X-Cube Model on Generic Lattices: New Phases and Geometric Order

Fracton order is a new kind of quantum order characterized by topological excitations which exhibit remarkable mobility restrictions and a robust ground state degeneracy (GSD) which can increase exponentially with system size. In this manuscript, we present a generic lattice construction (in three dimensions) for a generalized X-cube model of fracton order, where the mobility restrictions of the subdimensional particles inherit the geometry of the lattice. This helps explain a previous result that lattice curvature can produce a robust GSD, even on a manifold with trivial topology. We provide explicit examples to show that the (zero temperature) phase of matter is sensitive to the lattice geometry. In one example, the lattice geometry confines the dimension-1 particles to small loops, which allows the fractons to be fully mobile charges, and the resulting phase is equivalent to 3+1D toric code. However, the phase is sensitive to more than just lattice curvature; different lattices without curvature (e.g. cubic, stacked kagome, or even just a rotated cubic lattice) also result in different phases of matter, which are separated by phase transitions. (Models on different lattices can be compared by adding trivial gapped qubits so that both models share the same Hilbert space.) Thus, the long distance physics of the X-cube model (i.e. its phase, subdimensional particles, and GSD) is highly sensitive to lattice geometry. This greatly contrasts (liquid) topologically ordered models, such as toric code, which are blind to lattice geometry (and only sensitive to topology). We therefore propose that fracton orders should be regarded as a geometric order.