Anomaly matching for phase diagram of massless $\mathbb{Z}_N$-QCD

We elucidate that the phase diagram of massless $N$-flavor QCD under $\mathbb{Z}_N$ flavor-twisted boundary condition (massless $\mathbb{Z}_N$-QCD) is constrained by an 't Hooft anomaly involving two-form gauge fields. As a consequence, massless $\mathbb{Z}_N$-QCD turns out to realize persistent order at any temperatures and quark chemical potentials, namely, the symmetric and gapped phase is strictly forbidden. This is the first result on the finite-$(T,\mu)$ phase diagram in QCD-type theories based on anomaly matching related to center and discrete axial symmetries.