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

Abstract

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 $\phi$ and a unique $\phi \cdot \phi$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If ${\Delta }_{\phi }=(d-2+\eta )/2$ is the dimension of the order-parameter field $\phi$, and ${\Delta }_{\epsilon }=d-1/\nu$ is the RG dimension of the energy operator $\epsilon$, which can be identified with $[\phi \cdot \phi ]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality ${\Delta }_{\epsilon }\ge 2{\Delta }_{\phi }$, 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 ${\Phi }^4$ 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.

15 pages

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