FCC, Checkerboards, Fractons, and QFT

TL;DR

This work constructs a unified continuum field theory framework for fracton phases on a three-dimensional FCC lattice by starting from an XY-tetrahedral lattice model and gauging its subsystem momentum symmetry to obtain a $U(1)$ gauge theory, then showing a dual formulation in terms of a three-field $oldsymbol{ m extphi}$-theory. It extends this to a $oldsymbol{ m Z_N}$ gauge theory and demonstrates a BF-type duality structure, with explicit continuum Lagrangians and a detailed analysis of fluxes, global symmetries, and mobility-restricted defects such as fractons and lineons. The paper also establishes connections to anisotropic theories, relates the Z_N checkerboard model to the X-cube family for $N=2$, and explains the breakdown of this equivalence for $N>2$ through fusion rules and symmetry considerations. Overall, it provides a comprehensive continuum-field-theory perspective that unifies several fracton models, computes ground-state degeneracies, and reveals new dualities and relationships among checkerboard, X-cube, and anisotropic constructions, with potential implications for understanding long-distance fracton dynamics and topological quantum order.

Abstract

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 $\mathbb{Z}_N$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ 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 $N$.

Figures (3)

• 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 \ref{['H']}.
• Figure 2: The lattice models in Section \ref{['sec:XYtetra']}, Section \ref{['sec:Alattice']}, and Section \ref{['sec:ZNlattice']} are formulated on an FCC lattice (see Figure (a)). Equivalently, we can formulate it on the checkerboard of a cubic lattice (see Figure (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 Section \ref{['sec:XYtetra']}, 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 Figure (c) of the checkerboard. For the $U(1)$ and $\mathbb{Z}_N$ FCC gauge theories of Section \ref{['sec:Alattice']} and Section \ref{['sec:ZNlattice']}, 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 \ref{['checkerboardH']} of the $\mathbb Z_N$ checkerboard model. The cube shown here is any shaded cube of the checkerboard in Figure \ref{['fig:U1-CB']}(b).