Orthogonal Einstein–de Haas Effect

• The orthogonal Einstein–de Haas effect is a magneto-mechanical phenomenon where anisotropic spin–lattice coupling induces mechanical rotation perpendicular to the magnetization change.
• Spin–orbit coupling and crystal symmetry breaking generate a tensorial gyromagnetic ratio, yielding antisymmetric contributions and distinctive experimental signatures.
• Continuum and many-body models show that this effect spans solids and ultracold dipolar gases, enabling tunable angular momentum transfer and non-perturbative quantum resonances.

The orthogonal Einstein–de Haas (OEdH) effect is a recently established phenomenon in which a change of magnetization along one spatial direction induces a mechanical rotation about an axis orthogonal to the direction of magnetization. This effect generalizes the classical Einstein–de Haas effect, in which the rotation is parallel to the magnetization axis, by allowing the coupling between magnetization and lattice angular momentum to possess a tensorial, and often anisotropic, structure. The OEdH effect is fundamentally rooted in the interplay between spin–orbit coupling (SOC), crystal point-group symmetry, and collective angular momentum transfer involving spin, orbital, and lattice degrees of freedom. Recent theoretical developments provide explicit frameworks for the microscopic and continuum mechanisms that mediate the OEdH effect in both solids and quantum gases, with distinctive experimental signatures accessible in state-of-the-art measurement platforms (Xue et al., 24 Jan 2026, Acharya et al., 5 Oct 2025, Mentink et al., 2018, Ebling et al., 2017).

1. Tensorial Structure of the Gyromagnetic Ratio

The primary distinguishing feature of the OEdH effect is the emergence of an anisotropic, generally nondiagonal gyromagnetic tensor $\gamma_{ij}$ that relates the rate of change of magnetization $\dot{M}_j$ to the induced mechanical rotation rate $\dot{\theta}_i$: $\dot{\theta}_i = \gamma_{ij}\,\dot{M}_j$ where classically, for the standard Einstein–de Haas effect, $\gamma_{ij}$ reduces to a scalar $\gamma_0\delta_{ij}$ so that rotation occurs about the direction of magnetization change. In the OEdH regime, $\gamma_{ij}$ has antisymmetric and anisotropic contributions such that a change in $M_x$ can generate a rotation about $y$, and more generally

$\gamma_{ij} = |M|\,(p^{-1})_{ij}$

with $p_{ij}$ the coupling matrix connecting magnetization orientation $\hat{m}$ to phonon angular momentum $L_{\rm ph,i} = p_{ij}\,\hat{m}_j$ (Xue et al., 24 Jan 2026).

Microscopically, this anisotropy arises from symmetry constraints imposed by the spin group and the presence of SOC, which generate a molecular Berry curvature and enable antisymmetric and symmetric-traceless tensor contributions to $p_{ij}$. The decomposition

$$p_{ij} = p_0\delta_{ij} + p^s_{ij} + \varepsilon_{ijk}d_k$$

identifies the trace ($p_0$) as the isotropic, conventional channel, and the antisymmetric ($d_k$) and symmetric-traceless ($p^s_{ij}$) parts as the source of the OEdH effect (Xue et al., 24 Jan 2026).

2. Microscopic Mechanisms: Spin–Orbit Coupling and Symmetry

The OEdH effect fundamentally relies on the presence of spin–orbit coupling and the breaking of full rotational symmetry by the crystal point group. Starting from a full electron–lattice Hamiltonian incorporating local SOC,

$$\hat{H} = \frac{\hat{p}_e^2}{2m_e} + U(\bm{r}_e,\{\bm{R}_i\}) + J(\{\bm{R}_i\})\hat{\bm{m}}\cdot\bm{\sigma} + \frac{\hbar}{m_e^2c^2} \hat{\bm{O}}\cdot\bm{\sigma} + H_{\rm ph}$$

and integrating out electronic degrees of freedom generates Berry vector potentials and curvatures in the phonon sector. The resulting phonon Hamiltonian features angular momentum terms linear in atomic displacement and velocity: $$\hat{H}_{\rm ph} = \sum_i \frac{\bm{p}_i^2}{2} + \frac{1}{2}\bm{u}_i^T D^{(i)}\bm{u}_i + \sum_i \bm{\Omega}_i\cdot(\bm{u}_i \times \dot{\bm{u}}_i)$$ The explicit dependence of $\bm{\Omega}_i$ on the direction of magnetization and point group (e.g., $C_{1h}$ symmetry) results in a dipolar scaling of the induced phonon angular momentum and, hence, enables the OEdH effect. The tensor $p_{ij}$ encapsulates these microscopic mechanisms, with leading-order $p_{ij}\propto \lambda$ (SOC strength), and higher odd orders in the spin–conserving case, while the spin-flip component yields $p_{ij}\propto \lambda^2$ (Xue et al., 24 Jan 2026).

3. Continuum and Micromagnetic Models

In a continuum framework, the OEdH effect is realized via the balance of total angular momentum, including the coupling between magnetization per unit mass $m$, lattice (material) spin $\Omega$, stress, and couple-stress. The angular momentum conservation law,

$$\gamma^{-1}\rho\,\dot{\mathcal{M}} = \mathrm{div}\,\Lambda - X:T + \rho K$$

requires augmentation with a spin-inertia term $a$ for full dynamic consistency. The induced torque density in an orthogonal geometry—i.e., with $\bm{M}\perp \bm{\omega}$—is

$$\tau = \omega \times (c_1\,m) = c_1\,\Omega_z\,\hat{e}_z \times \hat{e}_x = c_1\,\Omega_z\,\hat{e}_y$$

such that the mechanical torque is perpendicular to both the rotation and magnetization axes. Reversal of $m$ thus induces a mechanical response in the orthogonal direction, with observable rotation angles consistent with classic Einstein–de Haas measurements when evaluated for realistic sample parameters (Acharya et al., 5 Oct 2025).

4. Quantum Many-Body and Non-Perturbative Descriptions

A fully quantum-mechanical account employs a rotationally invariant many-body Hamiltonian consisting of electron, phonon, and electron–phonon interaction terms: $H = H_e + H_p + H_{ep}$ with explicit angular momentum resolving structure, permitting the definition and tracking of total, spin, orbital, and phonon angular momenta. The angulon quasiparticle formalism provides an exact mapping for a magnetic impurity interacting with a phonon bath, yielding a variational state at fixed total angular momentum. The crucial result is the prediction of new, non-perturbative resonances in the angular-momentum transfer spectrum, specifically in geometries where the driving magnetization is orthogonal to the mechanical rotation axis.

The angular momentum current about the orthogonal axis is computed via

$$\tau_z(t) = d\langle J_z \rangle/dt = -i\langle [J_z, H] \rangle$$

and the corresponding susceptibility $\chi_{zx}^{(J)}(\omega)$ shows signatures of ultrafast angular momentum transfer ($\sim$10 fs) mediated by the high-frequency angulon resonance, in contrast to slower, conventional channels (Mentink et al., 2018).

5. Orthogonal Einstein–de Haas in Dipolar Quantum Gases

Analogs of the OEdH effect manifest in ultracold two-component dipolar Fermi gases, where the anisotropic dipole–dipole interaction induces spin–orbit transfer via mean field and exchange mechanisms. The transfer of $S_z$ to orbital angular momentum $L_{\rm tot}$ is accompanied by a pronounced “twisting” motion: $L_z \equiv L_\uparrow - L_\downarrow$ with $|L_\uparrow|, |L_\downarrow| \gg |L_{\rm tot}|$, indicating counter-rotation of spin subcomponents far exceeding the net transfer. This is physically rooted in the Fermi surface deformation mechanism and controllable via the $s$-wave scattering length $a_s$ and external magnetic field $B$: $$\text{sign}(L_z) = \text{sign}[E_{\rm dir}(0) + E_{\rm Z}(0) - E_{\rm c}(\infty)]$$ allowing the sign and magnitude of the twisting (orthogonal transfer) to be tuned experimentally (Ebling et al., 2017).

6. Experimental Manifestations and Measurement Platforms

Measurement of the OEdH effect requires sensitivity to angular momentum transfer orthogonal to the magnetization. In crystalline solids, nanomechanical resonators operating in the MHz–GHz regime enable direct detection of EdH torques in phase quadrature with the conventional $m\times B$ torque. Application of orthogonal RF magnetic fields excites and distinguishes the EdH and cross-product torque channels, facilitating quadrature separation in lock-in detection. Hysteresis and Barkhausen features, as well as resonance splitting, reveal the underlying spin–lattice coupling and the anisotropy of the gyromagnetic tensor (Mori et al., 2020).

In ultracold gases, the OEdH effect is observable via real-space and time-of-flight imaging, revealing characteristic spiral or crescent-shaped distortions of the spin-resolved clouds, the sign and magnitude of which can be reversed by tuning $a_s$ or $B$. The effect also produces a splitting of the scissor mode spectrum proportional to the twisting torque.

Ultrafast pump–probe spectroscopies—magneto-optical Kerr, time-resolved XMCD, and direct mechanical detection by femtosecond laser-driven cantilever torsion—probe the OEdH effect on the ultimate timescales set by phonon–angulon resonance, confirming non-perturbative angular momentum transfer orthogonal to the applied field on the femtosecond scale (Mentink et al., 2018, Xue et al., 24 Jan 2026).

7. Theoretical and Physical Implications

The OEdH effect establishes that magneto-mechanical coupling in solids and quantum gases is not fundamentally constrained to the parallel channel, but is generically tensorial in the presence of SOC and reduced symmetry. The connection between microscopic Berry curvature, point group representation, and the structure of $p_{ij}$ relates directly to observable rotational dynamics under spin manipulation. This motivates extensions to complex spin order, antiferromagnets, and systems with higher multipolar coupling, connecting spintronics, ultrafast magnetism, and quantum gas research. Future experiments targeting the measurement of off-diagonal $\gamma_{ij}$ will further clarify the role of anisotropy in spin–lattice angular momentum transfer (Xue et al., 24 Jan 2026, Ebling et al., 2017, Acharya et al., 5 Oct 2025, Mentink et al., 2018).