标题： 基于对称性的二维双曲晶体上狄拉克费米子的理论：与旋联络的耦合 显示英文标题

标题： Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection

摘要： 离散的费米子和玻色子模型在双曲晶格中自 metamaterial 平台中双曲晶格的实验实现以来，在多个领域引起了广泛关注，激发了双曲晶体学的发展。 然而，一个基本且具有实验意义的问题仍未得到解决：在弯曲空间中传播的费米子通过自旋连接与底层几何耦合，这是广义协变性所要求的——这一特性尚未被纳入对双曲晶体的研究中。 在此，我们引入了一个基于对称性的框架，用于二维双曲晶格上的狄拉克费米子，通过离散自旋连接显式地包含自旋-曲率耦合。 从庞加莱圆盘的连续对称性出发，我们分类了不可约表示并构建了一个对称适应的基，建立了与连续狄拉克理论的直接对应关系。 我们表明，这种连续理论预测在任意有限曲率的$D-$维双曲空间中具有$2\leq D \leq 4$，这表明狄拉克费米子在弱耦合下对相互作用驱动的不稳定性具有增强的敏感性。 随后，我们推导了由 Schläfli 符号$\{p,q\}$表征的晶格的离散平移和旋转对称性的显式形式，并通过平行传输显式构建了离散自旋连接，表示为跳跃相位。 我们的结果为在 metamaterial 平台上实现自旋-曲率效应以及在双曲几何中系统研究相关狄拉克相位开辟了道路。 显示英文摘要

摘要： Discrete fermionic and bosonic models for hyperbolic lattices have attracted significant attention across a range of fields since the experimental realization of hyperbolic lattices in metamaterial platforms, sparking the development of hyperbolic crystallography. However, a fundamental and experimentally consequential aspect remains unaddressed: fermions propagating in curved space inherently couple to the underlying geometry via the spin connection, as required by general covariance - a feature not yet incorporated in studies of hyperbolic crystals. Here, we introduce a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices, explicitly incorporating spin-curvature coupling via a discrete spin connection. Starting from the continuous symmetries of the Poincar\'e disk, we classify the irreducible representations and construct a symmetry-adapted basis, establishing a direct correspondence to the continuum Dirac theory. We show that this continuum theory predicts a finite density of states at zero energy for any finite curvature in $D-$dimensional hyperbolic space with $2\leq D \leq 4$, suggesting enhanced susceptibility of Dirac fermions to interaction-driven instabilities at weak coupling. We then derive explicit forms of discrete translational and rotational symmetries for lattices characterized by Schl\"afli symbols $\{p,q\}$, and explicitly construct the discrete spin connection, represented as hopping phases, via parallel transport. Our results pave the way for experimental realization of spin-curvature effects in metamaterial platforms and systematic numerical studies of correlated Dirac phases in hyperbolic geometries.