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) ${\mathrm{\Phi }}^4$ theories with a multicomponent scalar field $\mathit{\phi }$ and a unique $\mathit{\phi }·\mathit{\phi }$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If ${\mathrm{\Delta }}_{\phi }=(d-2+\eta )/2$ is the dimension of the order-parameter field $\mathit{\phi }$, and ${\mathrm{\Delta }}_{\varepsilon }=d-1/\nu$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[\mathit{\phi }·\mathit{\phi }]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality ${\mathrm{\Delta }}_{\varepsilon }\ge 2{\mathrm{\Delta }}_{\phi }$, 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 ${\mathrm{\Phi }}^4$ theories close to four dimensions, by exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW ${\mathrm{\Phi }}^4$ 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.

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• Conformal field theory
• Critical phenomena
• Phase transitions
• Renormalization group
• Statistical field theory
• Lattice models in statistical physics
• Landau-Ginzburg-Wilson theory