Gap parameters and collective modes for ³He-B in the presence of a strong magnetic field

Abstract

First we determine the 4 × 4 matrix Green's function for a p-wave pairing superfluid in a magnetic field where the order parameter is given by a real 3 × 3 matrix. For the B phase we take an order parameter which is equal to δ₁ times a 3 × 3 matrix, yielding a rotation of angle θ about the z-axis and a dilatation δ₂/δ₁ along the z-axis. Then the self-consistency equation for the 4 × 4 matrix self-energy reduces to three equations for δ₁, δ₂, and cos θ, and a fourth equation for the renormalized Larmor frequency. We find that, for increasing field, δ₁ increases and δ₂ decreases with respect to the zero-field gap δ_00. Above a (temperature-dependent) critical field we find δ₂ = 0 and α₂ = 90° corresponding to the planar state of lowest dipole energy. The correlation functions for the order parameter collective modes are calculated with the help of a previous theory. The results can be expressed in terms of six universal functions describing internal magnetization and virtual excitations of pairs of quasiparticles with all spin orientations. The complete set of eigenfrequencies as functions of the field is calculated for T = 0 and q = 0. The longitudinal NMR frequency is found to be almost independent of the field. We find a splitting of the pair-vibration frequencies (8/5) ^1/2δ₀₀ and (12/5)^1/2δ₀₀ which is linear in the field. This splitting is caused by fluctuations involving the spin-singlet component of the anomalous propagator. The splitting of the pair-vibration frequency (12/5)^1/2δ₀₀ (of the order 6H MHz, H in kG) should be observable by sound absorption experiments in strong magnetic fields.