From coupled wires to coupled layers: Model with three-dimensional fractional excitations

We propose a systematic approach to constructing microscopic models with fractional excitations in three-dimensional (3D) space. Building blocks are quantum wires described by the ($1+1$)-dimensional conformal field theory (CFT) associated with a current algebra $\mathfrak{g}$. The wires are coupled with each other to form a 3D network through the current-current interactions of ${\mathfrak{g}}₁$ and ${\mathfrak{g}}₂$ CFTs that are related to the $\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g}\ensuremath{\supset}{\mathfrak{g}}₁\ifmmode\times\else\texttimes\fi{}{\mathfrak{g}}₂$. The resulting model can be viewed as a layer construction of a 3D topologically ordered state, in which the conformal embedding in each wire implements the anyon condensation between adjacent layers. Local operators acting on the ground state create pointlike or looplike deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU{(2)}₁\ifmmode\times\else\texttimes\fi{}SU{(2)}₁\ensuremath{\supset}U{(1)}₄\ifmmode\times\else\texttimes\fi{}U{(1)}₄$. We show that the model possesses extensively degenerate ground states on a torus with deconfined quasiparticles, and that appropriate local perturbations lift the degeneracy and yield a 3D ${Z}₂$ gauge theory with a fermionic ${Z}₂$ charge.