LAUSR

Significance Recent experiments in circuit quantum electrodynamics and electric circuit networks have demonstrated the coherent propagation of wave-like excitations on hyperbolic lattices. The negative curvature of space that underlies such lattices invalidates the familiar Bloch theorem of ordinary solid-state physics. In this work, we prove rigorous Bloch theorems for hyperbolic lattices through the identification of appropriate periodic boundary conditions. Unlike the ordinary Bloch theorem for crystalline lattices, our hyperbolic Bloch theorem is generally nonabelian in nature and involves infinitely many Brillouin zones for a single lattice. Our work initiates a chapter in band theory and establishes deep connections between condensed matter physics and algebraic geometry.