长程李-杨与渗流临界性的$6-ε$展开
The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality
AI总结：
本文在$d=6-ε$维度下构建长程$φ^3$场论的微扰展开，通过单圈RG分析确定渗流临界性的交叉阈值$σ_*=2$，消除长程与短程临界指数$η$的表观不连续性，还得到李-杨及$q<2$的Potts普适类的相关指数。
AI中文摘要：
渗流中从长程（LR）到短程（SR）临界性的交叉问题一直悬而未决，因为此前在$ε'=3σ-d$展开内的重整化群（RG）分析将反常维度固定为$η=2-σ$，而短程渗流在$d=6$附近的$η_{\rm SR}<0$。随后Sak的匹配条件将交叉点置于$σ=2$以上，超出了长程相互作用占主导的区域。在空间维度$d=6-ε$中，我们为长程$φ^3$场论构建了微扰展开，并在微扰可及的非经典区域$0<δ<ε/3$中进行了单圈RG分析，其中$δ=2-σ$。我们推导了临界指数$η$和$ν$的单圈修正，这些修正呈现出对$ε$和$δ$的非平凡依赖关系；它们在长程上临界线处约化到平均场值，并在$σ→2$时连续恢复短程$6-ε$结果。这些结果支持交叉阈值$σ_*=2$，并消除了该框架内长程与短程$η$值之间的表观不连续性。相同方法还得到了长程李-杨普适类及$q<2$的$q$态Potts普适类的反常指数和边界指数。
英文摘要：
The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$, whereas SR percolation has $η_{\rm SR}<0$ near $d=6$. Sak'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 $q<2$.
