Maciejko Joseph, Rayan Steven
Department of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada;
Theoretical Physics Institute, University of Alberta, Edmonton, AB T6G 2E1, Canada.
Proc Natl Acad Sci U S A. 2022 Mar 1;119(9). doi: 10.1073/pnas.2116869119.
Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional (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 an 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 correspond 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.
双曲晶格是一种新型的合成量子物质形式，其中粒子有效地在二维（2D）双曲空间的离散镶嵌上跳跃，双曲空间是一种具有均匀负曲率的非欧几里得空间。为了描述在这样的晶格上跳跃的单粒子本征态和本征能量，先前基于代数几何的思想构建了能带理论的双曲推广。在这种双曲能带理论中，本征态是自守函数，布里渊区是一个高维环面，紧致化晶胞的雅可比行列式被理解为一个高亏格的黎曼曲面。有三个重要问题尚未得到解答：对于平移通常不满足对易关系的非欧几里得晶格，是否可以期望能带理论成立；是否可以严格建立形式上的布洛赫定理；以及双曲能带理论是否能够描述实验中实现的有限晶格。在本工作中，我们同时解决了这三个问题。通过为有限但任意大的晶格制定周期性边界条件，我们表明可以严格证明一个广义布洛赫定理，但它可能涉及也可能不涉及非阿贝尔平移群的高维不可约表示（irrep），这取决于晶格几何结构。高维irrep对应于高阶稳定全纯向量丛的模空间中的点，这进一步将布里渊区的概念推广到超出雅可比行列式的范围。对于一大类有限晶格，只出现一维irrep，并且先前发展的双曲能带理论变得精确。
