Comment on the Coupling of Zero Sound

Features in the zero sound attenuation near the pair-breaking edge in superfluid 3He-B have been observed in large magnetic fields. Schopohl and Tewordt [J. Low Temp. Phys. 57, 601 (1984)] claim that the J = 1−,M = ±1 order-parameter collective modes couple to zero sound as a result of the distortion of the equilibrium order parameter by a magnetic field; they identify the new features with these modes. However, we show that, when the effect of gap distortion on the collective modes is properly taken into account, the collective mode equations of Schopohl and Tewordt yield no direct coupling of zero sound to the J = 1− modes. Thus, the identification of the absorption features reported by Ling, Saunders and Dobbs [Phys. Rev. Lett. 59, 461 (1987)] near the pair-breaking edge with the J = 1− modes is not clearly established. * Present address: Department of Physics, Ohio State University, 174 West 18th Ave., Columbus, OH 43210, USA. Two features, a “peak” and an “anti-peak”, have been observed1,2,3 in the attenuation spectrum of zero sound in superfluid 3He-B near the pair-breaking edge (ω = 2∆) in strong magnetic fields. Schopohl and Tewordt4 (ST) have identified these features with the order-parameter collective modes that are odd under the particle-hole transformation, have total angular momentum J = 1 and magnetic quantum numbers M = ±1. In zero field elementary symmetry arguments show that the J = 1− modes do not couple to zero sound.5 To identify the observed features with the J = 1−,M = ±1 collective modes it must be shown that these modes have frequencies corresponding to the observed positions in the spectrum, and that they couple to zero sound in the presence of a magnetic field. Schopohl and Tewordt show that in a magnetic field, ~ H = Hẑ, the J = 1− modes have frequencies ω1−,M = 2∆−M g ωL , (1) where ωL is the effective Larmor frequency and g is the Landé g-factor for these modes. The value for g calculated by ST agrees with the value deduced from the splitting of the two features seen by Ling, Saunders and Dobbs.2 In addition, ST argue that the J = 1−, M = ±1 modes couple to zero sound in a magnetic field. In a field the B-phase order parameter is no longer the isotropic Balian-Werthamer (BW) state with J = 0, but has the form ~ ∆(p̂) = (∆1 p̂x,∆1 p̂y,∆2 p̂z) (2) where the anisotropy (“gap distortion”) is quadratic in the field, i.e. ∆1 − ∆2 ∼ H2. Schopohl and Tewordt claim that the J = 1−,M = ±1 modes couple to zero sound in a magnetic field because of this gap distortion. However, ST neglected the distortion of the collective modes by the field. When gap distortion is properly included in the time-dependent gap equation the coupling between zero sound and the J = 1− modes considered by ST vanishes. Thus, the identification of the attenuation features observed by Ling, et al. with the J = 1− modes is not established. Further inconsistencies between the observations of Ref. 2 and the theory of ST are pointed out in Ref. 3. The J = 1− modes are excitations of the the l = 1 component of the imaginary part of the order parameter, d−μ (p̂, ω, ~q ) = d − μj(ω, ~q )p̂j , the dynamics of which are described by