In this work we use Floquet theory to theoretically study the influence of monochromatic circularly and linearly polarized light on the Hofstadter butterfly-induced by a uniform perpendicular magnetic field-for both the kagome and triangular lattices. In the absence of the laser light, the butterfly has fractal structure with inversion symmetry about magnetic flux φ = 1/4, and reflection symmetry about φ = 1/2. As the system is exposed to an external laser, we find circularly polarized light deforms the butterfly by breaking the mirror symmetry at flux φ = 1/2. By contrast, linearly polarized light deforms the original butterfly while preserving the mirror symmetry at flux φ = 1/2. We find the inversion symmetry is always preserved for both linear and circular polarized light. For linearly polarized light, the Hofstadter butterfly depends on the polarization direction. Further, we study the effect of the laser on the Chern number of lowest band in the off-resonance regime (laser frequency is larger than the bandwidth). For circularly polarized light, we find that low laser intensity will not change the Chern number, but beyond a critical intensity the Chern number will change. For linearly polarized light, the Chern number depends on the polarization direction. Our work highlights the generic features expected for the periodically driven Hofstadter problem on different lattices.
Mercury, lead, and cadmium are among the most toxic and carcinogenic heavy metal ions (HMIs), posing serious threats to the sustainability of aquatic ecosystems and public health. There is an urgent need to remove these ions from water by a cheap but green process. Traditional methods have insufficient removal efficiency and reusability. Structurally robust, large surface‐area adsorbents functionalized with high‐selectivity affinity to HMIs are attractive filter materials. Here, an adsorbent prepared by vulcanization of polyacrylonitrile (PAN), a nitrogen‐rich polymer, is reported, giving rise to PAN‐S nanoparticles with cyclic π‐conjugated backbone and electronic conductivity. PAN‐S can be coated on ultra‐robust melamine (ML) foam by simple dipping and drying. In agreement with hard/soft acid/base theory, N‐ and S‐containing soft Lewis bases have strong binding to Hg2+, Pb2+, Cu2+, and Cd2+, with extraordinary capture efficiency and performance stability. Furthermore, the used filters, when collected and electrochemically biased in a recycling bath, can release the HMIs into the bath and electrodeposit on the counter‐electrode as metallic Hg0, Pb0, Cu0, and Cd0, and the PAN‐S@ML filter can then be reused at least 6 times as new. The electronically conductive PAN‐S@ML filter can be fabricated cheaply and holds promise for scale‐up applications.
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