A conjecture on the lower bound of the length-scale critical exponent $\nu$ at continuous phase transitions

A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $\nu$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $\Phi^4$ theories with a multicomponent scalar field ${\varphi}$ and a unique ${\varphi}\cdot {\varphi}$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $\Delta_\varphi=(d-2+\eta)/2$ is the dimension of the order-parameter field ${\varphi}$, and $\Delta_\varepsilon=d-1/\nu$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[{\varphi}\cdot {\varphi}]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $\Delta_\varepsilon \ge 2 \Delta_\varphi$, which implies $\nu \ge (2-\eta)^{-1}$ and $\gamma = (2-\eta)\nu\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $\epsilon$-expansion results for generic LGW $\Phi^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $\Phi^4$ theories. In particular, since unitarity requires $\eta\ge 0$, the above inequality implies $\nu\ge 1/2$ for unitary theories. This lower bound is more restrictive than $\nu > 1/d$, derived by noting that $\nu=1/d$ characterizes the singular finite-size behavior at first-order transitions.