论文标题
双曲线理论
Hyperbolic band theory
论文作者
论文摘要
Bloch波，晶体动量和能带的概念通常被视为具有交换符号的晶体材料的独特特征。尽管没有交换性翻译对称性，但我们从代数几何形状中利用了代数几何形状的思想来构建Bloch理论的第一个双曲线概括，这是由于最近对电路量子电动力学中双曲线晶格的实现的动机。对于在双曲线晶格电位中传播的量子粒子，我们构建了一个连续的本征菌家族，该家族在离散但非交通性的双曲线翻译基团（晶格的紫红色群）中获得类似Bloch的相位因子。随着Aharonov-bohm相与该组相关的较高基因riemann表面的循环，晶体动量的双曲类似物的产生。这种晶体动量生活在较高维的布里渊区圆环，即黎曼表面的雅各布式，可以在其上计算一组离散的连续能带。
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 the first 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.
