Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory at $θ=π$ from imaginary $θ$ simulations

Abstract

The spontaneous breaking of CP symmetry in 4D SU($N$) pure Yang-Mills theory at $θ=π$ has recently attracted much attention in the context of the higher-form symmetry and the 't Hooft anomaly matching condition. Here we use Monte Carlo simulations to study the $N=2$ case, which is interesting since it is the case opposite to the large-$N$ limit, where explicit calculations are available. In order to circumvent the severe sign problem due to the $θ$ term for real $θ$, we first obtain results at imaginary $θ$, where the sign problem is absent, and make an analytic continuation to real $θ$. We use the stout smearing in defining the $θ$ term in the action to be used in our simulations. Thus we obtain the expectation value of the topological charge and the deconfining temperature at $θ=π$, and provide an evidence that the CP symmetry, which is spontaneously broken at low temperature, gets restored \emph{strictly above} the deconfining temperature. This conclusion is consistent with the anomaly matching condition and yet differs from the prediction in the large-$N$ limit.