  Fracton order is a new kind of quantum order characterized by topological
excitations that exhibit remarkable mobility restrictions and a robust ground
state degeneracy (GSD) which can increase exponentially with system size. In
this paper, 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+1)-dimensional toric code. However, the phase is
sensitive to more than just lattice curvature; different lattices without
curvature (e.g. cubic or stacked kagome lattices) also result in different
phases of matter, which are separated by phase transitions. Unintuitively
however, according to a previous definition of phase [Chen, Gu, Wen 2010], even
just a rotated or rescaled cubic lattice results in different phases of matter,
which motivates us to propose a new and coarser definition of phase for gapped
ground states and fracton order. The new equivalence relation between ground
states is given by the composition of a local unitary transformation and a
quasi-isometry (which can rotate and rescale the lattice); equivalently, ground
states are in the same phase if they can be adiabatically connected by varying
both the Hamiltonian and the positions of the degrees of freedom (via a
quasi-isometry). In light of the importance of geometry, we further propose
that fracton orders should be regarded as a geometric order.
