More Exotic Field Theories in 3+1 Dimensions

TL;DR

This work develops and analyzes 3+1D nonrelativistic continuum field theories tied to fracton physics, organizing them around exotic global and gauge symmetries, and identifying their lattice realizations. It introduces the XY-cube (φ) theory with momentum and winding quadrupole symmetries and a self-duality, then builds two families of tensor gauge theories for B, C (and their Z_N variants), highlighting fracton defects with restricted mobility and rich dualities among U(1) and Z_N sectors. The paper also connects these theories to established fracton models such as the X-Cube and the plaquette Ising model, via lattice constructions and continuum BF-type descriptions, demonstrating how global symmetries can constrain dynamics and stabilize gapped phases. Overall, the results provide a coherent framework for understanding nonstandard continuum descriptions of fracton physics, with explicit lattice realizations, flux quantization, and ground-state degeneracies that reveal deep connections between symmetry, topology, and mobility constraints in higher-dimensional systems.

Abstract

We continue the exploration of nonstandard continuum field theories related to fractons in 3+1 dimensions. Our theories exhibit exotic global and gauge symmetries, defects with restricted mobility, and interesting dualities. Depending on the model, the defects are the probe limits of either fractonic particles, strings, or strips. One of our models is the continuum limit of the plaquette Ising lattice model, which features an important role in the construction of the X-cube model.

Figures (3)

• Figure 1: The relation between the two $U(1)$ gauge theories $A,\hat{A}$ of paper2, the two non-gauge theories $\phi,\hat{\phi}$ of paper2, and the two $U(1)$ gauge theories $C,\hat{C}$. The solid lines mean that there is a $\mathbb{Z}_N$ gauge theory whose $BF$ Lagrangian uses these two fields. The double lines mean that the two theories are dual to each other. The arrows, for example, $\phi\rightarrow A$, mean that the former is the Higgs field of the latter. Each solid line and arrow gives a gapped $\mathbb{Z}_N$ theory, with certain equivalences. We use the same color for the same $\mathbb{Z}_N$ gauge theory. In total, there are three different $\mathbb{Z}_N$ gauge theories. $\mathbb{Z}_N$$A$ or $\hat{A}$-theory in paper3: $(\phi\rightarrow A )=( \hat{\phi} \rightarrow \hat{A} )= (A-\hat{A})$, $\mathbb{Z}_N$$C$-theory of Section \ref{['sec:ZNC']}: $(A\rightarrow C )= (\hat{\phi}-C)$, $\mathbb{Z}_N$$\hat{C}$-theory of Section \ref{['sec:ZNhatC']}: $(\hat{A}\rightarrow \hat{C}) = (\phi -\hat{C})$.
• Figure 2: The global symmetries of the $U(1)$$A$-theory, the $U(1)$$C$-theory, the $\mathbb{Z}_N$$C$-theory, and their relations. The electric tensor symmetry of the $A$-theory is gauged and therefore it is absent in the $\mathbb{Z}_N$ gauge theory. Note that the $U(1)$$C$-theory does not have a magnetic symmetry.
• Figure 3: The global symmetries of the $U(1)$$\hat{A}$-theory, the $U(1)$$\hat{C}$-theory, the $\mathbb{Z}_N$$\hat{C}$-theory, and their relations. The electric dipole symmetry of the $\hat{A}$-theory is gauged and therefore it is absent in the $\mathbb{Z}_N$ gauge theory. Note that the $U(1)$$\hat{C}$-theory does not have a magnetic symmetry.