Fracton and topological order in the XY checkerboard toric code

Abstract

We introduce the XY checkerboard toric code. It represents a generalization of the $\mathbb{Z}_2$ toric code with two types of star operators with $x$ and $y$ flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of an sub-extensive number of $\mathbb{Z}_2$ conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with $\mathbb{Z}_2$ topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase like sub-extensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.