Beyond the Freshman's Dream: Classical fractal spin liquids from matrix cellular automata in three-dimensional lattice models

Abstract

We construct models hosting classical fractal spin liquids on two realistic three-dimensional (3D) lattices of corner-sharing triangles: trillium and hyperhyperkagome (HHK). Both models involve the same form of three-spin Ising interactions on triangular plaquettes as the Newman-Moore (NM) model on the 2D triangular lattice. However, in contrast to the NM model and its 3D generalizations, their degenerate ground states and low-lying excitations cannot be described in terms of scalar cellular automata (CA), because the corresponding fractal structures lack a simplifying algebraic property, often termed the 'Freshman's dream'. By identifying a link to matrix CAs -- that makes essential use of the crystallographic structure -- we show that both models exhibit fractal symmetries of a distinct class to the NM-type models. We devise a procedure to explicitly construct low-energy excitations consisting of finite sets of immobile defects or "fractons", by flipping arbitrarily large self-similar subsets of spins, whose fractal dimensions we compute analytically. We show that these excitations are associated with energetic barriers which increase logarithmically with system size, leading to "fragile" glassy dynamics, whose existence we confirm via classical Monte Carlo simulations. We also discuss consequences for spontaneous fractal symmetry breaking when quantum fluctuations are introduced by a transverse magnetic field, and propose multi-spin correlation function diagnostics for such transitions. Our findings suggest that matrix CAs may provide a fruitful route to identifying fractal symmetries and fracton-like behaviour in lattice models, with possible implications for the study of fracton topological order.

24 pages, 14 figures

References in corpus (6)

• Glassy dynamics of kinetically constrained models
• Local stabilizer codes in three dimensions without string logical operators
• Fate of partial order on the trillium and distorted windmill lattices
• Construction of Fractal Order and Phase Transition with Rydberg Atoms
• Theoretical study of quantum spin liquids in $S=1/2$ hyper-hyperkagome magnets: classification, heat capacity, and dynamical spin structure factor
• The influence of dimension on the relaxation process of East-like models

Cited by in corpus (6)

• A Classical Chiral Spin-Liquid from Chiral Interactions on the Pyrochlore Lattice
• Multipole groups and fracton phenomena on arbitrary crystalline lattices
• Boundary conditions dependence of the phase transition in the quantum Newman-Moore model
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Frustration and chirality in three-dimensional trillium lattices: Insights and Perspectives
• Multistage development of short-range spin correlations and weak magnetic order in the two coupled trillium lattices of K2Fe2(MoO4)(PO4)2