Spin models from nonlinear cellular automata

We study classical and quantum spin models derived from one-dimensional cellular automata (CA) with nonlinear update rules, focusing on rules 30, 54 and 201. We argue that the classical models, defined such that their ground states correspond to allowed trajectories of the CA, are frustrated and can be described in terms of local defect variables. Including quantum fluctuations through the addition of a transverse field, we study their ground state phase diagram and quantum phase transitions. We show that the nonlinearity of the CA rule leads to a quantum order-by-disorder mechanism, which selects a particular (rule-dependent) spatial structure for small transverse fields, with spontaneous breaking of the translation symmetry in some cases. Using numerical results for larger fields, we also observe a first-order quantum phase transition into a quantum paramagnet, as in previous studies of spin models based on linear CA rules.