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

TL;DR

This paper establishes a method linking $d$-dimensional cellular automata trajectories to ground states of $(d+1)$-dimensional classical spin models, analyzing their quantum phase transitions and illustrating with specific examples.

Contribution

It introduces a novel approach connecting cellular automata to classical spin models and explores their quantum phase transitions, including finite size scaling effects.

Findings

Explicit construction of classical spin models from elementary cellular automata

Analysis of quantum phase transitions in associated spin models

Demonstration of finite size scaling sensitivity in quantum critical behavior

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.