Thermomagnonic Torques

• Thermomagnonic torques are defined as nonconservative forces on magnetic order parameters induced by thermal gradients, mediated by magnon flows.
• They encompass entropic, spin-transfer, dissipative, DMI-induced, and Berry-curvature contributions that drive dynamics in domain walls, skyrmions, and altermagnets.
• These torques underpin potential spin caloritronic applications such as heat-driven racetrack memories, oscillators, and thermal diodes in low-dissipation devices.

Thermomagnonic torques are nonconservative generalized forces exerted on the magnetization or order parameter of magnetic materials by a thermal gradient, mediated by the flow and statistical properties of magnons (quantized spin waves). These torques play a central role in the field of spin caloritronics, enabling thermal control of magnetic textures, domain walls, skyrmions, and even uniform magnetization without direct charge currents. Theoretical frameworks based on the stochastic Landau–Lifshitz–Gilbert (LLG) equation, linear-response theory, and kinetic approaches have systematically classified distinct types of thermomagnonic torques, including those of entropic, spin-transfer, and topological origin. The following sections delineate the microscopic modeling, taxonomy, effects on texture dynamics, device implications, and generalizations of these torques.

1. Theoretical Frameworks and Microscopic Origin

In insulating magnets well below the Curie point, the stochastic LLG equation governs the magnetization dynamics subjected to both effective fields and thermal noise:

$$s\,\left[ 1 + \alpha\,\mathbf{n} \times \right] \dot{\mathbf{n}} = \mathbf{n} \times \left( \mathbf{h} + \mathbf{h}^{\rm th} \right)$$

with $s$ the local spin density, $\alpha$ the Gilbert damping, $\mathbf{n}$ the unit vector along the local spin direction, and $\mathbf{h}^{\rm th}$ a Langevin random field respecting the fluctuation–dissipation theorem (Kim et al., 2015). The core mechanism underlying thermomagnonic torques is the separation of slow texture dynamics ($\mathbf{n}^{(0)}$) and fast magnon fluctuations ($\delta\mathbf{n}$):

$$\mathbf{n}(\mathbf{r}, t) = \sqrt{1 - |\delta\mathbf{n}|^2} \,\mathbf{n}^{(0)}(\mathbf{r}, t) + \delta\mathbf{n}(\mathbf{r}, t)$$

Upon averaging to second order in fluctuations, magnon occupation modulates both the spin density and the effective exchange stiffness locally, while the thermally induced non-equilibrium magnon currents give rise to additional spin-transfer terms (Kim et al., 2015, Kovalev et al., 2011, Kovalev, 2014). A general description thus emerges in terms of reactive (field-like), dissipative (antidamping), and entropic (statistical occupation–induced) contributions, each identifiable within Onsager reciprocal relations and linear-response matrices.

2. Taxonomy of Thermomagnonic Torques

Distinct classes of thermomagnonic torques have been identified across various systems:

• Entropic Torque ("statistical" or "adiabatic" type): Originates from the spatial variation of the magnon-induced reduction in the effective spin density and exchange stiffness. Explicitly,

$$\boldsymbol\tau^{\rm ex} = -\frac{2\hbar A}{s} \mathbf{n}^{(0)} \times \left[ (\nabla\rho) \cdot \nabla \right] \mathbf{n}^{(0)}$$

where $\rho(\mathbf{r})$ is the local magnon density (Kim et al., 2015).

• Magnonic Spin-Transfer Torque (STT): Arises from the divergence of the magnonic spin current $\mathbf{J}^s$,

$$\boldsymbol\tau^{\rm st} = \hbar \left( \mathbf{J} \cdot \nabla \right)\mathbf{n}^{(0)} + \frac{\hbar A}{s} \mathbf{n}^{(0)} \times \left[ (\nabla\rho) \cdot \nabla \right]\mathbf{n}^{(0)}$$

The first term is analogous to classical current-driven STT, with the magnon current replacing electronic flow (Kim et al., 2015).

• β-type (Dissipative or Nonadiabatic) Correction: Quantifies the viscous coupling due to magnon spin mistracking across textures, central to phenomena such as the skyrmion Hall (Magnus) effect under $\nabla T$ (Kovalev, 2014). The coefficient $\beta/\alpha \approx d/2$, with $d$ the dimension, for thermal magnons (Kovalev, 2014).
• Dzyaloshinskii-Moriya Interaction (DMI)–Induced Torques: In systems with structural inversion asymmetry, a magnonic field-like torque (FLT) of the form $\mathbf{m} \times (\mathbf{z} \times \mathbf{j}_m)$ and a damping-like torque (DLT) $$\mathbf{m} \times [(\mathbf{z} \times \mathbf{j}_m) \times \mathbf{m}]$$ arise, with $\mathbf{j}_m$ the thermally generated magnon current (Manchon et al., 2014, Wang et al., 2016).
• Berry-Curvature (Topological) Thermomagnonic Torque: In topologically nontrivial magnetic systems (e.g., kagome lattices with DMI), the torque can acquire a geometric (Berry) contribution, with interband (intrinsic, antidamping-like) and intraband (field-like) terms distinguished via Kubo formalism (Kovalev et al., 2015).
• Nonlocal and Entropic Torques in Altermagnets: In altermagnets, the theory predicts both a spin-splitter magnonic torque (tensorial, anisotropic, and orientation-dependent) and an entropic torque, with the key form

$$\boldsymbol\tau_{\rm ss} = -\frac{2}{s}\,\mathbf{n} \times \left[ (\hat{\alpha}^s \cdot \nabla T) \cdot \nabla \right]\mathbf{n}, \qquad \boldsymbol\tau_{\rm ent} = -\frac{1}{s} (\eta^A - \eta^B)\, \mathbf{n} \times \left[ (\nabla T \cdot \nabla)\mathbf{n} \right]$$

encapsulating strong anisotropic dependencies on the direction of $\nabla T$ (Schwartz et al., 16 Dec 2025).

3. Dynamics of Magnetic Textures under Thermomagnonic Torques

Thermomagnonic torques induce rich dynamical behaviors in domain walls, skyrmions, and other solitonic textures.

• Domain Wall Motion: For a 1D Bloch wall, the velocity under temperature gradient $\partial_x T$ is

$v = \frac{\hbar A}{s^2 \alpha} \partial_x \rho$

with the direction and magnitude controlled by the interplay and relative signs of entropic and spin-transfer terms (Kim et al., 2015, Kovalev et al., 2011). In DMI systems, DMI-induced FLT can dominate over both exchange entropic and DMI entropic torques, leading to wall velocities an order of magnitude greater (Wang et al., 2016).

• Skyrmion Dynamics: Skyrmion motion in response to thermomagnonic torques is governed by both longitudinal (drift toward the hot side) and transverse (Magnus or Hall) components. The velocity components reflect both the direct transfer of magnonic angular momentum and the viscous β-type correction; the Hall angle sign is set by $(\beta-\alpha)$ (Kovalev, 2014).

Texture	Driving Torque	Motion	Reference
Domain wall	Entropic + STT + DMI	Longitudinal, orientation tunable	(Kim et al., 2015, Wang et al., 2016)
Skyrmion	STT + β term	Longitudinal + Hall (Nernst)	(Kovalev, 2014)
Domain wall (alt.)	Spin-splitter + entropic	Longitudinal + precessional, anisotropic	(Schwartz et al., 16 Dec 2025)

• Anisotropy and Rectification: Certain symmetry classes, such as altermagnets, enable orientation-dependent (anisotropic) thermomagnonic response. For specific angles, the spin-splitter torque vanishes, restoring precession-free, fast texture propagation suitable for racetrack applications (Schwartz et al., 16 Dec 2025). In DMI-rich nanostrips, tuning the DMI strength or the domain magnetization reverses the motion or rectifies spin current (thermal diode action) (Wang et al., 2016).

4. Onsager Reciprocity and Heat Pumping

Thermomagnonic torques and associated spin/heat currents respect Onsager reciprocity. The generalized kinetic equations account for coupled magnon and heat flows, and their cross-effects:

$$\mathfrak{s}(1 + \alpha\,\mathbf{m} \times)\dot{\mathbf{m}} + \mathbf{m} \times \mathbf{H}_{\rm eff} = p \left[ \partial_i \mathbf{m} + \beta\,\mathbf{m} \times \partial_i \mathbf{m} \right] j_i + p_{1} \left[ \partial_i \mathbf{m} + \beta_{1}\,\mathbf{m} \times \partial_i \mathbf{m} \right] \frac{\partial_i T}{T}$$

(Kovalev et al., 2011)

Experimentally, domain-wall motion induced by $\nabla T$ and reciprocal heat flows due to moving interfaces (magnonic Peltier effect) have been formulated. The same phenomenological coefficients govern both processes due to Onsager symmetry, allowing the possibility of extracting or harvesting heat via dynamically controlled magnetic textures (Kovalev et al., 2011).

5. Thermomagnonic Torques in Layered and Topological Systems

• Superconductor/Ferromagnet Hybrids: In S/F bilayers, proximity-induced Zeeman splitting and spin-coherent triplet states amplify thermomagnonic non-adiabatic torques on magnetic domain walls by up to $10^3$ over metallic systems. The key mechanism involves the divergence of thermally-induced spin current in the S layer acting on the F layer, with strongly enhanced wall velocities even for sub-Kelvin biases (Bobkova et al., 2019).
• Spin-Valve and Trilayer Heterostructures: In F/AF/F and N/F/N-F/N/F spin valves, magnonic spin currents mediate nonlocal torques that can switch the downstream F magnet via a Slonczewski-type term, with threshold temperature gradients $\nabla T_{\rm th} < 1$ K/nm at 300 K (Cheng et al., 2018, Bender et al., 2015). Noncollinear devices allow stabilization of self-oscillations, with characteristic angular “wavy” dependence of the torque (Luc et al., 2014).
• Topological Magnon Bands and Berry-Curvature Torques: In systems with nontrivial magnon band topology (kagome/pyrochlore lattices with DMI), a Berry-curvature driven intrinsic torque arises. The Kubo formula separates field-like (intraband) and anti-damping (interband) responses, both linked to measurable spin-Nernst signals and orientation-dependent switching fields (Kovalev et al., 2015).

6. Phenomenology in Altermagnets and Strongly Anisotropic Magnets

Recent theoretical work extends thermomagnonic torque concepts to insulating altermagnets, where unique “spin-splitter” torques arise due to sublattice structure and broken symmetry (Schwartz et al., 16 Dec 2025). Two leading contributions—the spin-splitter magnonic torque and the anisotropic entropic torque—are tensorial in form and manifest strong dependence on the orientation of the applied temperature gradient and the crystallography. This anisotropy enables control over domain-wall precession and the skyrmion Hall effect, enabling functional paradigms such as racetrack motion without transverse deflection by aligning $\nabla T$ along specific crystal axes.

7. Device Concepts and Experimental Considerations

Thermomagnonic torques provide a pathway for electrically silent spintronic devices, including:

• Heat-driven domain-wall and skyrmion racetrack memories, using temperature gradients for deterministic, rectified motion (Wang et al., 2016, Schwartz et al., 16 Dec 2025).
• Spin-torque oscillators and switching devices in insulating valves, solely driven by thermal gradients rather than charge currents, with measurable FMR linewidth shifts scaling with $\delta T$ (Bender et al., 2015).
• Thermal diodes, rectifying magnonic spin current via DMI or crystallographic orientation, by leveraging the sign reversibility of thermomagnonic torques (Wang et al., 2016).
• Cryogenic superconducting logic elements that exploit S/F-enhanced torques to achieve large domain wall velocities at minimal dissipation (Bobkova et al., 2019).
• Spin Seebeck and inverse spin Hall–based loopbacks for electrical detection of thermally induced texture motion, enabling direct experimental validation of thermomagnonic effects (Kovalev, 2014, Kovalev et al., 2015).

A plausible implication is that further enhancement and anisotropic control of thermomagnonic torques may enable domain-wall logic and topologically protected interconnects with unprecedented energy efficiency, provided material engineering can maximize magnon coherence and minimize damping in device architectures.

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References:

• (Kim et al., 2015) Landau-Lifshitz theory of the thermomagnonic torque
• (Manchon et al., 2014) Magnon-Mediated Dzyaloshinskii-Moriya Torque in Homogeneous Ferromagnets
• (Kovalev et al., 2015) Spin torque and Nernst effects in Dzyaloshinskii-Moriya ferromagnets
• (Wang et al., 2016) Thermally induced magnonic spin current, thermomagnonic torques and domain wall dynamics in the presence of Dzyaloshinskii-Moriya interaction
• (Flebus et al., 2016) Local thermomagnonic torques in two-fluid spin dynamics
• (Kovalev, 2014) Skyrmionic spin Seebeck effect via dissipative thermomagnonic torques
• (Bobkova et al., 2019) Thermally induced spin-transfer torques in superconductor/ferromagnet bilayers
• (Cheng et al., 2018) Magnonic Spin-Transfer Torque in Ferromagnet/Antiferromagnet/Ferromagnet Trilayer
• (Kovalev et al., 2011) Thermomagnonic spin transfer and Peltier effects in insulating magnets
• (Luc et al., 2014) Sustained RF oscillations from thermally induced spin-transfer torque
• (Bender et al., 2015) Thermally-driven spin torques in layered magnetic insulators
• (Schwartz et al., 16 Dec 2025) Theory of thermomagnonic torques in altermagnets