Physical test scenario 3 - single spin precession-nutation dynamics due to uniaxial and weak cubic anisotropy

[AdamsMP92]

Motivation

The effective macrospin model in Adams et al., Phys. Rev. B 110, 054415 (2024), provides two useful dynamics test cases:

1. an exact analytical trajectory for $k_4=0$;
2. a perturbative physics-regression test for small $k_4\neq0$.

The first case enables a strict pointwise comparison with the numerical trajectory. The second checks whether the expected weak nutational mode, frequency ratio, and amplitudes are reproduced.

$$\mathcal{H} = -k_2 m_z^2 + k_4 \left( m_x^4+m_y^4+m_z^4 \right).$$

This model provides two useful test scenarios:

1. an exact analytical test for $k_4=0$;
2. an approximate physics-regression test for small $k_4\neq0$.

The first case allows a strict pointwise comparison with an exact trajectory. The second case tests whether the expected weak nutational mode is reproduced.

Model

The normalised magnetisation is parametrised as

$$\mathbf{m} = \begin{bmatrix} \sin\theta\cos\phi\\\ \sin\theta\sin\phi\\\ \cos\theta \end{bmatrix}.$$

The undamped dynamics are governed by

$$\frac{d\mathbf{m}}{d\tau} = \mathbf{m}\times\mathbf{b}_{\mathrm{eff}},$$

with

$$\mathbf{b}_{\mathrm{eff}} = 2k_2m_z\mathbf{e}_z - 4k_4 \left( m_x^3\mathbf{e}_x + m_y^3\mathbf{e}_y + m_z^3\mathbf{e}_z \right).$$

Exact test for $k_4=0$

For $k_4=0$, the polar angle remains constant,

$$\theta(\tau)=\theta_0,$$

while the azimuthal angle evolves linearly,

$$\phi(\tau) = \phi_0-\nu_{p,0}\tau,$$

with

$$\nu_{p,0} = 2k_2\cos\theta_0.$$

The exact trajectory is therefore

$$\mathbf{m}(\tau) = \begin{bmatrix} \sin\theta_0\cos\left(\phi_0-\nu_{p,0}\tau\right)\\\ \sin\theta_0\sin\left(\phi_0-\nu_{p,0}\tau\right)\\\ \cos\theta_0 \end{bmatrix}.$$

A possible analytical reference implementation is

import numpy as np


def exact_uniaxial_trajectory(tau, theta0, phi0, k2):
    """Return the exact undamped trajectory for k4 = 0."""
    tau = np.asarray(tau, dtype=float)

    nu_p = 2.0 * k2 * np.cos(theta0)
    phi = phi0 - nu_p * tau

    return np.column_stack(
        (
            np.sin(theta0) * np.cos(phi),
            np.sin(theta0) * np.sin(phi),
            np.full_like(tau, np.cos(theta0)),
        )
    )

The normalised simulated magnetisation can then be compared directly with the analytical trajectory:

m_exact = exact_uniaxial_trajectory(
    tau=times,
    theta0=theta0,
    phi0=phi0,
    k2=k2,
)

np.testing.assert_allclose(
    m_simulated,
    m_exact,
    rtol=...,
    atol=...,
)

This exact case tests:

• conservation of the polar angle;
• the sign of the azimuthal motion;
• the precession frequency;
• and conservation of the magnetisation norm.

Approximate test for small $k_4$

For small but finite $k_4$, the dynamics contain a weak nutational modulation.

The approximate precession and nutation frequencies are

$$\nu_p = \cos\theta_0 \left[ 2k_2+ \left( 3-7\cos^2\theta_0 \right)k_4 \right],$$

and

$$\nu_c=4\nu_p.$$

The corresponding amplitudes are

$$a_\theta = \frac{k_4\sin^3\theta_0}{4\nu_p},$$

and

$$a_\phi = \frac{k_4\cos\theta_0\sin^2\theta_0}{4\nu_p} = a_\theta\cot\theta_0.$$

The approximate angular trajectory is

$$\theta(\tau) \approx \theta_0 + a_\theta \left[ \cos(4\phi_0) - \cos\left(4\phi_0-4\nu_p\tau\right) \right],$$

and

$$\phi(\tau) \approx \phi_0-\nu_p\tau + a_\phi \left[ \sin\left(4\phi_0-4\nu_p\tau\right) - \sin(4\phi_0) \right].$$

Because this result is perturbative, it should not be used as a strict pointwise equality test. Instead, the numerical trajectory can be analysed to extract:

• the dominant precession frequency;
• the nutation frequency;
• the polar nutation amplitude;
• and the azimuthal modulation amplitude.

Possible assertions are

assert np.isclose(
    nu_p_simulated,
    nu_p_analytical,
    rtol=...,
)

assert np.isclose(
    nu_c_simulated / nu_p_simulated,
    4.0,
    rtol=...,
)

assert np.isclose(
    a_theta_simulated,
    a_theta_analytical,
    rtol=...,
)

assert np.isclose(
    a_phi_simulated,
    a_phi_analytical,
    rtol=...,
)

The tolerances should account for both the perturbative approximation error and the numerical integration error.

Suggested test hierarchy

1. $k_4=0$: exact pointwise trajectory comparison;
2. small $k_4$: comparison of the precession frequency;
3. small $k_4$: verification of $\nu_c/\nu_p\approx4$;
4. small $k_4$: comparison of the nutation amplitudes.

The $k_4=0$ case provides a strict solver test, whereas the finite-$k_4$ case provides a nontrivial physics-regression test.

Expected value

These tests would validate:

• undamped anisotropy-driven precession;
• the implementation of uniaxial anisotropy;
• the implementation of cubic anisotropy;
• the sign and magnitude of the precession frequency;
• the emergence of the weak nutational mode;
• and the characteristic relation $\nu_c\approx4\nu_p$.