Anomalous Hall Conductivity as an Effective Means of Tracking the Floquet Weyl Nodes in Quasi-One-Dimensional $β$-Bi$_4$I$_4$

Abstract

While Floquet engineering offers a powerful paradigm for manipulating topological phases, particularly Floquet Weyl semimetals, establishing an experimentally feasible strategy for tracking the dynamic evolution of such states remains a significant challenge. Here, we propose that the anomalous Hall effect (AHE), as a sensitive, all-electrical probe, can be used to track Floquet Weyl nodes. Using first-principles calculations and symmetry analysis on the quasi-one-dimensional material $β$-Bi$_4$I$_4$, we demonstrate that circularly polarized light breaks time-reversal symmetry, driving the system from a trivial insulator into a Floquet Weyl semimetal phase characterized by a nonzero Berry curvature flux. Crucially, by continuously tuning the polarization phase $\varphi$ of the driving field, we show that the trajectory of the induced Weyl nodes is highly controllable, leading to their migration and eventual annihilation at high-symmetry points. We reveal that the anomalous Hall conductivity maps directly onto this topological evolution, serving as a definitive fingerprint for the generation and dynamics of Weyl nodes.

Figures (4)

• Figure 1: (a) Crystal structure of $\beta$-Bi$_4$I$_4$ (conventional unit cell). (b) Bulk Brillouin zone (BZ) and the projected (001) surface BZ, with high-symmetry points indicated. (c) Electronic band structure calculated using the modified Becke-Johnson (mBJ) exchange potential.
• Figure 2: (a) Evolution of the band structure of $\beta$-Bi$_4$I$_4$ under circularly polarized light with increasing light intensity $\mathrm{e}\mathrm{A}_{0}/\hbar$. (b) Band dispersions at $\mathrm{e}\mathrm{A}_{0}/\hbar = 0.056$ and $0.180$ Å$^{-1}$, respectively. Regarding the color coding: under optical driving, each original band splits into two branches. The red lines denote the inner branches that approach and intersect to form Weyl nodes, while the blue lines denote the outer branches that move away. The inset in the left panel of (b) confirms that there is no band crossing at the $M$ point. In these gapless phases, the system hosts a single pair of Floquet Weyl nodes $W^{\pm}$, located near the $M$ point and the $L$ point. Here, the superscript $\pm$ denotes the chirality of the Weyl node is $\pm1$. The Weyl nodes with positive and negative chirality are marked in red and green, respectively.
• Figure 3: (a), (b) Surface spectrum on the (001) surface of $\beta$-Bi$_4$I$_4$ driven by circularly polarized light. The field amplitudes are $eA_{0}/\hbar = 0.054$ and $0.186$ Å$^{-1}$, respectively. (c), (d) Corresponding isoenergy contours on the (001) surface calculated at the energy midpoint between the Weyl nodes. Red and green dots mark the projected Weyl nodes with positive and negative chirality, respectively. The light blue dots labeled $T$ and $H$ in (c) and (d) indicate the specific momenta where the surface spectrum in (a) and (b) were sampled. The lines dotted in gray denote the routes shown in (a), (b).
• Figure 4: (a) Trajectories of Floquet Weyl nodes in reciprocal space ($k_{1}$-$k_{3}$) driven by light with a tunable phase difference $\varphi$. The left and right panels correspond to field amplitudes $eA_{0}/\hbar = 0.056$ and $0.168$ Å$^{-1}$, respectively. $\varphi$ defines the phase delay between the orthogonal linear components. Solid black dots mark the merging points of Weyl nodes with opposite chirality at high-symmetry $M$ or $L$ points when $\varphi = 90^{\circ}$. (b) Calculated anomalous Hall conductivity $\sigma_{xy}$ as a function of chemical potential for various $\varphi$ in the light intensity of $eA_{0}/\hbar = 0.056$ Å$^{-1}$. Note that $\sigma_{xy}$ decreases and eventually vanishes as $\varphi$ increases. (c) Schematic phase diagram illustrating the topological transition driven by the polarization state of the light.