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

Konstantinos Sfairopoulos*, Luke Causer, Jamie F. Mair, and Juan P. Garrahan

  • School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom

  • *Contact author: ksfairopoulos@gmail.com
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Phys. Rev. E 113, 064112 – Published 3 June, 2026

DOI: https://doi.org/10.1103/n9l3-gfyt

Abstract

We show how the trajectories of d-dimensional cellular automata (CA) can be used to systematically construct the ground states of (d+1)-dimensional classical spin models, and we characterize 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 counterpropagating 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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Phys. Rev. E 113, 064112– Published 3 June, 2026
  • Received 21 April 2025
  • Accepted 13 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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