Fractal Symmetric Phases of Matter

TL;DR

This paper explores novel fractal symmetry protected topological phases in spin systems, revealing unique edge degeneracies and fracton excitations, and extends these concepts to three-dimensional systems with higher form symmetries.

Contribution

It introduces new fractal symmetry protected topological phases using a decorated defect approach and extends the framework to 3D systems with higher form fractal symmetries.

Findings

Identification of fractal symmetry protected topological phases

Edge degeneracies with projective fractal symmetries

Existence of symmetry protected fractons and fracton topologically ordered phases

Abstract

We study spin systems which exhibit symmetries that act on a fractal subset of sites, with fractal structures generated by linear cellular automata. In addition to the trivial symmetric paramagnet and spontaneously symmetry broken phases, we construct additional fractal symmetry protected topological (FSPT) phases via a decorated defect approach. Such phases have edges along which fractal symmetries are realized projectively, leading to a symmetry protected degeneracy along the edge. Isolated excitations above the ground state are symmetry protected fractons, which cannot be moved without breaking the symmetry. In 3D, our construction leads additionally to FSPT phases protected by higher form fractal symmetries and fracton topologically ordered phases enriched by the additional fractal symmetries.