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 $U\left(1\right)$ gauge theory on the fcc lattice. Surprisingly, this $U\left(1\right)$ gauge theory is dual to the original spin system. We also analyze a similar ${\mathbb{Z}}_N$ gauge theory on that lattice. All these systems are fractonic. The $U\left(1\right)$ theories are gapless and the ${\mathbb{Z}}_N$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics. Our ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2$ X-cube model. We find new models and new relations between known models. The ${\mathbb{Z}}_N$ fcc gauge theory can be realized by coupling three copies of an anisotropic model of lineons and planons to a certain exotic ${\mathbb{Z}}_2$ gauge theory. Also, although for $N=2$ this model is dual to two copies of the ${\mathbb{Z}}_2$ X-cube model, a similar statement is not true for higher $N$.

Figure 1

(a) The fcc lattice on which the XY tetrahedral model is defined. The black sites are the corners, while the red, green, and blue sites are at the face centers on the $yz, zx$, and $xy$ planes, respectively. The distance between two nearest-neighbor sites is $a/\sqrt{2}$. Each site participates in eight tetrahedra; or equivalently, in each cube, there are eight tetrahedra. (b) The tetrahedron of the interaction term in the second line of (2.2).

Figure 2

The lattice models in Secs. 2, 4, and 7a are formulated on an fcc lattice [see panel (a)]. Equivalently, we can formulate it on the checkerboard of a cubic lattice [see panel (b)]. The sites and the tetrahedra of the fcc lattice are mapped to the shaded cubes and sites of the checkerboard, respectively. The distance between two nearest-neighbor sites in the fcc lattice is $a/\sqrt{2}$, while it is $a/2$ on the checkerboard. For the XY-tetrahedral model of Sec. 2, the phase variables and their conjugate momenta are placed on the sites of the fcc lattice, or equivalently the shaded cubes of the checkerboard. Their interactions are associated with the tetrahedra of the fcc lattice, or equivalently, the four shaded cubes that share the same site in panel (c) of the checkerboard. For the $U\left(1\right)$ and ${\mathbb{Z}}_N$ fcc gauge theories of Secs. 4 and 7a, both the gauge parameters and the magnetic interactions are associated with the sites of the fcc lattice, or equivalently, the shaded cubes of the checkerboard. The gauge fields are placed on the tetrahedra, or equivalently, the sites of the checkerboard.

Figure 3

(a) The $G_c$ (Gauss law) term, and (b) the $L_c$ term in the Hamiltonian (7.11) of the ${\mathbb{Z}}_N$ checkerboard model. The cube shown here is any shaded cube of the checkerboard in Fig. 2.