# Classifying local fractal subsystem symmetry protected topological phases

## TL;DR

This paper investigates 2D fractal subsystem symmetry-protected topological phases, revealing that only a limited set of phases are realizable by local Hamiltonians, with their existence depending on symmetry structure.

## Contribution

It provides a classification of possible SPT phases in 2D with fractal symmetries, highlighting conditions under which non-trivial phases can or cannot exist.

## Key findings

- Few SPT phases are realizable by local Hamiltonians.
- The existence of non-trivial phases depends on the symmetry's spatial structure.
- An exhaustive enumeration of realizable phases is provided.

## Abstract

We study symmetry-protected topological (SPT) phases of matter in 2D protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases no non-trivial SPT phases are possible at all. In cases where non-trivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.