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.
Highlights
Order or disorder is a basic concept to classify classical and quantum phases of matter in modern physics
This theory has a ZNkþ1 one-form symmetry, and it is spontaneously broken in the color-flavor locked (CFL) phase, i.e., the CFL phase is topologically ordered if k ≠ 0
We have studied a UðNÞ gauge theory with N-flavor scalar fields whose Uð1Þ charge is Nk þ 1 (k ∈ Z)
Summary
Order or disorder is a basic concept to classify classical and quantum phases of matter in modern physics. It turned out that the CFL phase of QCD is not topologically ordered This is because the emergent discrete twoform symmetry is unbroken due to the interaction between vortices and massless Nambu-Goldstone bosons. We discuss a possible appearance of topological order in quantum field theories that have non-Abelian vortices. Unlike the CFL phase of QCD [47], we find that the CFL phase in the UðNÞ gauge theories is topologically ordered if k ≠ 0, while the previously considered UðNÞ theories with k 1⁄4 0 [30,31,32,33] are not This is because the system has a ZNkþ one-form symmetry, and it is spontaneously broken in the Higgs phase, which means that this phase has a topological order. We here review the appearance of topological order in the low-energy effective theory of the Abelian Higgs model in (3 þ 1) dimensions [6]. We show that there is an emergent Zk two-form symmetry in addition to the Zk one-form symmetry in the original action, and both of the Zk symmetries are spontaneously broken, by calculating correlation functions of Wilson loops and surface operators [48,49,50,51]
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