Quantum distillation in QCD

We propose a grading protocol which assigns global symmetry associated phases to states in the Hilbert space. Without modifying the Hilbert space, this changes the state sum, a process that we call quantum distillation. We describe the image of quantum distillation in path integral in terms of (non-dynamical) flavor holonomy dependence of (dynamical) gauge holonomy potentials, in QCD with $N_f=N_c$ fundamental and one massive adjoint fermion on $\mathbb R^3 \times S^1$. The compactified theory possesses an exact zero-form color-flavor center symmetry for a special choice of flavor holonomy (under which Polyakov loop is charged), despite the absence of one-form center-symmetry. We prove that the CFC symmetry is stable at small-$\beta$. This is the opposite of the high-temperature limit of thermal theory and a dramatic manifestation of quantum distillation. We show chiral symmetry breaking at small $S^1$ and that the vacuum structure of the theory on $\mathbb R^4$ and $\mathbb R^3 \times S^1$ are controlled by the same mixed 't Hooft anomaly condition.