The 4-$\epsilon$ Expansion for Long-range Interacting Systems

The establishment of the Wilson-Fisher fixed point (WFP) for $O(n)$ spin models in $d=4-\epsilon$ dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as $\propto 1/r^{d+\sigma}$, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the $\epsilon$-expansion calculations up to the two-loop level. We show that, as long as $\sigma<2$, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with $d/2 < \sigma < 2$, the critical exponents, including the anomalous dimension, are functions of $\epsilon$, $\delta=2-\sigma$ and $n$, which reduce to the exact results in the limiting cases $\epsilon \to 0$, $\delta \to 0$ or $n \to \infty$. Our $(4-\epsilon)$-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at $\sigma_*=2$, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.