Conjecture on the lower bound of the length-scale critical exponent ν 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 $ν$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $φ$ and a unique $φ\cdot φ$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $Δ_φ=(d-2+η)/2$ is the dimension of the order-parameter field $φ$, and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[φ\cdot φ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon \ge 2 Δ_φ$, which implies $ν\ge (2-η)^{-1}$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $ε$-expansion results for generic LGW $Φ^4$ 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 $Φ^4$ theories. In particular, since unitarity requires $η\ge 0$, the above inequality implies $ν\ge 1/2$ for unitary theories. This lower bound is more restrictive than $ν> 1/d$, derived by noting that $ν=1/d$ characterizes the singular finite-size behavior at first-order transitions.