Exotic $\mathbb{Z}_N$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory

TL;DR

The paper develops a universal continuum framework for fracton physics in 3+1D by constructing three dual $\mathbb{Z}_N$ tensor gauge theories (via $A$, $\hat{A}$, and a BF-type formulation) and showing their equivalence to the X-cube model. It analyzes lattice realizations, continuum Lagrangians, and a rich set of exotic global symmetries (tensor, dipole, and their $\mathbb{Z}_N$ manifestations), deriving exact dualities among the descriptions. Ground-state degeneracy is counted through winding modes, yielding $N^{2L^x+2L^y+2L^z-3}$ on a torus, consistent across lattice and continuum pictures. The results illuminate how discontinuous fields can be harnessed in non-Lorentzian QFT to robustly encode fracton phenomenology, and they connect toric-code–like topological orders to the X-cube fracton phase via IR dualities. Overall, the work provides a cohesive, dual description of a central fracton model and clarifies the role of symmetries, defects, and nonlocal operators in these 3+1D theories.

Abstract

Following our earlier analyses of nonstandard continuum quantum field theories, we study here gapped systems in 3+1 dimensions, which exhibit fractonic behavior. In particular, we present three dual field theory descriptions of the low-energy physics of the X-cube model. A key aspect of our constructions is the use of discontinuous fields in the continuum field theory. Spacetime is continuous, but the fields are not.

Figures (7)

• Figure 1: Relations between the four theories in paper2. The $U(1)$ tensor gauge theory of $A$ is dual to the non-gauge $\hat{\phi}$-theory, while the $U(1)$ tensor gauge theory of $\hat{A}$ is dual to the non-gauge $\phi$-theory. The $\phi$-theory is the Higgs field for the $U(1)$ gauge symmetry of $A$, while the $\hat{\phi}$-theory is the Higgs field for the $U(1)$ gauge symmetry of $\hat{A}$.
• Figure 2: (a) The term $L_{[xy]z},L_{[zx]y},L_{[yz]x}$ in the Hamiltonian, which are products of the $U$'s on the plaquettes. (b) Gauss law constraint $G$, which is a products of 12 $V$'s on the plaquettes. The shaded faces stand for $U$ and $V$ on the plaquette in (a) and (b), respectively. We suppress the orientation of these plaquette variables.
• Figure 3: The global symmetries of the $U(1)$$A$-theory, the $\phi$-theory, and the $\mathbb{Z}_N$ tensor gauge theory and their relations. The momentum dipole symmetry of the $\phi$-theory is gauged and therefore it is absent in the $\mathbb{Z}_N$ tensor gauge theory. The magnetic tensor symmetry of the $A$-theory is absent in the $\mathbb{Z}_N$ tensor gauge theory because of the constraint \ref{['EB0']}.
• Figure 4: (a) The term $\hat{L}$ in the Hamiltonian, which is a product of the $\hat{U}$'s of the 12 links around a cube. (b) The three Gauss law constraints $\hat{G}^{[xy]z}=\hat{G}^{[zx]y}=\hat{G}^{[yz]x}=1$, which are products of the $\hat{V}$'s on the links. The solid lines stand for $\hat{U}$ and $\hat{V}$ on the link in (a) and (b), respectively. We suppress the orientation of these link variables.
• Figure 5: The global symmetries of the $U(1)$$\hat{A}$-theory, the $\hat{\phi}$-theory, and the $\mathbb{Z}_N$ tensor gauge theory and their relations. The momentum tensor symmetry of the $\hat{\phi}$-theory is gauged and therefore it is absent in the $\mathbb{Z}_N$ tensor gauge theory. The magnetic dipole symmetry of the $\hat{A}$-theory is absent in the $\mathbb{Z}_N$ tensor gauge theory because of the constraint \ref{['hatEB0']}.