Coupled Ising fixed points near four dimensions

The quartic interaction $g_1{\varphi }_1^4+g_2{\varphi }_2^4+\lambda {\varphi }_1^2{\varphi }_2^2$ includes the Gaussian fixed point, single and double Ising model Wilson-Fisher fixed points, and an O(N) model fixed point with $N=2$. A rotation of two decoupled identical Ising models produces a second coordinate representation of their renormalization-group fixed point.

The orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ takes $g({\varphi }_1^4+{\varphi }_2^4+6{\varphi }_1^2{\varphi }_2^2)$ to $2g({\psi }_{+}^4+{\psi }_{-}^4)$. These are the same two decoupled Ising models expressed in rotated fields.