Phonon Rotoelectric Effect in Noncentrosymmetric Crystals

• Phonon Rotoelectric Effect is the generation of net phonon angular momentum via a linear coupling with an electric field in magnetic, noncentrosymmetric crystals.
• It emerges when both time-reversal and inversion symmetries are broken individually but their product is preserved, setting it apart from thermal Edelstein effects.
• Kubo formalism and toy models quantify the effect, revealing temperature-dependent behavior and order-of-magnitude estimates analogous to magnetoelectric phenomena.

The phonon rotoelectric effect is the generation of net phonon angular momentum by an applied electric field in magnetic, noncentrosymmetric crystals possessing combined time-reversal and inversion (TR×I) symmetry. Unlike the phonon thermal Edelstein effect, in which a temperature gradient produces a nonequilibrium phonon angular momentum under broken inversion symmetry and preserved time-reversal symmetry, the rotoelectric effect arises only when both time-reversal and inversion are individually broken but their product is conserved. This phenomenon is symmetry-distinct from thermo-phononic analogues and is characterized by a linear coupling between the electric field and the phonon angular momentum, mediated by a specific response tensor with magnetoelectric symmetry properties (Hamada et al., 2020).

1. Definition and Phenomenological Framework

Microscopically, each normal-mode phonon in a crystal carries a "microscopic" angular momentum given by

$$l_{\sigma,\alpha}(\mathbf k) =\hbar\,\epsilon_{\sigma}^\dagger(\mathbf k)\,M_\alpha\,\epsilon_{\sigma}(\mathbf k)\,,$$

where $\epsilon_{\sigma}(\mathbf k)$ is the polarization vector for the phonon mode $\sigma$ at wavevector $\mathbf k$, and the operator $M_\alpha$ encodes angular momentum components. The equilibrium angular momentum density is

$$J^{\rm ph}_\alpha =\frac1V\sum_{\mathbf k,\sigma} l_{\sigma,\alpha}(\mathbf k)\left[f_0(\omega_{\sigma}(\mathbf k))+\tfrac12\right],$$

with $f_0(\omega)$ the Bose occupation factor.

While the phonon thermal Edelstein effect features a linear response of $J^{\rm ph}_\alpha$ to temperature gradient $\nabla T$ in noncentrosymmetric, time-reversal symmetric crystals, the rotoelectric effect occurs in a different symmetry class. In crystals with both TR and inversion ($I$) broken but TR×I preserved, the total equilibrium phonon angular momentum and the phonon thermal Edelstein tensor vanish identically: $$J^{\rm ph}_\alpha = 0,\qquad \alpha_{\alpha\beta} = 0.$$ However, a coupling to an external electric field is permitted: $J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta,$ where $\beta_{\alpha\beta}$ is an axial response tensor analogous to the magnetoelectric tensor in multiferroics.

2. Symmetry Considerations

The emergence and selection rules of the phonon rotoelectric effect are determined by distinct transformation properties of involved vectors under inversion and time-reversal:

• Under inversion ($I$): The electric field $\mathbf E$ is a polar vector and flips sign, while the phonon angular momentum $J^{\rm ph}$ is an axial vector and does not transform.
• Under time-reversal (TR): $J^{\rm ph}$ flips sign, while $\mathbf E$ remains unchanged.

If TR is preserved and $I$ is broken, then $l_\sigma(\mathbf k) = -l_\sigma(-\mathbf k)$, leading to vanishing equilibrium sum but permitting a response to $\nabla T$. If both $I$ and TR are broken but TR$\times$I is preserved, then $l_\sigma(\mathbf k) \equiv 0$ for all modes and wavevectors—precluding any thermal Edelstein effect. In this symmetry class, both $\mathbf E$ (polar) and $J^{\rm ph}$ (axial) transform identically under TR$\times$I, allowing the linear rotoelectric coupling $J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta$ but forbidding any $\alpha_{\alpha\beta} \nabla T$ term (Hamada et al., 2020).

3. Linear Response and Kubo Formalism

The rotoelectric effect can be formulated within the zero-frequency, zero-wavevector Kubo response theory. The relevant operators are: $$\hat J^{\rm ph}_\alpha = \sum_{\mathbf k, \sigma} \hbar\,\epsilon_\sigma^\dagger(\mathbf k)\,M_\alpha\,\epsilon_\sigma(\mathbf k)\,\left(a_{\mathbf k\sigma}^\dagger a_{\mathbf k\sigma}+\tfrac12\right)$$ for the phonon angular momentum and

$$\hat P_\beta = \sum_{\ell\kappa}q_\kappa\,\hat u_{\ell\kappa,\beta}$$

for the ionic polarization. The response tensor is then

$$\beta_{\alpha\beta} = \lim_{\omega\to0}\frac{1}{\hbar\omega} \Im\,\langle\langle \hat J^{\rm ph}_\alpha;\,\hat P_\beta\rangle\rangle_{\omega} = \frac1{\hbar}\int_0^\infty dt\, \langle [\,\hat J^{\rm ph}_\alpha(t),\,\hat P_\beta(0)\,]\rangle_0,$$

so that $J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta$ in the limit of weak fields. While a full microscopic Kubo evaluation is not performed in the cited work, toy-model calculations serve as concrete realizations of this formalism (Hamada et al., 2020).

4. Temperature Dependence of $\alpha_{\alpha\beta}$ and $\beta_{\alpha\beta}$

The temperature dependencies of the phonon thermal Edelstein tensor $\alpha_{\alpha\beta}$ and the rotoelectric tensor $\beta_{\alpha\beta}$, as derived under the Debye approximation for acoustic modes in three-dimensional crystals, are summarized as follows:

Response	High-$T$ limit	Low-$T$ limit
$\alpha_{\alpha\beta}$	$\propto T^0$ (constant)	$\propto T^3$
$\beta_{\alpha\beta}$	$\propto T^{-1}$	Constant $+ \mathcal{O}(T^5)$

At high temperatures ($\hbar\omega \ll k_B T$), $\beta_{\alpha\beta}$ decays as $T^{-1}$ due to the vanishing of the leading $T$-dependent part by symmetry, with only a $1/T$ piece surviving. In the low-temperature regime, the equilibrium (zero-point) angular momentum—activated by $\mathbf E$—produces a constant contribution to $\beta_{\alpha\beta}$, with corrections scaling as $T^5$ (Hamada et al., 2020).

5. Toy Model Realization and Quantitative Estimates

The effect is elucidated using a two-dimensional spring-mass model featuring three ions per unit cell and antiferromagnetic ordering, with spin-phonon coupling mediated via a Raman interaction: $$H_{\rm SPI} = -g\sum_{l\kappa} \mathbf S_\kappa\cdot\left(\mathbf u_{l\kappa}\times m_\kappa\dot{\mathbf u}_{l\kappa}\right).$$ Application of an electric field $E_x$ perturbs the equilibrium positions, endowing each phonon band with nonzero $l_{\sigma,z}(\mathbf k)$ and yielding $J^{\rm ph}_z = \beta_{zx} E_x \neq 0$, linearly proportional to both the shift $\tilde d \propto E_x$ and $E_x$ itself. No such angular momentum arises for $E\parallel\hat y$, consistent with model symmetries.

A numerical parameter set typical of real magnets such as $\mathrm{Cr_2O_3}$ yields the order-of-magnitude estimate: $$J^{\rm ph} \sim \frac{(\varepsilon-\varepsilon_0)E\,g\hbar^2}{ea\,\omega_D} \sim 10^{-8}\hbar\ \text{per unit cell at } T\sim300~\mathrm{K}$$ for $\omega_D \sim 10\,\mathrm{THz}$ and $g\hbar\sim1\,\mathrm{cm}^{-1}$, assuming ionic displacements induced by applied fields of order $10$ V/mm (Hamada et al., 2020).

6. Relation to Magnetoelectric and Edelstein Effects

The phonon rotoelectric effect is the phononic analogue of the magnetoelectric effect in multiferroics, with $\beta_{\alpha\beta}$ possessing identical symmetry transformation properties to the magnetoelectric tensor $\alpha^{\rm ME}$. By contrast, the phonon thermal Edelstein effect is the direct phonon counterpart of the electronic Edelstein effect, appearing only under broken $I$ and preserved TR symmetry. In the combined TR$\times$I symmetry scenario, the rotoelectric effect is the exclusive allowed phonon angular momentum response, reflecting a different fundamental origin based on symmetry-enforced zero-point lattice rotations rather than nonequilibrium distribution shifts (Hamada et al., 2020).