Phononic dynamical axion in magnetic Dirac insulators

In cosmology, the axion is a hypothetical particle that is currently considered as candidate for dark matter. In condensed matter, a counterpart of the axion (the “axion quasiparticle”) has been predicted to emerge in magnetoelectric insulators with fluctuating magnetic order and in charge-ordered Weyl semimetals. To date, both the cosmological and condensed-matter axions remain experimentally elusive or unconfirmed. Here, we show theoretically that ordinary lattice vibrations can form an axion quasiparticle in Dirac insulators with broken time- and space-inversion symmetries, even in the absence of magnetic fluctuations. The physical manifestation of the phononic axion is a magnetic-field-induced phonon effective charge, which can be probed in optical spectroscopy. By replacing magnetic fluctuations with lattice vibrations, our theory widens the scope for the observability of the axion quasiparticle in condensed matter.

Figure 1

Modulus of the axionic phonon effective charge as a function of the scalar Dirac mass $m$, for fixed axial mass $m_5$. Curves were obtained at zero temperature from Eqs. (26) and (28). The dotted lines represent the values of $m$ for which $\mathrm{\Delta }=\hslash q_0$, where $\mathrm{\Delta }$ is the energy gap of the insulator and $q_0$ is the phonon frequency. (Left) $|m_5|>\hslash q_0/2$. (Middle) $|m_5|=\hslash q_0/2$. (Right) $|m_5|<\hslash q_0/2$. The parameter values are $B_0=1\mathrm{T}$ for the magnetic field, $\hslash q_0=10\mathrm{meV}$ for the phonon frequency, $g\approx 1\mathrm{eV}/\AA$ for the optical deformation potential, ${\mathcal{V}}_c=1{\mathrm{nm}}^3$ for the unit-cell volume and $\eta =0.01\hslash q_0$ for the frequency broadening factor (exception: $\eta =0$ for the solid-black line).

Figure 2

Modulus of the axionic phonon effective charge as a function of the axial mass $m_5$, for fixed scalar mass $m$. Curves were obtained at zero temperature from Eqs. (26) and (28). The dotted lines represent the values of $m_5$ for which $\mathrm{\Delta }=\hslash q_0$, where $\mathrm{\Delta }$ is the energy gap of the insulator and $q_0$ is the phonon frequency. (Left) $\left|m\right|>\hslash q_0/2$. (Middle) $\left|m\right|=\hslash q_0/2$. (Right) $\left|m\right|<\hslash q_0/2$. The parameter values are the same as for Fig. 1.

Figure 3

Real and imaginary part of the axionic phonon effective charge as a function of $\tilde{m}=m/\hslash q_0$ for different values of the dimensionless temperature $t={\left(\beta \hslash q_0\right)}^{-1}$ and for fixed $m_5=4\mathrm{meV}$. Curves were obtained from Eqs. (26) and (27) by numerical integration. The doted lines represent the values of $\tilde{m}$ for which $\mathrm{\Delta }=\hslash q_0$, where $\mathrm{\Delta }$ is the energy gap of the insulator and $q_0$ is the phonon frequency ($\hslash q_0=10\mathrm{meV}$). The parameter values are the same as for Fig. 1.

Figure 4

Phonon contribution to the light absorption coefficient at zero temperature, as a function of the photon frequency $\nu$. Curves were calculated from Eq. (37). The doted line corresponds to $\nu ={\omega }_0$, where ${\omega }_0$ is the transverse (and infrared active) optical phonon frequency. The variable $\phi$ represents the angle between the conventional phonon effective charge (${\mathbf{Q}}_0$) and the external magnetic field (${\mathbf{B}}_0$). The parameter values are $B_0=1\mathrm{T}$ (with the exception of a curve for which $B_0=0$), $Q_0=e$ for the conventional phonon effective charge, $Q_{\theta }=0.1e$ for the axionic phonon effective charge, and $\eta =0.01{\omega }_0$ for the frequency broadening factor.