Latent-Geometry Correspondence: Unraveling Hidden Symmetry Groups with Graph Theory

Seeking symmetry where it is not manifest, physicists have explored hidden symmetries throughout classical to quantum systems. A framework of generalized geometric symmetries, dubbed "latent symmetries," whose existence can be elegantly discerned entirely algebraically, has fueled this endeavor anew. However, latent symmetries become obvious only in a suitably chosen effective Hamiltonian. In this Letter, we drastically advance the understanding of latent symmetries by establishing a correspondence between latent and geometric symmetries. We show that any system with latent symmetry can be similarity transformed into an equivalent system with explicit geometric symmetry and that the converse holds under mild conditions. In other words, latent symmetries are geometric symmetries in disguise. On a practical level, this latent-geometry correspondence implies that phenomena associated with spatial symmetries can be further engineered in latently symmetric systems. We demonstrate this by experimentally realizing higher-order topological insulators (HOTI) protected by latent symmetry groups. Our Letter provides a unified perspective on latent and geometric symmetry, and it paves the way for utilizing latent symmetries in designing and engineering exotic matter in classical waves, solid-state materials, and quantum systems.