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

## TL;DR

This paper introduces a systematic method to construct 3D models with fractional excitations using coupled quantum wires and conformal field theory, leading to topologically ordered states with deconfined quasiparticles.

## Contribution

It presents a novel layer construction approach for 3D topological phases via conformal embeddings and anyon condensation in coupled wire models.

## Key findings

- Constructed a solvable 3D model with fractional excitations.
- Demonstrated ground state degeneracy and deconfined quasiparticles.
- Showed how local perturbations lead to a 3D $Z_2$ gauge theory with fermionic charge.

## Abstract

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 $\mathfrak{g} \supset \mathfrak{g}_1 \times \mathfrak{g}_2$. 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 point-like or loop-like deconfined excitations depending on the branching rule. We demonstrate our construction for a simple solvable model based on the conformal embedding $SU(2)_1 \times SU(2)_1 \supset U(1)_4 \times U(1)_4$. 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.