaces lead us to expect that the vector $\stackrel{^}{l}$, which describes the orbital angular momentum of a Cooper pair in an "axial" state, is anchored normal to the walls. Some consequences of these results on various experimental problems (fourth sound, monodomain production, Josephson couplings, behavior in thin capillaries) are briefly discussed.

DOI
10.1103/physreva.9.2676

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Although our discussion in this section will be largely phenomenological we must anticipate the major result of the microscopic theory which will be developed in Parts II and III; in particular we must make use of the fact that superfluidity in Fermi systems arises from the formation of a special kind of bound state in which electrons (or He quasiparticles) on (or near) the Fermi surface having oppositely directed momenta, + p and – p , form (collectively) bound pairs. For the vast majority of superfluid electron systems (metals), the pairs form in a state with no net orbital angular momentum, an l = 0 or s-wave state which we will refer to as conventional fermion superfluidity; by the Pauli principle the electrons must then be in a singlet spin state (s = 0). In the decade or so following the development of the BCS theory, and especially after the experimental discovery of the superfluid phases of liquid He, theorists explored various types of pairing in systems with a more complicated orbital and spin structure in which the gap can vary, in both magnitude and phase, with position on the Fermi surface. In the case of odd l pairing, the pairs form (again due to the Pauli principle) in a triplet s = 1 rather than a singlet spin state. The energy gap associated with such states may have zeros (i.e., vanish) at points or lines on the Fermi surface, and as a result the number of excitations at low temperatures varies as a power of the temperature, rather than exponentially.

Cambridge University Press eBooks•Jan 12, 1999•0 citations
A discussion of an anisotropic (ppp state) Fermi superfluid with a spatially varying order parameter is presented. We restrict ourself to the Landau-Ginzburg region and consider spatial variations governed by the linear gap equation. The kernel of the linearized gap equation is shown to be related to a current-current correlation function of quasiparticles in the normal state. This relation permits the calculation of the two coherence lengths \ensuremathξL{\ensuremath{\xi}}_{L}\ensuremathξL​ and \ensuremathξT{\ensuremath{\xi}}_{T}\ensuremathξT​ defined earlier in the framework of a phenomenological Landau-Ginzburg theory, and a discussion of boundary conditions for the order parameter at interfaces. A significant anisotropy in the coherence lengths is found (\ensuremathξL=3\ensuremathξT{\ensuremath{\xi}}_{L}=\sqrt{3}{\ensuremath{\xi}}_{T}\ensuremathξL​=3​\ensuremathξT​), which reflects itself in the nature of the supercurrents. The boundary conditions at specularly and diffusely reflecting surfaces lead us to expect that the vector \stackrel{^}{l}, which describes the orbital angular momentum of a Cooper pair in an "axial" state, is anchored normal to the walls. Some consequences of these results on various experimental problems (fourth sound, monodomain production, Josephson couplings, behavior in thin capillaries) are briefly discussed.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
The Ginzburg–Landau (GL) equation is derived for the spatially inhomogeneous order parameter (OP) of the He3 type superfluid Fermi liquid (SFL) (composed of electrically neutral fermions with triplet p-wave pairing) for a uniform flow of the normal component of the SFL with a velocity vn≪2Tc/pF (Tc is the temperature of phase transition from the normal to the superfluid state, and pF is the Fermi momentum). The derivation is based on the integrodifferential equation for OP, which is valid at temperatures near Tc in the absence of strong-coupling effects. The main results are the corrections for the coefficients of the OP gradients in the GL equation (caused by the fact that vn≠0), which are calculated in the perturbation theory. These corrections, which indicate the deformation of the OP due to the SFL flow with the velocity vn, can be not only comparable for vn≪2Tc/pF, but also greater than the so-called nontrivial strong-coupling corrections to these coefficients, depending on the value of vn, and hence can be of practical importance for describing He–B3 at low pressures, when the strong-coupling effects are weak.
The purpose of this work is to study superfluid properties of a neutral fermion system with pairing in a non-zero angular momentum state, in particular, with 3P2 pairing expected in neutron stars. The gap equation and the equation of the Ginzburg-Landau (G-L) type are derived for general pairing. We then specialize to the case of 3P2 pairing and present the G-L equation for the five component order parameter in the form of a non-linear field equation for a spin 2 field. The expressions for the free energy, the current density and the angular momentum density are found in terms of the order parameter. With the help of the G-L equation some features of vortex states are discussed.
The spectra of fermionic excitations, pairing correlations, and edge currents confined near the boundary of a chiral ppp-wave superfluid are calculated to leading order in $\ensuremath{\hbar}/{p}_{f}\ensuremath{\xi}$. Results for the energy- and momentum-resolved spectral functions, including the spectral current density, of a chiral ppp-wave superfluid near a confining boundary are reported. The spectral functions reveal the subtle role of the chiral edge states in relation to the edge current and the angular momentum of a chiral ppp-wave superfluid, including the rapid suppression of Lz(T){L}_{z}(T)Lz​(T) for $0\ensuremath{\lesssim}T\ensuremath{\ll}{T}_{c}$ in the fully gapped two-dimensional chiral superfluid. The edge current and ground-state angular momentum are shown to be sensitive to boundary conditions, and as a consequence the topology and geometry of the confining boundaries. For perfect specular boundaries, the edge current accounts for the ground-state angular momentum, ${L}_{z}=(N/2)\ensuremath{\hbar}$, of a cylindrical disk of a chiral superfluid with N/2N/2N/2 fermion pairs. Nonspecular scattering can dramatically suppress the edge current. In the limit of perfect retroreflection, the edge states form a flat band of zero modes that are nonchiral and generate no edge current. For a chiral superfluid film confined in a cylindrical toroidal geometry, the ground-state angular momentum is, in general, nonextensive, and can have a value ranging from ${L}_{z}>(N/2)\ensuremath{\hbar}$ to ${L}_{z}<\ensuremath{-}(N/2)\ensuremath{\hbar}$ depending on the ratio of the inner and outer radii and the degree of backscattering on the inner and outer surfaces. Nonextensive scaling of Lz{L}_{z}Lz​, and the reversal of the ground-state angular momentum for a toroidal geometry, would provide a signature of broken time-reversal symmetry of the ground state of superfluid 3{}^{3}3He-AAA, as well as direct observation of chiral edge currents.
Although our discussion in this section will be largely phenomenological we must anticipate the major result of the microscopic theory which will be developed in Parts II and III; in particular we must make use of the fact that superfluidity in Fermi systems arises from the formation of a special kind of bound state in which electrons (or He quasiparticles) on (or near) the Fermi surface having oppositely directed momenta, + p and – p , form (collectively) bound pairs. For the vast majority of superfluid electron systems (metals), the pairs form in a state with no net orbital angular momentum, an l = 0 or s-wave state which we will refer to as conventional fermion superfluidity; by the Pauli principle the electrons must then be in a singlet spin state (s = 0). In the decade or so following the development of the BCS theory, and especially after the experimental discovery of the superfluid phases of liquid He, theorists explored various types of pairing in systems with a more complicated orbital and spin structure in which the gap can vary, in both magnitude and phase, with position on the Fermi surface. In the case of odd l pairing, the pairs form (again due to the Pauli principle) in a triplet s = 1 rather than a singlet spin state. The energy gap associated with such states may have zeros (i.e., vanish) at points or lines on the Fermi surface, and as a result the number of excitations at low temperatures varies as a power of the temperature, rather than exponentially.