The crossover region between long-range and short-range interactions for the critical exponents

TL;DR

The paper addresses how critical exponents cross from long-range to short-range behavior as the interaction decay exponent $\sigma$ traverses the crossover value $\sigma^*=2-\eta_{SR}$. It develops a scaling framework for the crossover, arguing that the propagator obeys a universal form $G(k,\delta)\approx \delta^{m}\mathcal{H}(\log k\,\delta)$ with $m=1$, and ties this to the linear vanishing of the renormalization factor $Z$ at $\sigma^*$. Through a conformal-bootstrap–inspired argument and a $1/N$ expansion of the long-range $O(N)$ model, it shows that $Z^{-1}$ indeed vanishes linearly as $\sigma\to\sigma^*$ and provides explicit expressions illustrating the crossover. A numerical check in two-dimensional long-range percolation reveals logarithmic corrections to the standard power laws at the crossover and data collapses consistent with the proposed scaling function, strengthening the proposed picture of the crossover region.

Abstract

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation.

Figures (5)

• Figure 1: Logarithm of the susceptibility as a function of link density for three different values of $\sigma$ for different system size. Full lines are $\overline{\log(\chi)}$ and dashed lines are $\log(\overline{\chi})$. In the inset, data for $\sigma=1.5$ are plotted versus the scaling variable $\rho = \overline{\log(S_1/S_2)}$. Upper curves are for larger sizes.
• Figure 2: Data at $\sigma=\sigma^*=2-\eta_{SR}$. Left: maximum of $\chi$ versus system size $L$ in a log-log scale; the green curve proportional to $L^{\sigma^*}/\log(L)$ fits data better than the best power law $L^a$(with $a=1.63$) shown with a blue dotted line. Right: scaling of the susceptibility in the whole critical region according to simple power law (lower blue curves) and with logarithmic corrections (upper red curves).
• Figure 3: A check of the scaling in Eq.(\ref{['scalingChi']}).
• Figure 4: Local slopes $sl(L) \equiv \partial\log(\chi)/\partial\log(L)$ converge to the asymptotic behavior (shown with black lines) with logarithmic corrections close to the crossover point $\sigma=\sigma^*=2-\eta_{SR}$.
• Figure 5: Scaling of the local slopes $sl(L) \equiv \partial\log(\chi)/\partial\log(L)$ is consistent with the asymptotic exponent predicted by Sak, $as(\sigma) = \min(\sigma,\sigma^*)$, with logarithmic corrections at $\sigma=\sigma^*$, although smallest sizes still show correction to scaling. The black curve is the scaling function obtained with the simplest choice $F(z)=(\exp(z)-1)^{-1}$ and has no fitting parameters at all.