Searching for Fracton Orders via Symmetry Defect Condensation

TL;DR

This paper develops a set of constraints to identify and classify fracton phases and subsystem symmetry-protected topological states across different dimensions, revealing new phases and enriching known models through defect condensation analysis.

Contribution

It introduces a generalized framework of constraint equations for fracton phases via symmetry defect condensation, connecting dualities with subsystem symmetries and solving them to find new and known phases.

Findings

Identifies gapped fracton phases in 3D including known and new models.

Recovers known fracton phases like Haah's code and introduces symmetry-enriched variants.

Finds new SSPT states in 2D and confirms cohomology solutions in 1D.

Abstract

We propose a set of constraints on the ground-state wavefunctions of fracton phases, which provide a possible generalization of the string-net equations used to characterize topological orders in two spatial dimensions. Our constraint equations arise by exploiting a duality between certain fracton orders and quantum phases with "subsystem" symmetries, which are defined as global symmetries on lower-dimensional manifolds, and then studying the distinct ways in which the defects of a subsystem symmetry group can be consistently condensed to produce a gapped, symmetric state. We numerically solve these constraint equations in certain tractable cases to obtain the following results: in $d=3$ spatial dimensions, the solutions to these equations yield gapped fracton phases that are distinct as conventional quantum phases, along with their dual subsystem symmetry-protected topological (SSPT) states. For an appropriate choice of subsystem symmetry group, we recover known fracton phases such as Haah's code, along with new, symmetry-enriched versions of these phases, such as non-stabilizer fracton models which are distinct from both the X-cube model and the checkerboard model in the presence of global time-reversal symmetry, as well as a variety of fracton phases enriched by spatial symmetries. In $d=2$ dimensions, we find solutions that describe new weak and strong SSPT states, such as ones with both line-like subsystem symmetries and global time-reversal symmetry. In $d=1$ dimension, we show that any group cohomology solution for a symmetry-protected topological state protected by a global symmetry, along with lattice translational symmetry necessarily satisfies our consistency conditions.