A simple anisotropic three-dimensional quantum spin liquid with fracton topological order

TL;DR

This paper introduces a new three-dimensional spin model exhibiting fracton topological order with unique immobile excitations, subextensive degeneracy, and boundary phenomena, expanding understanding of quantum spin liquids.

Contribution

The authors present a novel anisotropic 3D spin model with fracton topological order, demonstrating properties distinct from layered toric codes and analyzing boundary effects.

Findings

Exhibits fracton topological order with immobile pointlike excitations.

Displays a subextensive ground state degeneracy on a 3-torus.

Shows zero energy surface modes on certain boundaries.

Abstract

We present a three-dimensional cubic lattice spin model, anisotropic in the $\hat{z}$ direction, that exhibits fracton topological order. The latter is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground state degeneracy: On an $L_x\times L_y\times L_z$ three-torus, it has a $2^{2L_z}$ topological degeneracy, and an additional non-topological degeneracy equal to $2^{L_xL_y-2}$. The fractons can be combined into composite excitations that move either in a straight line along the $\hat{z}$ direction, or freely in the $xy$ plane at a given height $z$. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either the $\hat{x}$ or $\hat{y}$ directions, and their absence on the surfaces normal to $\hat{z}$. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.