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Physical review. B, Condensed matter

Response functions and collective modes of superfluid B3 in strong magnetic fields

R. S. Fishman, James A. Sauls

Published May 1, 1986• 80 citations
Publisher

Abstract

The field dependence of the magnetic susceptibility and collective mode frequencies of superfluid $^{3}\mathit{B}$ are calculated perturbatively in (\ensuremath{\gamma}H/\ensuremath{\Delta}) using the quasiclassical theory. With the aid of this perturbation theory, we interpret the gap distortion, nonlinear susceptibility, and collective excitations of the order parameter in terms of additional correlations between Cooper pairs that are induced by a magnetic field. We also calculate the dispersion splittings of the real squashing modes in a magnetic field. In addition to the nonlinear Zeeman shifts of these modes, which occur in strong magnetic fields (\ensuremath{\gamma}H\ensuremath{\sim}0.10\ensuremath{\Delta}), there are low-field (\ensuremath{\gamma}H\ensuremath{\sim}${q}^{2}$${v}_{f}^{2}$/\ensuremath{\Delta}) nonlinearities of the collective mode frequencies resulting from the field dependence of the quantization axis. Our results for all of these response properties of $^{3}\mathit{B}$ depend upon a small number of material parameters; thus measurements of these properties can provide detailed information on the quasiparticle interactions.

DOI
10.1103/physrevb.33.6068

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The field dependence of the magnetic susceptibility and collective mode frequencies of superfluid 3B^{3}\mathit{B}3B are calculated perturbatively in (\ensuremath{\gamma}H/\ensuremath{\Delta}) using the quasiclassical theory. With the aid of this perturbation theory, we interpret the gap distortion, nonlinear susceptibility, and collective excitations of the order parameter in terms of additional correlations between Cooper pairs that are induced by a magnetic field. We also calculate the dispersion splittings of the real squashing modes in a magnetic field. In addition to the nonlinear Zeeman shifts of these modes, which occur in strong magnetic fields (\ensuremath{\gamma}H\ensuremath{\sim}0.10\ensuremath{\Delta}), there are low-field (\ensuremath{\gamma}H\ensuremath{\sim}{q}^{2}$${v}_{f}^{2}/\ensuremath{\Delta}) nonlinearities of the collective mode frequencies resulting from the field dependence of the quantization axis. Our results for all of these response properties of 3B^{3}\mathit{B}3B depend upon a small number of material parameters; thus measurements of these properties can provide detailed information on the quasiparticle interactions.
We have conducted an ultrasonic pulse-time-of-flight investigation of the squashing mode of superfluid 3B^{3}\mathit{B}3B in a magnetic field. All five of the expected Zeeman sublevels for this J=${2}^{\mathrm{\ensuremath{-}}}$ mode were observed. The investigation was done at a very low temperature (T/${\mathit{T}}_{\mathit{c}}$0.35) to reduce the quasiparticle broadening that tends to smear the collective modes and makes resolution of the individual sublevels impossible. Variation of the pressure was used to sweep the collective-mode frequencies through the operating frequency, 137.6 MHz of the resonant-sound transducers. Both group-velocity spectroscopy and pulse-shape analysis were used for identification of different sublevels. Performing experiments at different orientations of the sound propagation with respect to magnetic-field direction (parallel, perpendicular, 55\ifmmode^\circ\else\textdegree\fi{}, and 13\ifmmode^\circ\else\textdegree\fi{}) was the key to observation of all five modes and their identification. The Land\'e g factor was obtained from the linear dependence of the Jz{\mathit{J}}_{\mathit{z}}Jz​=+2 sublevel on magnetic field and compared with theory.
The effects of the dipole interaction on the equilibrium and nonequilibrium properties of 3B^{3}\mathit{B}3B are calculated using the quasiclassical theory. In zero field, the dipole interaction distorts the l=1 gap, generates an l=3 gap, and induces a new rotated planar phase next to the normal-state transition. The field dependence of the Leggett angle ${\ensuremath{\theta}}_{L}$ is calculated to order (\ensuremath{\gamma}H/\ensuremath{\Delta})2{)}^{2})2, including the effect of nonquasiparticle corrections. Using the equilibrium properties as input, we calculate the zero-field NMR frequencies in the collisionless regime. A calculation of the field dependence of the longitudinal NMR frequency reveals the existence of a Fermi-liquid oscillation term due to the out-of-phase motion of the normal and superfluid components. Finally, we compare the collisionless and hydrodynamic theories of NMR.
Measurements of the attenuation of zero sound in superfluid 3B^{3}\mathit{B}3B show coupling to a collective mode in applied magnetic fields at frequencies \ensuremath{\nu}\ensuremath{\sim}2\ensuremath{\Delta}(T), which is identified as the J${=1}^{\mathrm{\ensuremath{-}}}$ mode. The attenuation as a function of reduced temperature in zero magnetic field is subtracted from that measured in magnetic fields up to 80 mT to reveal both the Jz{J}_{z}Jz​=+1 and Jz{J}_{z}Jz​=-1 components. The measured G factor is in good agreement with theory. The Jz{J}_{z}Jz​=+1 resonance occurs in the pair excitation continuum and subtracts from the total attenuation, resulting in an antipeak.
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Parametric excitation yields new results, which cannot be obtained by conventional linear spectroscopy, on the collective modes in superfluid 3B^{3}\mathit{B}3B. In our ultrasonic experiments, performed in a chamber with four quartz crystals, we have excited the real squashing (J=2+{2}^{+}2+) modes by two simultaneous sound waves which are mutually either parallel, orthogonal, or antiparallel. Qualitatively these results are in agreement with the theoretical dispersion relation of the J=2+{2}^{+}2+ modes, and quantitatively we have been able to extract the collective-mode velocities from the line splittings in zero magnetic field.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.