Critical Exponents for Long-Range $${O(n)}$$ O ( n ) Models Below the Upper Critical Dimension

Abstract

We consider the critical behaviour of long-range $O(n)$ models ($n \ge 0$) on ${\mathbb Z}^d$, with interaction that decays with distance $r$ as $r^{-(d+\alpha)}$, for $\alpha \in (0,2)$. For $n \ge 1$, we study the $n$-component $|\varphi|^4$ lattice spin model. For $n =0$, we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension $d_c=2\alpha$. For dimensions $d=1,2,3$ and small $\epsilon>0$, we choose $\alpha = \frac 12 (d+\epsilon)$, so that $d=d_c-\epsilon$ is below the upper critical dimension. For small $\epsilon$ and weak coupling, to order $\epsilon$ we prove existence of and compute the values of the critical exponent $\gamma$ for the susceptibility (for $n \ge 0$) and the critical exponent $\alpha_H$ for the specific heat (for $n \ge 1$). For the susceptibility, $\gamma = 1 + \frac{n+2}{n+8} \frac \epsilon\alpha + O(\epsilon^2)$, and a similar result is proved for the specific heat. Expansion in $\epsilon$ for such long-range models was first carried out in the physics literature in 1972. Our proof adapts and applies a rigorous renormalisation group method developed in previous papers with Bauerschmidt and Brydges for the nearest-neighbour models in the critical dimension $d=4$, and is based on the construction of a non-Gaussian renormalisation group fixed point. Some aspects of the method simplify below the upper critical dimension, while some require different treatment, and new ideas and techniques with potential future application are introduced.