Peeking into the θ vacuum

Abstract

We propose a subvolume method to study the θ dependence of the free energy density of the four-dimensional SU(N) Yang-Mills theory on the lattice. As an attempt, the method is first applied to SU(2) Yang-Mills theory at T=1.2Tc to understand the systematics of the method. We then proceed to the calculation of the vacuum energy density and obtain the θ dependence qualitatively different from the high temperature case. The numerical results combined with the theoretical requirements provide the evidence for the spontaneous CP violation at θ=π, which is in accordance with the large N prediction and indicates that the similarity between 4d SU(N) and 2d CPN−1 theories does not hold for N=2.