Crystallography of hyperbolic lattices
Crystallography of hyperbolic lattices
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DOI：
10.1103/physrevb.105.125118
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发表时间：
2022-03-15
影响因子：
3.7
通讯作者：
Thomale, Ronny
中科院分区：
文献类型：
--
作者：
Boettcher, Igor;Gorshkov, Alexey, V;Thomale, Ronny
双曲晶格是一个革命性的平台，用于在弯曲空间中进行全息和量子物理的桌面模拟，并促进有效的量子纠错码。它们的基本几何是非欧几里德的，布洛赫定理的缺乏排除了通常不可或缺的能带理论的直接应用，以研究双曲格上的模型哈密顿。最近的洞察双曲带理论的启发，我们开始双曲晶格的晶体学。我们发现，许多双曲晶格具有隐藏的晶体结构，其特征在于单位细胞，双曲布拉维晶格，和相关的对称群。使用高亏格黎曼曲面和Fuchsian群的数学框架，我们导出了一系列示例双曲{p，q}格及其双曲Bravais格，包括五个无限族和几个与电路量子电动力学和拓扑电路实验相关的图形。这极大地简化了双曲格点上紧束缚哈密顿能谱的计算，从图上的精确对角化到求解不可约表示的有限方程组。这一成就的重要性需要与传统的欧几里得晶体学在固体研究中所起的至关重要的作用相比较。我们通过构造和对角化有限维布洛赫波哈密顿量来证明这种方法的高潜力。
Hyperbolic lattices are a revolutionary platform for tabletop simulations of holography and quantum physics in curved space and facilitate efficient quantum error correcting codes. Their underlying geometry is non-Euclidean, and the absence of Bloch's theorem precludes the straightforward application of the often indispensable energy band theory to study model Hamiltonians on hyperbolic lattices. Motivated by recent insights into hyperbolic band theory, we initiate a crystallography of hyperbolic lattices. We show that many hyperbolic lattices feature a hidden crystal structure characterized by unit cells, hyperbolic Bravais lattices, and associated symmetry groups. Using the mathematical framework of higher-genus Riemann surfaces and Fuchsian groups, we derive a list of example hyperbolic {p, q} lattices and their hyperbolic Bravais lattices, including five infinite families and several graphs relevant for experiments in circuit quantum electrodynamics and topolectrical circuits. This dramatically simplifies the computation of energy spectra of tight-binding Hamiltonians on hyperbolic lattices, from exact diagonalization on the graph to solving a finite set of equations in terms of irreducible representations. The significance of this achievement needs to be compared to the all-important role played by conventional Euclidean crystallography in the study of solids. We exemplify the high potential of this approach by constructing and diagonalizing finite-dimensional Bloch wave Hamiltonians.