The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

Abstract

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε&#39;=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$, whereas SR percolation has $η_{\rm SR}<0$ near $d=6$. Sak&#39;s matching condition then places the crossover above $σ=2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$, we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$, where $δ= 2-σ$. We derive the one-loop corrections to the critical exponents $η$ and $ν$, which acquire nontrivial dependence on $ε$ and $δ$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$. These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$-state Potts universality classes with $q<2$.