Spin models from nonlinear cellular automata

Abstract

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.

Figures (13)

• Figure 1: CA with nonlinear rules. Evolution from a single occupied site for (a) Rule 30 (chaotic, non-repeating pattern), (b) Rule 54 (periodic pattern) and (c) Rule 201 (note that this starts from a single occupied site and immediately flips to a nearly all-occupied configuration). Black squares denote occupied cells of the CA ($x=1$) or spin down in the spin model ($\sigma = -1$), while white squares are unoccupied ($x = 0$) or spin up ($\sigma = 1$).
• Figure 2: The ground states (or, equivalently, periodic CA trajectories) for Rule 201 for a $4 \times 4$ system size. As in Fig. \ref{['fig:Rule_cycles_nonlinear_rules']}, black and white squares are, respectively, down ($\sigma = -1$) and up ($\sigma = +1$) spins (or occupied and unoccupied cells of the CA). Ground states which are obtained by translations of these states are omitted.
• Figure 3: Similar to Fig. \ref{['fig:rule201_tiles']} for the ground states of the Rule 54 for a $4 \times 2$ system size.
• Figure 4: Similar to Fig. \ref{['fig:rule201_tiles']} for the ground states of the Rule 54 for a $4 \times 4$ system size.
• Figure 5: Similar to Fig. \ref{['fig:rule201_tiles']} for the ground states of the Rule 30 for a $4 \times 8$ system size.