Nonperturbative topological current in Weyl and Dirac semimetals in laser fields

We study nonperturbatively the anomalous Hall current and its high harmonics generated in Weyl and Dirac semimetals by strong elliptically polarized laser fields in the context of kinetic theory. We find a crossover between perturbative and nonperturbative regimes characterized by the electric field strength ${\mathcal{E}}^{*}=\frac{\mu \omega }{2e{v}_F}$ ($\omega$, laser frequency; $\mu$, Fermi energy; $v_F$, Fermi velocity). In the perturbative regime, the anomalous Hall current depends quadratically on the field strength $\left(\mathcal{E}\right)$, whereas the higher-order corrections, as well as high harmonics, vanish at zero temperature. In the nonperturbative regime, the anomalous Hall current saturates and decays as $(ln\mathcal{E})/\mathcal{E}$, while even-order high harmonics are generated when in-plane rotational symmetry is broken. Based on the analytical solution of the Boltzmann equation, we reveal the topological origin of the sharp crossover: the Weyl monopole stays inside or moves outside of the Fermi sphere, respectively, during its fictitious motion in the perturbative or nonperturbative regimes. Our findings establish a nonlinear response intrinsically connected to topology, characteristic of Weyl and Dirac semimetals.

Figure 1

Anomalous Hall current (${\mathit{j}}_{AH}$) in a Weyl semimetal (WSM) induced by elliptically polarized light (EPL). The circularly polarized component (CPL) of EPL generates a dc anomalous Hall current, while an additional linearly polarized (LP) component induces even-order topological high harmonic generation (HHG).

Figure 2

Illustration of the shifted distribution functions $f_0(\mathit{p}-e\mathit{\Delta }(0,u))$ contributing to the anomalous Hall current, ${\mathit{j}}_{AH}$ [Eq. (7)], induced by CPL in the (a) weak-field ($\mathcal{E}<{\mathcal{E}}^{*}$) and (b) strong-field ($\mathcal{E}>{\mathcal{E}}^{*}$) regimes. ${\mathcal{E}}^{*}$ is the maximum field value for which the monopole remains enclosed by the Fermi surface during its fictitious motion [Eq. (9)]. Circles represent the Fermi sphere shifted by $\mathit{\Delta }(0,u)=\mathit{A}\left(0\right)-\mathit{A}\left(u\right)$. In the weak-field regime, the overlap between shifted distribution and Weyl monopole $M$ enables perturbation theory to render the exact result of the current. In the strong-field regime, perturbation theory is only able to describe a small set of shifted distributions, denoted by $R_P$ (depicted by the blue arrow). The remaining distributions contributing to the current belong to $R_{NP}$ (red arrow) and are responsible for the nonperturbative behavior of ${\mathit{j}}_{AH}$.

Figure 3

(a) Anomalous Hall current density (${\mathit{j}}_{AH}$) for a single Weyl node ($\eta =1$) at 30 K, measured in units of $e{\mu }^3/v_F^2{\left(2\pi \hslash \right)}^3$, as a function of electric field $\mathcal{E}$, in units of ${\mu }^2/e\hslash v_F$, for different values of $\tau$, in units of ${\omega }^{-1}$. Circles result from numerical evaluation of Eq. (6), while solid lines stem from the analytical results presented in Eqs. (10) and (11, 12, 13). For numerical analysis, we set $\omega =1\mathrm{THz}$, $\mu =118\mathrm{meV}$, and $v_F=7.8\times {10}^5\text{m/s}$. The critical field strength ${\mathcal{E}}^{*}$, obtained from Eq. (9), is independent of $\tau$. (b) Absolute value of the difference between the anomalous Hall current (${\mathit{j}}_{AH}$) and $j_{{\mathcal{E}}^2}$, defined by Eq. (10), as a function of the electric field $\mathcal{E}$, in units of ${\mu }^2/e\hslash v_F$, for different values of $\tau$, in units of ${\omega }^{-1}$.

Figure 4

(a) Time dependence of the anomalous Hall current density (${\mathit{j}}_{AH}$) for a single Weyl node ($\eta =1$) at 30 K, measured in units of $e{\mu }^3/v_F^2{\left(2\pi \hslash \right)}^3$, for a period $T$ of the driving EPL with $2{\mathcal{E}}_x={\mathcal{E}}_y=\mathcal{E}$ ($\alpha =0.5$), for different values of $\mathcal{E}$ (in units of ${\mu }^2/e\hslash v_F$) and $\omega \tau =0.1$. (b) Field dependence of HHG, defined as an absolute value cosine Fourier coefficient $|a_n|$ of ${\mathit{j}}_{AH}$, for $n=0$, 2, and 4 with $\alpha =0.1$ at 30 K. Circles result from numerical evaluation of Eq. (6) for EPL, while solid lines are obtained from analytical evaluation of ${\mathit{j}}_{AH}$ up to cubic order in $\alpha$. (c) Ellipticity dependence of HHG with $\mathcal{E}=0.25$ at 30 K. Circles result from numerical evaluation of Eq. (6), while solid lines are obtained from analytical evaluation of ${\mathit{j}}_{AH}$ up to tenth order in $\alpha$. For numerical analysis, we considered $\mu =118\mathrm{meV}$ and $v_F=7.8\times {10}^5\text{m/s}$.