The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from algebraic geometry to construct a hyperbolic generalization of Bloch theory, despite the absence of commutative translation symmetries. For a quantum particle propagating in a hyperbolic lattice potential, we construct a continuous family of eigenstates that acquire Bloch-like phase factors under a discrete but noncommutative group of hyperbolic translations, the Fuchsian group of the lattice. A hyperbolic analog of crystal momentum arises as the set of Aharonov-Bohm phases threading the cycles of a higher-genus Riemann surface associated with this group. This crystal momentum lives in a higher-dimensional Brillouin zone torus, the Jacobian of the Riemann surface, over which a discrete set of continuous energy bands can be computed.

中文翻译：

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双曲带理论

布洛赫波、晶体动量和能带的概念通常被认为是具有交换平移对称性的晶体材料的独特特征。受最近在电路量子电动力学中实现双曲晶格的启发，我们利用代数几何的思想来构建布洛赫理论的双曲推广，尽管没有交换平移对称性。对于在双曲晶格势中传播的量子粒子，我们构建了一个连续的本征态族，这些本征态在离散但非交换的双曲平移群（晶格的 Fuchsian 群）下获得类 Bloch 相位因子。当一组 Aharonov-Bohm 相位穿过与该组相关的更高属黎曼曲面的循环时，出现了晶体动量的双曲线模拟。