Relations between short-range and long-range Ising models

We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions ($J\propto r^{-d-\sigma }$) on both a one-dimensional chain ($d=1$) and a square lattice ($d=2$). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters $(d,\sigma )$ to that of a short-range (SR) model in an equivalent dimension $D$. We then study the critical behavior of the $d=2$ LR model close to the lower critical $\sigma$, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.

Figure 1

Behavior of $\eta \left(\sigma \right)$ for a LR model in $d=2$ as proposed in different works: In the work of Fisher et al. [4] $\eta =2-\sigma$ up to $\sigma =2$, while for Sak [10] $\eta =max(2-\sigma ,{\eta }_{\mathrm{SR}}=\frac{1}{4})$, and the data of Picco [15] $\eta$ seem to interpolate smoothly between $2-\sigma$ and ${\eta }_{\mathrm{SR}}=\frac{1}{4}$.

Figure 2

Scale-invariant observable ${\chi }_L/L^{\sigma }$ (left) and Binder cumulant (right), computed at different $n={log}_{2}L$, as a function of the temperature $T$, at $\sigma =0.654533$. The curves at different sizes should cross at a temperature that approaches $T_c$ when $L$ grows. The Binder cumulant shows stronger corrections to scaling with respect to ${\chi }_L/L^{\sigma }$.

Figure 3

(Left) Quotient of the Binder parameter for $L$ and $2L$ at $T_L^{*}$, computed at the crossing temperature of ${\chi }_L/L^{\sigma }$. The straight line is the best fit using Eq. (17) with $\omega$ left as a free parameter. (Right) Quotient of the derivative of the Binder parameter at $T_L^{*}$. The straight line is the best fit as a function of $L^{-\omega }$ using Eq. (18), with $\omega$ determined from the previous fit and the intercept $2^{1/\nu }$ left as a free parameter.

Figure 4

The spin-spin correlation function for different sizes at $\sigma =1.75$ and $d=2$. The left panel shows raw data, while in the right panel boundary effects have been drastically reduced by plotting parametrically versus the variable $x^{\prime }\left(x\right)\equiv sin(\pi x/L)L/\pi$.

Figure 5

Data for the spin-spin correlation function $G\left(x\right)$ measured at $\sigma =1.75$ and $d=2$, rescaled by the asymptotic power law $x^{\prime }{\left(x\right)}^{\eta }$, in order to highlight the corrections to the asymptotic decay. (Left) $\eta =0.25$ and $\delta =0.3$. (Right) ${\eta }_P=0.332$ and ${\delta }_P=0.5$.

Figure 6

Log-log plot of the susceptibility at the maximum as a function of the size, for $\sigma =1.75$ and $d=2$. Two fits using $f\left(x\right)=L^{2-\eta }(a+b{L}^{-\delta })$ and $g\left(x\right)=L^{2-{\eta }_P}(a+b{L}^{-{\delta }_P})$ are shown, which are both compatible with the data.

Figure 7

$1/\left(\nu d\right)$ as a function of the exponent $\widehat{\sigma }=\sigma /d$ for HM in $d=1$ and the LR model in $d=1$ and $d=2$, as found in various works and in this work. The SR values follow the matching formula (19).

Figure 8

$1/\gamma$ as a function of the exponent $\widehat{\sigma }=\sigma /d$ in the non-mean-field region for HM in $d=1$ and the LR model in $d=1$ and $d=2$, as found in various works and in this work.