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

TL;DR

The paper analyzes a $(3+1)$-D $U(N)$ gauge-Higgs theory with $N$-flavor scalars carrying $U(1)$ charge $Nk+1$ and demonstrates that the color-flavor locked (CFL) phase hosts topological order. Using an Abelian duality, the low-energy dynamics are captured by a BF theory with a matrix coupling, revealing a spontaneously broken $\mathbb{Z}_{Nk+1}$ one-form symmetry and a dual two-form symmetry, with Wilson loops and vortex surface operators exhibiting nontrivial linking braiding and ground-state degeneracy dependent on spatial topology. The introduction of a theta term deforms correlation functions and indicates a global inconsistency-style anomaly that extends the theta periodicity to $2\pi(Nk+1)^2$, signaling robust topological structure. Compared with the CFL phase in $SU(N)$ theories, this $U(N)$ setup relies on gauging the $U(1)$ factor and larger $Nk+1$ charge to realize non-Abelian vortex-based topological order, paving the way for further exploration of vortex moduli and non-Abelian dualities in broader gauge groups.

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.

Figures (3)

• Figure 1: Graphical representation of $W({\cal C})| \Omega \rangle =| \Omega\rangle$ on $S^{3}$.
• Figure 2: Graphical representation of Eq. \ref{['eq:intersection']}:$V({\cal S}) W({\cal C}) V^{-1}({\cal S}) = e^{i\phi} W({\cal C})$.
• Figure 3: World sheet of a vortex (black line) and a world line of a particle (red line).