Limit cycles in quantum theories

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.