Linear and nonlinear optical responses in Kitaev spin liquids

We theoretically study THz-light-driven high-harmonic generation (HHG) in the spin-liquid states of the Kitaev honeycomb model with a magnetostriction coupling between spin and electric polarization. To compute the HHG spectra, we numerically solve the Lindblad equation, taking account of the dissipation effect. We find that isotropic Kitaev models possess a dynamical symmetry, which is broken by a static electric field, analogous to HHG in electron systems. We show that the HHG spectra exhibit characteristic continua of Majorana fermion excitations, and their broad peaks can be controlled by applying static electric or magnetic fields. In particular, the magnetic-field dependence of the HHG spectra drastically differs from those of usual ordered magnets. These results indicate that an intense THz laser provides a powerful tool to observe dynamic features of quantum spin liquids.

Figure 1

(a) Lattice structure of the Kitaev model. Blue, green, and red lines, respectively, correspond to $x, y$, and $z$ bonds. (b) Kitaev model with a dimerization on $x$ and $y$ bonds, which is caused by a static electric field $E_{\mathrm{dc}}$ along the $\tilde{x}$ direction. (c) Gapless itinerant fermion band of the Kitaev model at $J_{x,y,z}=J$ and $E_{\mathrm{dc}}=\kappa =0$. (d) Ground-state phase diagram of the Kitaev model in $(J_x,J_y,J_z)$ space at $\kappa =0$. Orange and blue areas are, respectively, gapless and gapped QSLs. Application of $E_{\mathrm{dc}}$ induces $(J_x,J_y)\to (J_x-{\mathcal{E}}_{\mathrm{dc}},J_y+{\mathcal{E}}_{\mathrm{dc}})$. (e)–(g) Density of states of itinerant fermions at $({\mathcal{E}}_{\mathrm{dc}},\kappa )=(0,0)$, (0.1,0), and (0,0.2) in $J<0$. For details of the density of states, see Figs. S8 and S9 of the supplemental material [57].

Figure 2

HHG spectra $I\left(\omega \right)$ in driven isotropic ($J_{x,y,z}=J$) Kitaev models with/without a dc electric field $E_{\mathrm{dc}}$ at $\kappa =0$ under a THz pulse of $\mathrm{\Omega }=2.0J$. (a) $I\left(\omega \right)$ as a function of $\omega$ at ${\mathcal{E}}_{\mathrm{dc}}=0$ and $0.1J$ under the irradiation of ${\mathcal{E}}_{\mathrm{ac}}=0.1J$. $I\left(\omega \right)$ is normalized with its maximum value. (b),(c) $(E_{\mathrm{ac}},E_{\mathrm{dc}})$ dependence of SHG [$I\left(2\mathrm{\Omega }\right)$] and FHG [$I\left(4\mathrm{\Omega }\right)$] spectra. The intensities, panels (b) and (c), are normalized with $I\left(\mathrm{\Omega }\right)$ at ${\mathcal{E}}_{\mathrm{ac}}={\mathcal{E}}_{\mathrm{dc}}=0.05J$.

Figure 3

$I\left(\mathrm{\Omega }\right)$ of the Kitaev models with $J_{x,y,z}=J$ and ${\mathcal{E}}_{\mathrm{dc}}=0.1J$ at (a) a weak laser intensity ${\mathcal{E}}_{\mathrm{ac}}/J={10}^{-3}$ and (b) strong intensities of ${\mathcal{E}}_{\mathrm{ac}}/J=0.1$, 0.15, and 0.2. Blue lines and dotted points are, respectively, results of the linear response (Kubo) theory and numerically solved master equation [57]. $I\left(\mathrm{\Omega }\right)$ is normalized with the maximum value of the Kubo formula. Panels (c) and (d) show DOSs of fermions corresponding to cases (a) and (b), respectively. (e) SHG [$I\left(2\mathrm{\Omega }\right)$] and (f) THG [$I\left(3\mathrm{\Omega }\right)$] of the Kitaev model with ${\mathcal{E}}_{\mathrm{dc}}=0.1J$ in the space $({\mathcal{E}}_{\mathrm{ac}},\mathrm{\Omega })$. The intensities, panels (e) and (f), are normalized with $I\left(\mathrm{\Omega }\right)$ at $({\mathcal{E}}_{\mathrm{ac}},\mathrm{\Omega })=(0.05J,2J)$. The corresponding DOSs are depicted in panels (g) and (h).

Figure 4

(a) $I\left(\mathrm{\Omega }\right)$ in an antiferromagnetic ($J<0$) Kitaev model with $\kappa =0.2\left|J\right|, {\mathcal{E}}_{\mathrm{dc}}=0$, and ${\mathcal{E}}_{\mathrm{ac}}={10}^{-3}\left|J\right|$. (b) $\kappa$ dependence of $I\left(\mathrm{\Omega }\right)$ in $\mathrm{\Omega }=2.0\left|J\right|, {\mathcal{E}}_{\mathrm{dc}}=0$, and ${\mathcal{E}}_{\mathrm{ac}}={10}^{-3}\left|J\right|$. Blue line and dotted points are, respectively, results of the linear response theory and the master equation [57]. $I\left(\mathrm{\Omega }\right)$ in (a) and (b) are normalized with the maximum value in panel (a). (c) $\mathrm{\Omega }$ and (d) $\kappa$ dependences of $\mathcal{D}\left(\omega \right)$ that, respectively, correspond to panels (a) and (b). (e) $I\left(\mathrm{\Omega }\right)$ and (f) $I\left(2\mathrm{\Omega }\right)$ of the Kitaev model with ${\mathcal{E}}_{\mathrm{dc}}={\mathcal{E}}_{\mathrm{ac}}=0.1\left|J\right|$. The intensities, panels (e) and (f), are normalized with $I\left(\mathrm{\Omega }\right)$ at $(\kappa ,\mathrm{\Omega })=\left(0.05\right|J|,2|J\left|\right)$.