Circular dichroism in higher-order harmonic generation: Heralding topological phases and transitions in Chern insulators

Topological materials are of interest to both fundamental science and advanced technologies because topological states are robust with respect to perturbations and dissipation. Experimental detection of topological invariants is thus in great demand but remains extremely challenging. Ultrafast laser-matter interactions, and in particular high-harmonic generation, meanwhile, were proposed several years ago as tools to explore the structural and dynamical properties of various matter targets. Here we show that the high-harmonic emission signal produced by a circularly polarized laser contains signatures of topological phases and transitions in the paradigmatic Haldane model. In addition to clear shifts of the overall emissivity and harmonic cutoff, the high-harmonic emission shows a unique circular dichroism, which exhibits clear changes in behavior at the topological phase boundary. Our findings pave the way to understand fundamental questions about the ultrafast electron-hole pair dynamics in topological materials via nonlinear high-harmonic generation spectroscopy.

Figure 1

Phase diagram and band structures of the Haldane model. (a) Phase diagram in the plane (${\varphi }_0,M_0/t_2$), with the different phases labeled by their Chern number $C$; the green line shows the boundary of the topological phase transition. The band-structure diagrams correspond to the points X, Y, Z, and Q as marked in (a), with the point Q at the phase transition showing the gapless Dirac cone at the $K$ point. (Y) and (Z) depict the band structure when the conduction-band topological invariants are $C=\pm 1$ at the phase points ${\varphi }_0=\pm \pi /2$ and $M_0=2.54t_2$. (Y) shows how the midinfrared laser-source oscillations (red-solid line) drive the topological material; this can be with both linear and circular polarizations. We also depict a physical cartoon of the electron-hole pair dynamics driven by a linearly polarized laser, i.e., creation, propagation, and annihilation or recombination by the black-dashed lines with arrows, and finally the subsequent harmonic emission (violet oscillations). The dashed lines in (a) indicate the cuts used for the parameter scans below. Panels (b) and (c) show the Brillouin zone and the real-space lattice of the Haldane model with the couplings in use.

Figure 2

(a)–(d) Dipole transition matrix elements over the Brillouin zone for trivial and topological phases for different gauges. We show the $x$ component in a color map ranging from dark blue to red, with white at zero. The hexagon shows the first BZ, with the $\mathrm{K}$ (${\mathrm{K}}^{\prime }$) points denoted by the white (yellow) circles. The insets in (c), (d) show cuts through the singularities: (c) along $k_y$, at fixed $k_x=-1.85$ (black), $-1.80$ (green), $-1.75$ (blue), and $-1.70\text{a.u.}$ (red), and (d) in vertical cuts at fixed $k_y=-0.55$ (black), $-0.50$ (green), $-0.45$ (blue), and $-0.40\text{a.u.}$ (red). The cyan circles highlight the discontinuities in the dipoles. (e)–(h) The resulting HHG spectra under an LCP laser driver, for gauges A and B (e), (g) and our variable-gauge method (f), (h), for both phases. We set $t_1=0.075\text{a.u.}, t_2=t_1/3, M_0=0.0635\text{a.u.}=2.54t_2$ and $a_0=1\text{Å}$, giving a band gap of $E_g=3.0\mathrm{eV}$, and we use a magnetic flux of (a), (b) ${\varphi }_0=0.06\mathrm{rad}$, giving $C=0$, and (c), (d) ${\varphi }_0=1.16\mathrm{rad}$, giving $C=+1$. For the HHG spectra, we use the pulse Eq. (25) with $E_0=0.0045\text{a.u.}, {\omega }_0=0.014\text{a.u.}=0.38\mathrm{eV}$ (corresponding to $\lambda =3.2\mu \mathrm{m}$), and with a Gaussian envelope lasting $N_c=14$ cycles at FWHM; we use a dephasing time of $T_2=220\text{a.u.}$ We integrate over a momentum grid of $N_x=873$ by $N_y=511$ points using a two-dimensional Simpson rule, with boundaries at $k_{x,\max }=\frac{2\pi }{\sqrt{3}{a}_0}$ and $k_{y,\max }=\frac{2\pi }{3a_0}$; for the SBE time integration our time step is $dt=0.1\text{a.u.}$

Figure 3

(a) Experimental [27] and (b) theoretical HHG spectra from an $\alpha$-quartz crystal. Both the theoretical and experimental data are averaged along the parallel and perpendicular harmonic emission with respect to a MIR $800\mathrm{nm}$ laser linearly polarized along the $\mathrm{\Gamma }-K$ crystal orientation [cf. Fig. 1]. (c) Parallel and (d) perpendicular harmonic emission from our $\alpha$-quartz toy model, showing the interband (blue), intraband (red), and total (green) currents. We use laser parameters following Ref. [27], with a wavelength ${\lambda }_0=800\mathrm{nm}$ giving a central frequency ${\omega }_0=0.056\text{a.u.}\left(1.53\mathrm{eV}\right)$, FWHM duration of ten cycles under a Gaussian envelope, and peak intensity of $I_0=3.2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. We use a honeycomb lattice with lattice constant $a_0\sim 4.8\text{Å}$, NN hopping $t_1=2.1\mathrm{eV}$, NNN hopping amplitude $t_2=0.36\mathrm{eV}$ with magnetic flux ${\varphi }_0=0$, and staggering potential $M=4.5\mathrm{eV}$, with a band gap $E_g\sim 9\mathrm{eV}$, and a dephasing time of $T_2=220\text{a.u.}=5.3\mathrm{fs}$.

Figure 4

High-order harmonics for trivial and nontrivial topological materials. We show the harmonic emission from equal band-gap instances from the trivial and topological phases driven by lasers with (a) linear polarization along $\mathrm{\Gamma }-K$ and (b) right-circular polarization; the violet areas indicate the difference in the emission between the two phases. We use $t_1=2.04\mathrm{eV}, t_2=t_1/3$ and $M_0=2.54t_2$, with lattice constant $a_0=1\text{Å}$, driven by a laser centered at $3.25\mu \mathrm{m}$ (${\omega }_0=0.014\text{a.u.}$) with a Gaussian envelope with 14-cycle FWHM and a peak electric-field strength of $E_0=0.0045\text{a.u.}$ ($I_0=7.0\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$), and we use a dephasing time of $T_2=5.2\mathrm{fs}$. The magnetic flux of the trivial and nontrivial phases, ${\varphi }_0=0.06$ and ${\varphi }_0=1.16\mathrm{rad}$, respectively, are chosen to give equal band gaps of $E_g\approx 3.0\mathrm{eV}$.

Figure 5

Circular dichroism as a signature of topological phase transitions in HHG. (a), (b) Harmonic spectra of equal band-gap instances from the trivial and the topological phase, respectively, driven by right- and left-circularly polarized lasers, with the difference between the two emissions shaded in green. (We use the same parameters as in Fig. 3.) (c), (d) Circular dichroism (CD), as defined in Eq. (46), for the magnetic-flux and staggering potential parameter scans marked with dashed lines in Fig. 1, for the plateau harmonics. The top horizontal axes show the energy band gap $E_g$ in atomic units (a.u.) as a function of ${\varphi }_0$ (c) and $M_0/t_2$ (d), respectively. The trivial and topological regions are shaded green and blue, with the critical region shaded red.

Figure 6

Variation of the harmonic spectrum for circularly-polarized drivers. (a), (b) Harmonic spectrum for the magnetic-flux parameter scan indicated with the red dashed line in Fig. 1, driven by LCP and RCP light, respectively, with ${\varphi }_0$ ranging over a 60-point scan of the interval $[0,\pi ]$ and the staggering potential held fixed at $M_0=2.54t_2$. At the critical region, where the band gap closes between the two phases [as indicated in (c), which charts the band gap], the harmonic emission increases, as carrier creation is much easier. The signature of the topological phase is in the circular dichroism [the difference between the RCP-driven signal in (b) and the LCP-driven signal in (a)], which we plot in Fig. 5 for the harmonics in the green box. (d)–(e) Identical plots for the staggering-potential scan marked by the blue dashed line in Fig. 1, varying $M_0/t_2$ at fixed magnetic flux of ${\varphi }_0=\frac{\pi }{2}$. The (c) and (f) depict the band gaps, $E_g$, as a function of ${\varphi }_0$ and $M_0/t_2$, respectively.

Figure 7

Role of the dephasing time in the high-harmonic emission and in its circular dichroism. (a), (b) High-harmonic generation spectra for various dephasing times $T_2$, for trivial and topological phases, respectively, driven by a right-circularly polarized laser. (c)–(f) Circular dichroisms for $3n+1$ (corotating) and $3n+2$ (contrarotating) harmonic orders [(c)–(f), respectively], for trivial (c), (e) and topological (d), (f) phases. The parameters for the material and the driving laser are the same as in Fig. 2 with a slightly altered peak field strength, $E_0=0.005\text{a.u.}$, and pulse duration, $N_c=16$.

Figure 8

High harmonics and circular dichroism from Wilson Mass Model for trivial, topological phases and transitions. (a) Total high-harmonic generation emissions for trivial (blue) and topological (red with violet shadow) phases. The Wilson mass model (WMM) parameters of lattice constant $a_0=3.33$ a.u. and others for trivial, $C=0$ (topological, $C=+1$) phase are hopping parameter $t_1=0.021$ a.u. (0.034 a.u.) with Wilson mass term $\mathcal{M}=\frac{10}{3}$, ($\frac{6}{5}$) and hopping ${\delta }_0=0.0434$ (0.0723). Those WMM parameters produce equal band gaps of $E_g\approx 3.0$ eV for trivial and the topological phases. (b) Depicts the topological phase transition for corotating harmonic orders ($3n+1$) by the normalized circular dichroism (CD) as a function of $\mathcal{M}$ at a fixed hopping parameter of $t_1=0.034$ a.u. The top horizontal axis points to the energy gap $E_g$ (atomic units are used) as a function of $\mathcal{M}$. The $\mathrm{CD}$ shows two different plateaus of $\mathrm{CD}\approx +1$ and $\mathrm{CD}\approx -1$ for, respectively, the topological invariants $C=+1$ and $C=-1$. On the contrary, for regions where the Chern number $C=0$, the CD is oscillating. The latter observations are absolutely consistent with the results provided by Haldane model in Fig. 5. The laser parameters are the same as in Fig. 5.

Figure 9

Trivial and topological high harmonic cutoff law for linear polarized driving fields. Total HHG emissions for trivial and topological phases, with the same HM parameters than those used in Fig. 2, as a function of the laser field strength $E_0$ are depicted in (a) and (b), respectively. (c), (e) The decomposition of the total harmonic emission for the intraband and interband current emissions of the trivial phase, respectively. (d), (f) The intraband and interband current emissions for the topological phase. Green lines mimic the linear cutoff for both trivial and topological phases. The laser field for these HHG calculations is linearly polarized along the ${\mathrm{K}}^{\prime }-\mathrm{\Gamma }-\mathrm{K}$ direction of the honeycomb hexagonal lattice.