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We show how the trajectories of $d$-dimensional cellular automata (CA) can beused to determine the ground states of $(d+1)$-dimensional classical spinmodels, and we characterise their quantum phase transition, when in thepresence of a transverse magnetic field. For each of the 256 one-dimensionalelementary CA we explicitly construct the simplest local two-dimensionalclassical spin model associated to the given CA, and we also describe thismethod for $d>1$ through selected examples. We illustrate our generalobservations with detailed studies of: (i) the $d=1$ CA Rule 150 and its $d=2$four-body plaquette spin model, (ii) the $d=2$ CA whose associated model is the$d=3$ square-pyramid plaquette model, and (iii) two counter-propagating $d=1$Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For thequantum spin models, we show that the connection to CAs implies a sensitivityon the approach to the thermodynamic limit via finite size scaling for theirquantum phase transitions.

Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions