Phonon Spin Angular Momentum

• Phonon spin angular momentum is the angular momentum tied to quantized lattice vibrations, emerging from polarization states driven by spin–orbit and spin–phonon interactions.
• Experimental probes like Raman spectroscopy and thermal gradient-induced torque measurements confirm its role in angular momentum transfer in chiral and magnetic systems.
• The theoretical framework decomposes phonon angular momentum into intrinsic (spin) and orbital components, influencing applications in spintronics, caloritronics, and ultrafast magnetization dynamics.

Phonon spin angular momentum refers to the angular momentum associated with the motion of quantized lattice vibrations, i.e., phonons, within solids. Traditionally, phonons are considered to be spinless excitations, as their quantization arises from a vector displacement field leading to three vibrational branches per $k$-point, but without intrinsic spin in the sense of elementary particles. However, recent theoretical and experimental studies demonstrate that in specific symmetry environments, and particularly in the presence of spin–phonon or spin–orbit interaction, phonons can possess well-defined angular momentum, primarily via their polarization states (e.g., circular, elliptical), which is often referred to as “phonon spin,” while also featuring an orbital component associated with the collective lattice motion. The investigation of phonon spin angular momentum has substantial implications for angular momentum conservation in condensed matter systems, spintronics, caloritronics, and magnetization dynamics, as well as for the rigorous interpretation of canonical experiments such as Einstein–de Haas.

1. Fundamental Principles and Mathematical Formalism

The total angular momentum of phonons, $J^{\mathrm{ph}}$, is defined via the atomic displacement $\mathbf{u}$ and the canonical momentum $\mathbf{p}$ operators. For a phonon mode characterized by wavevector $\mathbf{k}$ and branch $\sigma$ with polarization vector $\varepsilon_{\mathbf{k},\sigma}$, the modal angular momentum (for the $z$-component) is

$$l^{z}_{\mathbf{k},\sigma} = \hbar \varepsilon^{\dagger}_{\mathbf{k},\sigma} M \varepsilon_{\mathbf{k},\sigma}$$

where $M$ is the matrix representation of the $z$-component of the angular momentum operator in 3D space. The total phonon angular momentum is thus

$$J^{\mathrm{ph}}_z = \sum_{\sigma,\mathbf{k}} l^{z}_{\mathbf{k},\sigma} \left( f(\omega_{\mathbf{k},\sigma}) + \frac{1}{2} \right)$$

where $f$ is the Bose–Einstein distribution and the $1/2$ term represents the zero-point (vacuum) contribution.

In the presence of spin–phonon interactions, the Hamiltonian for a magnetic crystal is modified as: $$H = \frac{1}{2} (\mathbf{p} - \tilde{A}\mathbf{u})^{\mathrm{T}} (\mathbf{p} - \tilde{A}\mathbf{u}) + \frac{1}{2} \mathbf{u}^{\mathrm{T}} K \mathbf{u}$$ with $\tilde{A}$ encoding Raman-type spin–phonon couplings proportional to magnetization. The inclusion of such terms distorts the phonon polarization vectors such that $$l^{z}_{-\mathbf{k},\sigma} \neq -l^{z}_{\mathbf{k},\sigma}$$, thus breaking the cancellation in equilibrium and leading to a finite net phonon angular momentum in magnetic systems (Zhang et al., 2013).

A crucial distinction is made between “spin” and “orbital” parts of phonon angular momentum. In canonical treatments, the full angular momentum operator for collective lattice vibrations can be decomposed as: $L = L^{(1)} + L^{(2)}$ with $$L^{(1)} = \int d^3 r \, \rho \, \mathbf{r} \times \partial_t \mathbf{u}$$ (orbital part, corresponding to rigid rotation of the body) and $$L^{(2)} = \int d^3 r \, \rho \, \mathbf{u} \times \partial_t \mathbf{u}$$ (interpreted as the intrinsic phonon “spin” or local shearing/circular motions) (Garanin et al., 2015). Upon quantization, the $L^{(2)}$ component reveals that circularly polarized phonons carry angular momentum $\pm \hbar$ along their propagation direction.

2. Symmetry Requirements and Microscopic Origin

The existence and magnitude of phonon angular momentum depend critically on crystal symmetry:

• Centrosymmetric nonmagnetic crystals possess both inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries—the combined $\mathcal{PT}$ symmetry enforces $l(\mathbf{q}) = -l(\mathbf{q})$, so angular momentum vanishes away from high-symmetry points (Coh, 2019).
• Chiral or polar crystals with broken inversion symmetry and/or magnets with broken time-reversal symmetry admit finite angular momentum at generic $\mathbf{q}$. In noncentrosymmetric but $\mathcal{T}-$invariant (Class III) systems, displacement-induced antisymmetric force constants yield $l(\mathbf{q}) = -l(-\mathbf{q})$. In ferromagnets ($\mathcal{P}$ preserved, $\mathcal{T}$ broken, Class IV), velocity-induced (Berry-phase) force terms driven by spin–orbit interaction generate $l(\mathbf{q}) = l(-\mathbf{q})$ and produce chiral phonon bands even at $\mathbf{q}=0$ (Coh, 2019).

The explicit microscopic forms are: $$- M_i \frac{d^2 x_i^{\alpha}}{dt^2} = \sum_{j,\beta} F_{ij}^{\alpha\beta} x_j^{\beta} + G_{ij}^{\alpha\beta} \frac{d x_j^{\beta}}{dt}$$ where $F$ is the force-constant matrix and $G$ the velocity-force (Berry) matrix. The velocity-force contribution emerges from spin–orbit–entangled electronic states (Coh, 2019, Yip, 2023).

3. Temperature Dependence and Nonequilibrium Generation

In thermodynamic equilibrium and in the absence of time-reversal or inversion breaking, the angular momentum of phonons cancels upon summing over $\mathbf{k}$ due to $l(\mathbf{k}) = -l(-\mathbf{k})$. However, if time-reversal symmetry is broken by an applied magnetic field, internal magnetization, or crucially by a temperature gradient (nonequilibrium), an off-equilibrium phonon distribution results: $$f_{\sigma, \mathbf{k}} = f_0(\omega_{\sigma}(\mathbf{k})) - \tau v_{\sigma,i}(\mathbf{k}) \frac{\partial f_0}{\partial T} \frac{\partial T}{\partial x_i}$$ This population imbalance between $+\mathbf{k}$ and $-\mathbf{k}$ yields a finite net angular momentum. The resulting nonequilibrium phonon AM, e.g., under a temperature gradient $\nabla T$, is

$$J^{(\mathrm{ph})}_i = -\frac{\tau}{V} \sum_{\mathbf{k},\sigma} l_{\sigma,i}(\mathbf{k}) v_{\sigma,j}(\mathbf{k}) \frac{\partial f_0}{\partial T} \frac{\partial T}{\partial x_j} = \alpha_{ij} \frac{\partial T}{\partial x_j}$$

The relevant response tensor $\alpha_{ij}$ is nonzero only in low-symmetry (e.g., chiral, polar) materials (Hamada et al., 2018).

At low temperatures, the angular momentum is dominated by the zero-point contribution, while at higher $T$, thermal occupations gradually cancel this via population of modes with opposite angular momentum; $J^{\mathrm{ph}} \to 0$ in the classical high-$T$ limit (Zhang et al., 2013). Thus, finite phonon angular momentum is a quantum effect.

4. Angular Momentum Transfer: Einstein–de Haas, Spin Relaxation, and Experimental Manifestations

Angular momentum carried by phonons has direct implications for angular momentum conservation in solids: $$J^{\mathrm{tot}} = J^{\mathrm{lat}} + J^{\mathrm{ph}} + J^{\mathrm{spin}} + J^{\mathrm{orb}}$$ In Einstein–de Haas–like scenarios, changes in the lattice’s mechanical rotation are only equal and opposite to $\Delta J^{\mathrm{spin}} + \Delta J^{\mathrm{orb}}$ after subtracting $\Delta J^{\mathrm{ph}}$. For high-precision measurements or materials with strong spin–phonon coupling, failure to account for $J^{\mathrm{ph}}$ leads to systematic error in inferred electron angular momentum (Zhang et al., 2013, Rückriegel et al., 2019).

During spin relaxation events, the “spin” part of phonon angular momentum (locally circular lattice motion) constitutes the missing channel for $z$-component transfer. For an atomic spin relaxing from $|m\rangle$ to $|m-1\rangle$, the phonon spin accommodates $L^{(2)}_z = -(1 - e^{-\Gamma t}) \hbar$ (Garanin et al., 2015). Phonon angular momentum is also implicated in ultrafast angular momentum dissipation: during femtosecond laser-induced demagnetization, angular momentum lost by the electrons due to spin–orbit-mediated processes is transferred into phonon angular momentum, with TDDFT simulations confirming coherent excitation of optical phonons carrying finite AM (Mrudul et al., 23 Apr 2025). Thus, phonon AM is fundamental to ultrafast spin-lattice relaxation dynamics.

5. Experimental Probes and Observations

Several experimental strategies have been proposed and realized to detect or infer phonon angular momentum:

• Raman Spectroscopy: Spin–phonon interaction leads to splitting of degenerate phonon modes at high-symmetry points; the splitting magnitude is proportional to the spin–phonon coupling constant and to the magnetization, enabling indirect probes of phonon AM (Zhang et al., 2013).
• Thermal Gradient–Induced Torque: The first direct experimental observation of phonon angular momentum was achieved in single-crystal Te, where a thermal gradient along the chiral axis yielded a measurable mechanical torque on a cantilever (on the order of $10^{-11}$ N·m), reversing sign on flipping the temperature gradient. Polycrystals lacking net chirality produced no such torque, ruling out trivial artifacts (Zhang et al., 20 Sep 2024).
• Optical Phonon Hall and Edge Magnetization: Under a temperature gradient, the net transport of phonon angular momentum (phonon angular momentum Hall effect, PAMHE) leads to edge accumulation of AM and, in systems with nonzero Born effective charge, to edge magnetization detectable by magneto-optical effects (Park et al., 2020).
• Spin Seebeck and Chiral Phonon Activation: In noncentrosymmetric or chiral materials, off-diagonal elements in the phonon density matrix, enhanced by scattering and quantum transitions between phonon modes under thermal drive, result in large AM and corresponding magnetic moments, explaining the enhancement of spin Seebeck signals (Zhong et al., 2022).

6. Spin-Phonon Angular Momentum in Spintronic and Quantum Materials

Phonon spin angular momentum is directly implicated in thin-film and heterostructure magnetization dynamics, spintronics, and caloritronic applications. For instance, the transfer of angular momentum from surface acoustic waves (elliptically polarized, Rayleigh-type) to the magnetization of a Ni film via magnetoelastic coupling enables low-power, non-thermal phononic control of magnetization (Sasaki et al., 2020). Chiral phonons and magnon–phonon hybrids (“magnon polarons”) in antiferromagnets exhibit intrinsic ellipticity and angular momentum, with their distinct energies and helicities being externally tunable via light polarization, magnetic field, or temperature (Ning et al., 14 Oct 2024).

Switchable chiral phonon domains have been realized in ferrimagnetic insulators, where the phonon angular momentum can be flipped nonvolatilely by reversing magnetization, with the magnitude of the phonon’s effective magnetic moment exceeding that of a magnon near criticality. Domain boundaries are proposed to host topologically protected edge modes (Wu et al., 18 Jan 2025).

In materials with discrete rotational symmetry, nonlinear phonon–phonon coupling (rotational Umklapp processes) allows for coherent transfer and upconversion of phonon angular momentum, a process confirmed experimentally and supported by ab initio calculations (Minakova et al., 14 Mar 2025).

7. Theoretical and Conceptual Controversies

Despite robust evidence for angular momentum in phonon systems, the nature of “phonon spin” remains debated. Some analyses argue that angular momentum observed in circularly or elliptically polarized phonons should be interpreted entirely as orbital, given the scalar field origin of phonons and the absence of a true intrinsic spin degree of freedom (Tiwari, 2017, Tiwari, 2018). The analogy to photon spin is critiqued as physically misleading because, unlike the electromagnetic field, the displacement vector is not a gauge-dependent dynamical field. Intrinsic phonon spin is deemed unphysical for conventionally quantized elastic media, with only composite (orbital) angular momentum arising from collective motion.

Notably, in Cosserat-type (micropolar) elastic media allowing for local rotations and internal torque, new excitations (“Cosseratons”) with genuine spin one or two are theoretically possible and distinct from phonons, though not realized in conventional crystals (Tiwari, 2017).

8. Outlook and Open Questions

The ongoing elucidation of phonon spin angular momentum raises several avenues for research:

• Phonon–electron spin conversion: Chiral phonons can induce net spin-polarization in otherwise non-magnetic electron systems via vibronic spin–orbit interaction, establishing a vibrational route to spin filtering and spin-current generation (Fransson, 2022).
• Interface Transport and Conservation Laws: Across interfaces between chiral and achiral crystals, the sum of spin and orbital phonon angular momentum is conserved when correctly accounting for extrinsic orbital shifts (Imbert–Fedorov effect), establishing rigorous transport and boundary conditions (Suzuki et al., 13 Sep 2024).
• Berry Phase Effects: In strongly SOC-coupled materials, Berry phase and curvature effects, induced by pseudospin–phonon coupling, lead to splitting and chirality in acoustic modes and contribute to topological phenomena such as the phonon thermal Hall effect (Yip, 2023).
• Ultrafast Dynamics: During femtosecond demagnetization, phonon angular momentum serves as the immediate sink for angular momentum shed by the electronic system, a process that can be accessed by time-resolved electron diffraction and advanced quantum dynamics simulations (Mrudul et al., 23 Apr 2025).

Whether phonon angular momentum is best conceptualized as spin, orbital, or some hybrid object depends on the symmetry, microscopic mechanisms, and theoretical framework adopted. Recent experimental advances, especially the direct measurement of phonon AM by macroscopic torque, have established its reality and created opportunities for exploiting phonon angular momentum in solid-state devices. Future work will address the universality of these effects in complex materials, their integration with quantum information platforms, and their impact on nonequilibrium and topological quantum phenomena in the solid state.