Interaction Effects on the Zeeman Splitting of Collective Modes in SuperfluidHe3-B

Abstract

Exact calculations of the B-phase collective-mode frequencies in a magnetic field are presented, and they provide new information about the effective mass, the pairing interaction, and the Landau parameters. The calculation of the Zeeman splitting for the 2+ mode (real squashing mode) is consistent with the zero-field frequency and agrees quantitatively with the measurements of Avenel et al. 9 for a range of values of m* which includes some, but not all, of the differing experimental results for this important parameter.PACS numbers: 67.50.Fi Superfluid 3 He supports a number of orderparameter collective modes with temperaturedependent frequencies of the formwhere a and b are weakly temperature-dependent functions of order 1. The zero-sound dispersion relation co=c Q q crosses the dispersion relations for these modes when qv ¥ /A(T)~v F /c 0 «l, and hence in discussing their excitation by ultrasound one can generally neglect the dispersion of the collective modes. It is suggestive to think of the collective modes as excited states of the Cooper pairs, and to regard their excitation by zero sound as an ultrasonic Cooper-pair spectroscopy analogous to the optical spectroscopy of diatomic molecules. This analogy is strengthened by studies of zero-sound absorption in a magnetic field. For example, Schopohl and Tewordt 1 * 2 predicted that a small magnetic field should produce a linear "Zeeman splitting" of both the real and imaginary J=2 order-parameter modes in 3 He-i3 (the 2 + and 2 -modes, in our previous notation, 3 which we follow henceforth). The collective modes are then labeled by ffi j, their angular momentum projection along H, and their frequencies have the form u> 2± (m^( T, H) = u 2 JL7 1 ) + m J g 2 Ji7 i )w L {T, H\where oo L (T,H), the temperature-dependent effective Larmor frequency, is proportional to the magnetic field. The Zeeman splitting of the 2 -mode has yet to be observed, but the splitting of the 2 + mode has been measured by Avenel, Varoquaux, and Ebisawa; 4 their observation of a fivefold splitting of the new ultrasonic absorption line discovered by Giannetta et al. 5 and Mast et al. 6 established conclusively that this absorption line is associated with the 2+ mode. Equation (2) gives 6oo(T,H), the linear splitting of the modes at a fixed temperature, but experimentally one measures 6T(w, H), the splitting in temperature of the modes at a fixed frequency. If we expand Eq. (2) around T 0 , the temperature of the absorption line in zero field, we obtain oT ((l3Schopohl and Tewordt 7 calculated the Zeeman splittings of the 2 ± modes keeping only the Landau parameter F 0 a and assuming a pure 1 = 1 pairing interaction. With the help of a new identity given below [Eq. (14)], their results can be written as co L (T,H) = yH/[l + F 0^ + iY)].

Magneto-acoustic rotation of transverse waves in 3He–B

“…5 are the effective Larmor frequency, ω L = γ eff H, the condensate response, ρ s (ω, T + ) and the slope of the mode frequency, |Ω ′ + |T + . The remaining open parameter is the Landé g-factor which determines the vertical scale for Λ H ; the fit to the data gives g = 0.02 ± 0.002, which is in agreement with theoretical calculations of the g-factor that include attractive f-wave pairing correlations in 3 He-B [4].…”

Transverse waves in superfluid3He-B

“…To complete the calculation of the restoring force we solve for the order parameter response to a transverse current, (@],/Sg~u), from the inhomogeneous gap equation (37). We first carry out the p-wave projection to obtain, …”

Symmetry protected topological superfluid3He-B

“…The magnetic Zeeman field, however, gives rise to the splittings of J = 2 − collective modes as Ω 2− (q, H) = Ω 2− (q, 0) + M J g 2− ω L , where g 2− is the g-factor for J = 2 − modes and ω L is the effective Larmor frequency [339,351] for the left circularly polarized wave is deviated by the magnetic field from that for the right circularly polarized wave, C + = C − . The circular birefringence gives rise to the acoustic analogue of the magneto-optic Faraday effect in which the direction of a linearly polarized wave rotates along the direction of propagation with the period proportional to the inverse of the Zeeman splitting [349,350].…”