Fractal spectrum in twisted bilayer optical lattice
Abstract: The translation symmetry of a lattice is greatly modified when subjected to a perpendicular magnetic field [Zak, Phys. Rev. \textbf{134}, A1602 (1964)]. This change in symmetry can lead to magnetic unit cells that are substantially larger than the original ones. Similarly, the translation properties of a double-layered lattice alters drastically while two monolayers are relatively twisted by a small angle, resulting in large-scale moir\'{e} unit cells. Intrigued by the resemblance, we calculate the complete band structures of a twisted bilayer optical lattice and show that the geometric moir\'{e} effect can induce fractal band structures. The fractals are controlled by the twist angle between two monolayers and are closely connected to the celebrated butterfly spectrum of two-dimensional Bloch electrons in a magnetic field [Hofstadter, Phys. Rev. B \textbf{14}, 2239 (1976)]. We demonstrate this by proving that the twisted bilayer optical lattice can be mapped to a generalized Hofstadter's model with long-range hopping. Furthermore, we provide numerical evidence on the infinite recursive structures of the spectrum and give an algorithm for computing these structures.
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- We note that the phases in the cosine functions in H𝐤subscript𝐻𝐤H_{\mathbf{k}}italic_H start_POSTSUBSCRIPT bold_k end_POSTSUBSCRIPT play the role of the momentum along the fictitious y′superscript𝑦′y^{\prime}italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT-direction of the two-dimensional model. Thus only the Bloch states with ky′=0subscript𝑘superscript𝑦′0k_{y^{\prime}}=0italic_k start_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 of H2Dsubscript𝐻2𝐷H_{2D}italic_H start_POSTSUBSCRIPT 2 italic_D end_POSTSUBSCRIPT can be mapped to eigenstates of H𝐤subscript𝐻𝐤H_{\mathbf{k}}italic_H start_POSTSUBSCRIPT bold_k end_POSTSUBSCRIPT. Yet Bloch states with finite ky′subscript𝑘superscript𝑦′k_{y^{\prime}}italic_k start_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT are also valuable, as they correspond to eigenstates of TBOL with finite displacement between V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and V2subscript𝑉2V_{2}italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.
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