Odd fracton theories, proximate orders, and parton constructions
The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy nontrivial conditions on their low-energy properties when a combination of lattice translation and U(1) symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other subdimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that “odd” versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd Z 2 gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry breaking, thereby allowing us to identify a class of conventionally ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. In conclusion, condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions.