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We demonstrate a general gauging procedure of a pure matter theory on a lattice with a mixture of subsystem and global symmetries. This mixed symmetry can be either a semidirect product of a subsystem symmetry and a global symmetry, or a non-trivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G=mathbb{Z}_3^{text{sub}} rtimes mathbb{Z}_2^{text{glo}}$, $G=(mathbb{Z}_2^{text{sub}} times mathbb{Z}_2^{text{sub}}) rtimes mathbb{Z}_2^{text{glo}}$, $1to mathbb {Z}_2^text {sub}to Gto mathbb {Z}_2^text {glo}to 1$, and $1to mathbb {Z}_2^text {sub}to Gto K_4^text {glo}to 1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction of the gauging procedure provides an identification of the electric charges of these fracton orders with irreducible representations of the symmetry. Furthermore, by constraining the local Hilbert space, the magnetic fluxes with different geometry (tube-like and plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This algebraic structure leads to an identification of the magnetic fluxes to the conjugacy classes of the symmetry.
Based on several previous examples, we summarize explicitly the general procedure to gauge models with subsystem symmetries, which are symmetries with generators that have support within a sub-manifold of the system. The gauging process can be applie
We discuss the procedure for gauging on-site $mathbb{Z}_2$ global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, a choice of an Ab
We consider a 2D quantum spin model with ring-exchange interaction that has subsystem symmetries associated to conserved magnetization along rows and columns of a square lattice, which implies the conservation of the global dipole moment. In a certai
In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization group transformation for the model that utilizes toric code bilayers as an entanglement resource, and show how to exte