双曲格的自守Bloch定理
Automorphic Bloch theorems for hyperbolic lattices
摘要：
双曲晶格是一种新形式的合成量子物质，其中粒子有效地跳跃在2D双曲空间的离散镶嵌上，这是一种具有均匀负曲率的非欧几里德空间。为了描述单粒子的本征态和本征能在这样的晶格上跳跃，一个双曲的带理论的推广先前被构造，基于代数几何的思想。在这个双曲带理论中，本征态是自守函数，布里渊区是一个更高维的环面，紧化晶胞的雅可比矩阵被理解为更高亏格的黎曼曲面。有三个重要的问题没有得到解答：带理论是否可以被期望适用于非欧几里德格，其中平移通常不交换;形式的布洛赫定理是否可以严格建立;以及双曲带理论是否可以描述实验中实现的有限格。在目前的工作中，我们同时解决这三个问题。通过制定周期性边界条件有限，但任意大的晶格，我们表明，广义布洛赫定理可以严格证明，但可能会或可能不会涉及高维不可约表示（不可约）的nonabel平移群，这取决于晶格几何。高维非线性表示对应于高阶稳定全纯向量丛的模空间中的点，这进一步推广了雅可比矩阵之外的布里渊区概念。对于一个大类的有限格，只有一维irreps出现，和双曲带理论以前开发的成为准确的。
英文摘要：
Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of 2D hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved, but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps corrrespond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.
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