Peeking into the $\theta$ vacuum

Abstract

We propose a subvolume method to study the $\theta$ 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 $T=1.2T_c$ to understand the systematics of the method. We then proceed to the calculation of the vacuum energy density and obtain the $\theta$ 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 $\theta =\pi$, which is in accordance with the large $N$ prediction and indicates that the similarity between 4d SU($N$) and 2d CP${}^{N-1}$ theories does not hold for $N$=2.

arXiv comment6 pages, 8 figures, text clarified, references added, systematic errors estimated, argument on surface tension omitted, Fig.7 added to clarify the limitation of the method, no significant change in conclusion. published version

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• Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory at $\theta =\pi$ from imaginary $\theta$ simulations
• The $\theta$-angle and axion physics of two-color QCD at fixed baryon charge. Part I
• Generalized symmetry constraints on deformed 4d (S)CFTs
• The mixed $0$-form/$1$-form anomaly in Hilbert space: pouring new wine into old bottles
• First-order CP phase transition in two-flavor QCD at $\theta =\pi$ under electromagnetic scale anomaly via a Nambu-Jona-Lasinio description
• Grassmann tensor renormalization group for the massive Schwinger model with a $\theta$ term using staggered fermions
• Large $\theta$ angle in two-dimensional large $N$ ${\mathbb{CP}}^{N-1}$ model
• Generalized symmetry enriched criticality in (3+1)d