Phonon Angular Momentum: Mechanisms & Applications

• Phonon angular momentum is the rotational component of lattice vibrations characterized by circular or elliptical atomic trajectories in crystal structures.
• It arises from complex polarization vectors and symmetry breaking, particularly through the loss of time-reversal or inversion symmetry.
• Generation mechanisms via thermal gradients, electric fields, and terahertz excitation underpin emerging applications in spintronics and topological device engineering.

Phonon angular momentum refers to the rotational component of atomic motion in crystal lattices, where quantized lattice vibrations (phonons) can carry angular momentum due to circular or elliptical atomic trajectories. This concept is the bosonic analog of electronic or magnonic spin/orbital angular momentum, but it emerges from the symmetry properties and mode structure of phonons rather than an intrinsic spin degree of freedom. The microscopic origin, symmetry requirements, mechanisms for phonon angular momentum generation, and its conversion into other physical observables have become central to condensed matter research across spintronics, topological phases, and ultrafast magneto-mechanics.

1. Quantum-Mechanical Definition and Operator Formalism

In the most general form, the total phonon angular momentum in a crystal is expressed as

$$\hat{\mathbf{L}}_{\text{ph}} = \sum_{l,a} \mathbf{u}_{a l} \times M_a \dot{\mathbf{u}}_{a l}$$

where $\mathbf{u}_{a l}$ is the mass-weighted displacement of atom $a$ in unit cell $l$, $M_a$ is its mass, and the cross product reflects the local rotation about the equilibrium position (Zhang et al., 2024). In second quantization, one expands the displacement operator in phonon creation/annihilation operators, obtaining for mode $\sigma$ at $\mathbf{k}$: $$l_{\sigma,i}(\mathbf{k}) = \hbar\,\epsilon_{\sigma}^\dagger(\mathbf{k})\,M_i\,\epsilon_{\sigma}(\mathbf{k})$$ where $\epsilon_{\sigma}(\mathbf{k})$ is the normalized polarization vector, $M_i$ the generator of rotations about axis $i$, and the total angular momentum density is

$$J_i^{\rm ph} = \frac{1}{V}\sum_{\mathbf{k},\sigma} l_{\sigma,i}(\mathbf{k})\left[ f_0(\omega_{\sigma}(\mathbf{k})) + \frac{1}{2} \right]$$

with $f_0(\omega)$ the Bose–Einstein occupation (Hamada et al., 2018).

Crucially, phonon angular momentum arises when the polarization vectors are complex, signaling circular or elliptical internal motion. In equilibrium and in the presence of time-reversal symmetry (TRS), $$l_{\sigma,i}(\mathbf{k}) = -l_{\sigma,i}(-\mathbf{k})$$ and the net angular momentum vanishes due to mode pairing. Phonon angular momentum is thus a subtle measure of the lattice's quantum geometry and symmetry configuration (Coh, 2019).

2. Symmetry Requirements and Classification

Phonon angular momentum depends sensitively on crystal symmetry, particularly TRS and inversion ($\cal P$) (Coh, 2019). Coh’s formal symmetry classification organizes materials into five classes:

Class	$\cal P$	$\cal T$	$\cal PT$	$l(\mathbf{q})\neq 0$?	Microscopic Origin
I	✔	✔	✔	No	—
II	✘	✘	✔	No	—
III	✘	✔	✘	Yes	Force-constant ($F$)
IV	✔	✘	✘	Yes	Velocity-force ($G$)
V	✘	✘	✘	Yes	Both $F$ and $G$

In class III (broken inversion), phonon angular momentum is odd in $\mathbf{q}$, originating from noncentrosymmetric force constants. In class IV (broken TRS), it is even in $\mathbf{q}$ and arises from velocity-induced (Berry-phase) forces, such as spin–orbit or magnetic couplings. Class V admits both mechanisms (Coh, 2019).

For polar or chiral crystals, only certain tensor components of the phonon angular momentum response survive. In polar (but non-chiral) crystals, the antisymmetric part $\alpha_{ij}^{(A)} = -\alpha_{ji}$ (defining a polar vector) is nonzero, leading to perpendicular relations between heat current and angular momentum. In chiral crystals, the symmetric part $\alpha_{ij}^{(S)} = \alpha_{ji}$ can be finite, yielding parallel or complex tensorial relations (Hamada et al., 2018).

In contrast, in the presence of both inversion and TRS, every normal mode can be chosen real, and the angular momentum operator commutes with the phonon Hamiltonian, leading to strictly zero angular momentum in the vacuum and thermal states (Yi et al., 26 Jun 2025).

3. Mechanisms for Generation: Edelstein, Rotoelectric, and Optical Drives

Thermal Gradient (Phonon Edelstein Effect)

A static temperature gradient $\nabla T$ breaks TRS in the phonon nonequilibrium distribution, skewing the population and enabling a finite macroscopic phonon angular momentum: $$J_i^{\rm ph} = -\frac{\tau}{V}\sum_{\mathbf{k},\sigma} l_{\sigma,i}(\mathbf{k})\,v_{\sigma,j}(\mathbf{k})\,\frac{\partial f_0}{\partial T} \frac{\partial T}{\partial x_j} = \alpha_{ij}\,\frac{\partial T}{\partial x_j}$$ where $\tau$ is the phonon relaxation time and $v_{\sigma,j}$ the group velocity (Hamada et al., 2018, Choi et al., 2022). The tensor $\alpha_{ij}$ encodes symmetry-allowed coupling: in GaN, for $\nabla T\sim10$ K/$\mu$m, one estimates $M_x\sim10^{-4}$ A/m (Hamada et al., 2018). At low $T$, incorporating the mode-dependent $\tau_{q\nu}$ leads to divergence in the phonon angular momentum response, especially in low-frequency polar modes (Choi et al., 2022).

Electric Field (Phonon Rotoelectric Effect)

In crystals breaking both inversion and TRS but preserving their product, phonon angular momentum cannot be thermally generated. However, an applied electric field modifies equilibrium atomic positions via Born effective charge, introducing mode mixing and nonzero angular momentum: $J_\alpha^{\rm ph} = \beta_{\alpha\beta}\,E_\beta$ with $\beta_{\alpha\beta}$ fixed by the magnetic point group (Hamada et al., 2020). This is analogous to the magnetoelectric effect, and its temperature dependence saturates at low $T$ due to zero-point motion and decays as $T^{-1}$ at high $T$.

AC Fields (Terahertz Excitation)

Circularly polarized or phase-locked THz fields can induce angular momentum via resonance with doubly-degenerate optical modes at the zone center. In polar GaN, driving an $\mathrm{E}_1$ transverse optical doublet near $\sim16.6$ THz achieves $\mathcal{J}^z_{\rm ph} \sim \hbar$ per unit cell (Sun et al., 6 Jun 2025). The process is governed by off-diagonal matrix elements in the angular momentum operator and shows Lorentzian resonance with a linewidth set by phonon lifetime. Phase control yields sign-selective generation (e.g., $\sin\varphi$ dependence for ellipticity).

4. Phonon Angular Momentum Transfer, Conservation, and Measurement

Intrinsic Transfer Mechanisms

Conservation of angular momentum in solids implies that phonon angular momentum can be transferred not only to rigid-body rotation but also to other phonon modes and spin subsystems. Umklapp and nonlinear phonon–phonon processes enable coherent transfer between chiral modes, as shown in bismuth selenide via helical nonlinear phononics, where three-phonon processes under $C_3$ symmetry enforce quantized transfer rules $2l_{\rm IR}=l_R$ (modulo 3) (Minakova et al., 14 Mar 2025). These mechanisms underpin spin–lattice relaxation and dissipative processes.

Einstein–de Haas Effect and Partition

In magneto-mechanical experiments, conservation law reads: $$J_{\rm tot} = J_{\rm rigid} + J_{\rm ph} + J_{\rm spin} + J_{\rm orb}$$ where the partition between $J_{\rm rigid}$ (macroscopic rotation) and $J_{\rm ph}$ (vibrational angular momentum) can be quantified via the Eckart frame decomposition (Nie et al., 28 Jan 2026). In typical nanodisc Fe, Co, CrI$_3$ experiments, $\eta_L^{\rm ph} \sim 0.15$–$0.25$ and $\eta_E^{\rm vib} \gtrsim 0.8$–$0.9$, indicating most of the kinetic energy resides in phonons but angular momentum is dominated by rigid-body motion.

Interfacial Transfer and Boundary Phenomena

Phonon angular momentum can be injected across interfaces, e.g., into non-chiral hosts, via spin and orbital transfer routes (Suzuki et al., 2024). Continuity of total angular momentum flux polarized normal to a smooth interface is enforced by elastic unitarity (Fresnel coefficients), and the orbital part arises from wavepacket lateral shifts (analog of Imbert–Fedorov effect). Such injected phonon–spin currents may subsequently interact with electronic spin systems, giving rise to magnetoelectric signals.

5. Phononic Hall Effects, Magnetization, and Topological Phenomena

Phonon Angular Momentum Hall Effect

A temperature gradient in noncentrosymmetric or TRS-broken crystals induces a transverse phonon angular momentum current (PAMHE), analogous to the electronic or magnonic Hall effects: $j^{\rm PAM}_y = \sum_x \beta^z_{yx}(-\nabla_x T)$ with the Hall conductivity given by the phononic Berry curvature and thermal occupation (Park et al., 2020). PAM accumulates at edges, resulting in boundary magnetization if ions have finite Born effective charge. For typical parameters, estimated surface magnetizations reach $10^3$–$10^4$ A/m, detectable via magneto-optical or NV-center probes.

Chiral Phonons, Magnetic Switching, and Topological Edge Modes

In ferrimagnetic insulators such as Fe$_{1.75}$Zn$_{0.25}$Mo$_3$O$_8$, intrinsic magnetic order lifts the degeneracy between chiral phonon doublets. Branch splitting up to 20% of frequency, large effective moments ($>2\mu_B$), and nonvolatile switching via applied fields demonstrate tunable phononic magnetism (Wu et al., 18 Jan 2025). Topological phonon bands with edge states are predicted at domain boundaries, carrying unidirectional angular momentum flow and heat.

In magnon–phonon hybridized systems (e.g., FePSe$_3$ (Ning et al., 2024) and d-wave altermagnets (Bendin et al., 11 Nov 2025)), the angular momentum carried by phonons can be spontaneously generated via coupling to magnetic degrees of freedom. The resultant magnon polarons possess complex angular momentum textures and can mediate transverse angular momentum currents (phonon–splitter effect).

6. Special Cases, Quantum Fluctuations, and Controversies

Spin vs. Orbital Angular Momentum

Despite frequent references to "phonon spin," detailed analysis (McLellan, Tiwari) establishes that phonons strictly carry orbital angular momentum arising from spatial rotations of displacements, not intrinsic spin. Only in Cosserat-type elastic media, with internal torque degrees of freedom, can spin-1 or spin-2 excitations (Cosseratons) exist, and these are fundamentally distinct from phonons (Tiwari, 2017).

Conservation in Chiral and Screw-Symmetric Crystals

In chiral crystals, true phonon angular momentum is not a conserved quantum number due to the lack of pure rotation symmetry; only pseudo-angular momentum (from screw symmetry) remains strictly conserved (Kato et al., 2023). This distinction governs selection rules and characterizes phenomena such as rotational phonon–phonon scattering as seen in bismuth selenide (Minakova et al., 14 Mar 2025).

Quantum Coherence and Vacuum Fluctuations

In symmetric crystals, angular momentum expectation values vanish in the vacuum, but finite fluctuations exist due to off-diagonal coherence between orthogonally polarized modes (Yi et al., 26 Jun 2025). These rotational zero-point fluctuations can be probed via polarization-resolved spectroscopy and provide insight into quantum geometry and dynamical multiferroicity.

7. Applications, Experimental Realization, and Outlook

The ability to generate, manipulate, and measure phonon angular momentum opens direct pathways for phononic spintronics, ultrafast control of magnetic and topological order, non-reciprocal heat transport, and quantum information transfer. Demonstrated applications include deterministic magnetization switching via surface acoustic waves (Sasaki et al., 2020), phonon-driven torque detection in microcantilevers (Zhang et al., 2024), and engineered chiral edge currents in topologically ordered phases. Rapid advances in phase-resolved THz polarimetry, microscopy, and ab initio modeling continue to expand the material and device landscape.

Ongoing challenges include isolating phononic contributions in complex spin–lattice systems, harnessing divergence regimes at low temperatures/critical strain (Choi et al., 2022), and advancing theoretical frameworks for nontrivial symmetry classes and quantum coherence phenomena. The emergence of phonon angular momentum as an experimentally accessible and technologically relevant physical quantity heralds new paradigms in the control and utilization of lattice dynamics in solids.