Numerical evidence for a CP broken deconfined phase at $\theta =\pi$ in 4D SU(2) Yang-Mills theory through simulations at imaginary $\theta$

TL;DR

This paper provides numerical evidence that CP symmetry is spontaneously broken at low temperatures and restored at high temperatures in 4D SU(2) Yang-Mills theory at $ heta=\pi$, using simulations at imaginary $ heta$ to circumvent the sign problem.

Contribution

The study demonstrates CP symmetry breaking and restoration at different temperatures in 4D SU(2) Yang-Mills theory through novel numerical simulations at imaginary $ heta$.

Findings

CP symmetry is spontaneously broken at low temperature.

CP symmetry is restored above the deconfining temperature.

Critical temperatures for CP restoration and deconfinement are identified.

Abstract

We investigate the possibility of the spontaneous breaking of CP symmetry in 4D SU(2) Yang-Mills at $\theta=\pi$, which has recently attracted much attention in the context of the higher-form symmetry and the 't Hooft anomaly matching condition. Here we provide a numerical evidence that the CP symmetry is indeed spontaneously broken at low temperature and it gets restored above the deconfining temperature at $\theta=\pi$, which is consistent with the anomaly matching condition and yet differs from the situation predicted in the large-$N$ limit. We avoid the severe sign problem by performing simulations at imaginary $\theta$. We obtain the critical temperature of the CP restoration and that of deconfinement at $\theta=\pi$ by analytic continuation, which leads to the above conclusion.