标题： 任意规范场在具有实跃迁幅度的合成晶体中的变换群 显示英文标题

标题： Translation Groups for arbitrary Gauge Fields in Synthetic Crystals with real hopping amplitudes

摘要： 在[F. R. Lux和E. Prodan，Annales Henri Poincaré 25(8), 3563 (2024)]中引入的Cayley晶体是一类带有哈密顿量的晶格，其平移群$G$是通用的，可能是非交换的。我们证明这些系统自然实现了所谓的磁平移群到任意离散规范群的推广。单体动力学模拟了携带电荷叠加的粒子的动力学，每个电荷耦合到不同的静态规范场配置。可能的规范场类型由换位子子群$C \subset G$的不可约表示确定，而威尔逊环路配置——它们不必是均匀的——由$C$在$G$中的嵌入决定。假设$C$是有限的，深入分析了其他子群在塑造晶格几何和动力学中的作用。我们讨论了一个具有直接工程相关性的定理，对于任何循环规范群，该定理可以得到所有兼容的平移群。然后我们构造了等价于被非均匀磁通量穿过的正方形晶格的二维Cayley晶体示例。重要的是，Cayley晶体仅需实数跃迁幅度，并且可以在可扩展的几何结构中实现，能够容纳高于三维的动力学，从而使得实验探索以及在超材料、cQED和其他合成平台中的最终应用成为可能。 显示英文摘要

摘要： The Cayley-crystals introduced in [F. R. Lux and E. Prodan, Annales Henri Poincar\'e 25(8), 3563 (2024)] are a class of lattices endowed with a Hamiltonian whose translation group $G$ is generic and possibly non-commutative. We show that these systems naturally realize the generalization of the so-called magnetic translation groups to arbitrary discrete gauge groups. A one-body dynamics emulates that of a particle carrying a superposition of charges, each coupled to distinct static gauge-field configuration. The possible types of gauge fields are determined by the irreducible representations of the commutator subgroup $C \subset G$, while the Wilson-loop configurations - which need not be homogeneous - are fixed by the embedding of $C$ in $G$. The role of other subgroups in shaping both the lattice geometry and the dynamics is analyze in depth assuming $C$ finite. We discuss a theorem of direct engineering relevance that, for any cyclic gauge group, yields all compatible translation groups. We then construct two-dimensional examples of Cayley-crystals equivalent to square lattices threaded by inhomogeneous magnetic fluxes. Importantly, Cayley-crystals can be realized with only real hopping amplitudes and in scalable geometries that can fit higher-than-3D dynamics, enabling experimental exploration and eventual exploitation in metamaterials, cQED, and other synthetic platforms.