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

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 evidence that the CP symmetry, which is spontaneously broken at low temperature, gets restored 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.