The Hasenfratz-Maggiore-Niedermayer exact mass gap for the two-dimensional O(4) lattice model

Theorem 1.1: for the periodic infinite-volume state, the gap of the full Osterwalder-Schrader transfer operator, including rotation-invariant sectors and measured in original lattice time steps, satisfies $\lim_{\beta\to\infty}e^{\pi\beta}\beta^{-1/2}m_{\mathrm{lat}}(\beta)=32\exp(\pi/4-1/2)$. The input companion gives $c\sqrt\beta e^{-\pi\beta}\le m_{\mathrm{lat}}\le C\sqrt\beta e^{-\pi\beta}$ and a positive full gap at every $\beta>0$. Not shown: the corresponding exact constant for $O(3)$ or general $O(n)$, a continuum $O(4)$ theory, or any statement about nonperiodic Gibbs states.

Theorem 1.1: for the periodic infinite-volume state, the gap of the full Osterwalder-Schrader transfer operator, including rotation-invariant sectors and measured in original lattice time steps, satisfies $\lim_{\beta\to\infty}e^{\pi\beta}\beta^{-1/2}m_{\mathrm{lat}}(\beta)=32\exp(\pi/4-1/2)$. The input companion gives $c\sqrt\beta e^{-\pi\beta}\le m_{\mathrm{lat}}\le C\sqrt\beta e^{-\pi\beta}$ and a positive full gap at every $\beta>0$. Not shown: the corresponding exact constant for $O(3)$ or general $O(n)$, a continuum $O(4)$ theory, or any statement about nonperiodic Gibbs states.