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

• URL: http://arxiv.org/abs/2309.08059v1
• Date: Thu, 14 Sep 2023 23:03:14 GMT
• Title: Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions
• Authors: Konstantinos Sfairopoulos and Luke Causer and Jamie F. Mair and Juan P. Garrahan
• Abstract summary: We show how the trajectories of $d$-dimensional cellular automata (CA) can be used to determine the ground states of $(d+1)$-dimensional classical spin models.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show how the trajectories of $d$-dimensional cellular automata (CA) can be used to determine the ground states of $(d+1)$-dimensional classical spin models, and we characterise their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for $d>1$ through selected examples. We illustrate our general observations 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 the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite size scaling for their quantum phase transitions.

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