Fcc lattice, checkerboards, fractons, and quantum field theory

We consider XY-spin degrees of freedom on an fcc lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding U(1) gauge theory on the fcc lattice. Surprisingly, this U(1) gauge theory is dual to the original spin system. We also analyze a similar ZN gauge theory on that lattice. All these systems are fractonic. The U(1) theories are gapless and the ZN theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics. Our Z2 fcc gauge theory is the continuum limit of the well-known checkerboard model of fractons. Our continuum analysis leads to a straightforward proof of the known fact that this theory is dual to two copies of the Z2 X-cube model. We find new models and new relations between known models. The ZN fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic Z2 gauge theory. Also, although for N=2 this model is dual to two copies of the Z2 X-cube model, a similar statement is not true for higher N.