# Topological order in a color-flavor locked phase of $(3+1)$-dimensional $U(N)$ gauge-Higgs system

## TL;DR

This paper investigates a 3+1 dimensional $U(N)$ gauge theory with scalar fields, revealing topological order in the color-flavor locked phase through the spontaneous breaking of a $bZ_{Nk+1}$ one-form symmetry and the role of vortices in this phenomenon.

## Contribution

It demonstrates the presence of topological order in the CFL phase of a $U(N)$ gauge-Higgs system and links vortex world sheets to the generation of one-form symmetry.

## Key findings

- CFL phase exhibits topological order when $k
eq 0$
- The theory has a $bZ_{Nk+1}$ one-form symmetry that is spontaneously broken
- Vortex world sheets generate the one-form symmetry

## Abstract

We study a $(3+1)$-dimensional $U(N)$ gauge theory with $N$-flavor fundamental scalar fields, whose color-flavor locked (CFL) phase has topologically stable non-Abelian vortices. The $U(1)$ charge of the scalar fields must be $Nk+1$ for some integer $k$ in order for them to be in the representation of $U(N)$ gauge group. This theory has a $\mathbb{Z}_{Nk+1}$ one-form symmetry, and it is spontaneously broken in the CFL phase, i.e., the CFL phase is topologically ordered if $k\not=0$. We also find that the world sheet of topologically stable vortices in CFL phase can generate this one-form symmetry.