Systematic Constructions of Fracton Theories

Abstract

Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called ${\mathfrak{F}}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k-p)$ gauge parameters. The ${\mathbb{Z}}_2$ rank-two theory of type ${\mathfrak{F}}_2$, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the $\mathrm{U}(1)$ rank-two theory of type ${\mathfrak{F}}_2$ is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this ${\mathfrak{F}}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to ${\mathfrak{F}}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.

arXiv comment41 pages, 8 figures. v2: minor clarifications and references added. v3: expanded discussion of models on BCC lattices and with nonabelian symmetries, further minor clarifications and reference additions

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