A conjecture on the lower bound of the length-scale critical exponent $ν$ 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 $ν$ that should hold at continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $Φ^4$ theories with a multicomponent scalar field $\varphi_i$ (including some extensions with fermionic and gauge fields). If $Δ_\varphi=(d-2+η)/2$ is the dimension of the order parameter -- $\varphi_i$ in LGW models -- and $Δ_\varepsilon=d-1/ν$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[\sum_i \varphi_i^2]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $Δ_\varepsilon - 2 Δ_\varphi\ge 0$, which implies $1/ν\le 2-η$ and $γ= (2-η)ν\ge 1$. These inequalities are supported by general arguments for lattice models, exact relations for two-dimensional minimal conformal field theories, and are consistent with all 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 the bound $ν> 1/d$, which is derived by noting that $ν=1/d$ is the expected behavior at first-order transitions.

Figures (2)

• Figure 1: RG flow and FPs of the O($M$)$\otimes$O($N$) model in the large-$N$ limit. Tere are four FPs: the unstable Gaussian (G) and O($N$) symmetric (H) FPs, the stable chiral (C) FP and the unstable antichiral (A) FP.
• Figure 2: The three-dimensional RG flow of the cubic model for $N=2$ (left) and $N\ge 3$ (right), because $N_c\approx 2.9$ in three dimensions Aharony-76CPV-00PV-02HV-11Chester-etal-21Hasenbusch-23Hasenbusch-24.