Hallmarks of spin textures for high-harmonic generation in two-dimensional materials

TL;DR

The paper addresses how spin-orbit coupling and Berry curvature shape high-harmonic generation in two-dimensional non-centrosymmetric materials, with a focus on even-order harmonics. It develops HHG selection rules via dynamical symmetries and validates them with microscopic models, showing that spin textures breaking $C_2$ are required for finite even harmonics and that Berry curvature enables these harmonics under time-reversal symmetry; it also demonstrates how dynamical symmetry breaking can modulate higher-order harmonics. The work provides a framework to use HHG as a spectroscopic tool to detect rotational-symmetry breaking, spin textures, and dynamical phase transitions, with potential applications in ultrafast spintronics and symmetry-dependent nonlinear optics. It also discusses limitations such as neglected scattering, outlining directions for including relaxation effects in future studies.

Abstract

Spin-orbit coupling and quantum geometry are fundamental aspects in modern condensed matter physics, with their primary manifestations in momentum space being spin textures and Berry curvature. In this work, we investigate their interplay with high-harmonic generation (HHG) in two-dimensional non-centrosymmetric materials, with an emphasis on even-order harmonics. Our analysis reveals that the emergence of finite even-order harmonics necessarily requires a broken twofold rotational symmetry in the spin texture, as well as a non-trivial Berry curvature in systems with time-reversal invariance. This symmetry breaking can arise across various degrees of freedom and impact both spin textures and optical response via spin-orbit interactions. We also show that HHG is particularly sensitive to dynamical rotational-symmetry breaking, as even high-order components can be modulated by a time-dependent symmetry breaking. These findings underscore the potential of HHG as a tool for exploring electronic phases with broken rotational symmetry, as well as the associated phase transitions in two-dimensional materials, and provide novel perspectives for designing symmetry-dependent nonlinear optical phenomena.

Figures (3)

• Figure 1: Illustration of spin-texture patterns in momentum space for high-harmonic generation. Left: in 2D systems with spin textures invariant under twofold rotational symmetry, $\hat{\mathcal{C}}_2$, there is no emission of even-order harmonics. Right: for spin textures breaking $\hat{\mathcal{C}}_2$ symmetry, the emission of even-order harmonics is allowed.
• Figure 2: Spin textures and high-harmonic response for different physical configurations. Schematic illustration of the trigonal lattice (a) with threefold rotational symmetry and antiferromagnetic pattern (d) with broken twofold rotational symmetry. Both systems possess a vertical mirror symmetry $\mathcal{M}_x$ along the $x$-axis, and in both cases, the incident field (green arrow) is polarized perpendicular to this direction. Blue and orange arrows represent the expected polarizations for odd- and even-order harmonics, respectively. Panels (b) and (e) show the spin textures in momentum space corresponding to the physical configuration in (a) and (d), respectively. The color legend indicates the out-of-plane spin component $\langle \sigma_z \rangle$, with color blue, gray and red corresponding to $\langle \sigma_z \rangle=-1$, $0$ and $1$, respectively. Black circles mark spots in the spin textures where $\hat{\mathcal{C}}_2$ symmetry, as described by Eq. (\ref{['C2spin']}), is broken. Panels (c) and (f) show the HHG spectra (in logarithmic scale) for the time-reversal-invariant trigonal configuration and the antiferromagnetic pattern, respectively. The Hamiltonian parameters, in units of the hopping energy $t$, are: $\gamma_R=0.1$ for both cases; $\lambda=0.1$ in (c); and $\Delta\varepsilon=0.1$, $B=0.1$ in (f). The computations were performed at temperature $T=0$ with chemical potential $\mu=1.0$. In both cases, the incident field has intensity $|{\bf A}_0|=1.0$ and frequency $\Omega_{in}=0.5$. Polarizations of odd- (blue dots) and even-order (orange squares, with different shades of orange) harmonics are consistent with the outcomes of Eq. (\ref{['mirror']}). The insets show the intensity of even-order harmonics (in linear scale) as functions of the parameters controlling the $\hat{\mathcal{C}}_2$-symmetry breaking: the trigonal warping $\lambda$ in (c) and the charge imbalance $\Delta \epsilon$ in (f). The units and color legend of even-order harmonics in each inset are consistent with those of the corresponding panel.
• Figure 3: Dependence of even-order harmonics on the driving frequency. Amplitudes of the second- (blue), fourth- (green), sixth- (red) and eighth-order harmonics are shown as functions of the ratio between the driving frequency and the pump frequency. The parameters are set to $\Omega_{pump}=0.1$, $\gamma_R=0.02$, $B_z=0.15$, $B_x=0.1$ and $\mu=0.9$.