Universal Design and Physical Applications of Non-Uniform Cellular Automata on Translationally Invariant Lattices

Abstract: Lattice geometry profoundly shapes physical phenomena such as subsystem symmetry and directed percolation (DP). Among various lattice geometries, hyperbolic lattices are characterized by constant negative curvature and non-Abelian translation symmetry, offering a rich platform for investigating this geometry-physics interplay. However, the exponentially growing lattice size and nontrivial translation symmetry make approaches developed for Euclidean lattices incompatible, a limitation particularly evident in uniform cellular automata (CA). To resolve this, we develop a higher-order non-uniform cellular automata (NUCA) algorithm applicable to both translationally invariant regular Euclidean and hyperbolic lattices. In the algorithm, the non-uniform update rules incorporate nontrivial geometric data through a lattice-deforming procedure. We demonstrate the broad applicability of our algorithm to hyperbolic lattices through several applications on the hyperbolic ${5,4}$ lattice. By applying a linear NUCA, we generate subsystem symmetry-protected topological (SSPT) states and spontaneous subsystem symmetry-breaking states associated with regular or irregular subsystem symmetries unattainable on Euclidean lattices. We design the multi-point strange correlators to detect nontrivial SSPT states and derive a sufficient condition for non-Abelian translationally invariant NUCA-generated models. Furthermore, by generalizing the NUCA to non-uniform Clifford quantum cellular automata (CQCA), we generate subsystem symmetries of the hyperbolic cluster state, extending the established correspondence between translationally invariant CQCA and subsystem symmetries. Moreover, we simulate the DP process via a probabilistic NUCA that inherits the treelike structure of the lattice, and numerically estimate percolation thresholds and the phase diagram.