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Spin-wave spectra in the B-phase

See also the source file: he3_bspec.f.

Theory

At low frequencies and magnetic fields $\omega, \gamma H\ll\Delta$ there are four degrees of freedom in the $^3$He-B order parameter. Motion of the phase $\varphi$ is sound, and motions of the rotation matrix $R_{aj}$ (three degrees of freedom) are spin waves. In the discussion of spin waves there are three important terms in the hamiltonian: magnetic energy, energy of spin-orbit interaction and gradient energy: \begin{eqnarray} F_M &=& - ({\bf S} \cdot \gamma {\bf H}) + \frac{\gamma^2}{2\chi_B} {\bf S}^2,\\ F_{SO} &=& \frac{\chi_B\Omega_B^2}{15\gamma^2} \left[ R_{jj}R_{kk} + R_{jk}R_{kj}\right],\\ F_\nabla &=& \frac12 \Delta^2 \left[ K_1 (\nabla_j R_{ak})(\nabla_j R_{ak}) + K_2 (\nabla_j R_{ak})(\nabla_k R_{aj}) + K_3 (\nabla_j R_{aj})(\nabla_k R_{ak}) \right]. \end{eqnarray}

Linear equation for small spin oscillations in the uniform texture can be written as $$ \delta \ddot {\bf S} = [\delta \dot{\bf S}\times \gamma {\bf H}] + \hat{\bf\Lambda}\delta{\bf S}, \qquad \hat\Lambda_{ab} = \frac{\Delta^2\gamma^2}{\chi_B} \left[ K\ \delta_{ac}\ \nabla^2 - K'\ R^0_{aj}R^0_{bk} \nabla_j\nabla_k\right] - \Omega_B^2\ n_a n_b, $$

where $K=2K_1+K_2+K_3$ and $K'=K_2+K_3$.

This can be rewritten for a spin wave with frequency $\omega$, wave vector $\bf k$ and amplitude $\bf s$ as $$ -\omega^2 {\bf s} = i\omega\ [{\bf s}\times \gamma {\bf H}] + \hat{\bf\Lambda} {\bf s}, \qquad \hat\Lambda_{ab} = \frac{\Delta^2\gamma^2}{\chi_B} \left[ - K\ \delta_{ac}\ {\bf k}^2 + K'\ R^0_{cj}R^0_{ak} k_j k_k\right] - \Omega_B^2\ n_a n_b. $$

If effect of gradient and spin-orbit energies is small ($\Lambda\ll\omega,\omega_L$) then longitudinal and transverse spin modes can be separated and we have "simple" formula for magnon spectra: \begin{eqnarray} \omega(\omega-\gamma H) &=& c_\perp^2\ k^2 + (c_\parallel^2-c_\perp^2) ({\bf k\cdot\hat l})^2 + \frac12 \Omega_B^2 \sin^2\beta_n, \\ \omega^2 &=& C_\perp^2 k^2 + (C_\parallel^2-C_\perp^2) ({\bf k\cdot\hat l})^2 + \Omega_B^2 \cos^2\beta_n, \end{eqnarray} where spin wave velocities are introduced (see he3lib_grad section): \begin{equation} c_\perp^2 = \frac{\gamma^2\Delta^2}{\chi_B}(K-K'/2),\quad c_\parallel^2 = \frac{\gamma^2\Delta^2}{\chi_B} K,\quad C_\perp^2 = \frac{\gamma^2\Delta^2}{\chi_B} K,\quad C_\parallel^2 = \frac{\gamma^2\Delta^2}{\chi_B} (K-K'). \end{equation} These equations have three solutions for $\omega$: acoustic and optical transverse magnons and longitudinal magnons.

However, it is also possible to solve the full qubic equation without splitting transverse and longitudinal modes. This is "full" formula below. It should work for any values of $\gamma H$, $\Omega_L$, $\bf k$ if the texture is uniform (of changes much slower than the wavelength). The full equation have a simple form in the case of uniform NMR ($\bf k = 0$): $$ \omega^6 - \omega^4(\omega_L^2 + \Omega_B^2 ) + \omega^2\omega_L^2 \Omega_B^2 n_z^2 = 0. $$ There is a mode $\omega=0$ (acoustic magnons) and two modes with non-zero frequency.

Examples

Examples show magnon spectra in $T=0$ and $P=0$ (Legget frequency $\Omega_B/2\pi = 125.1$ kHz). Light curves are simple formula, dark ones are full formula.

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