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

TL;DR

This paper introduces Cayley-crystals, a class of lattices with non-commutative translation groups, capable of simulating arbitrary gauge fields with real hopping amplitudes, opening new avenues for experimental synthetic materials.

Contribution

It generalizes magnetic translation groups to arbitrary discrete gauge groups within Cayley-crystals, providing a framework for engineering complex gauge fields in scalable, real-hopping lattice systems.

Findings

Realization of gauge fields with non-commutative translation groups

Construction of 2D Cayley-crystals with inhomogeneous magnetic fluxes

Potential for experimental implementation in metamaterials and quantum platforms

Abstract

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.