One—Scale-Factor Universality for Critical Phenomena

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of $V_X=\frac{n_X{q}_X{K_C}^X}{g_X{l}_X}$, where $q_X$ is the coordination number, ${K_C}^X$ is the dimensionless inverse critical temperature, and $n_X$, $g_X$, and $l_X$ are the universality scale factors. $V_X$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{V_X}{l_X}$ is an exact invariant.