Field-theoretic origin of the double power-law critical correlation function and the universality-class structure of the long-range-to-short-range crossover in the two-dimensional long-range Ising model

The two-dimensional ferromagnetic Ising model with algebraically decaying couplings $J(r)\propto r^{-(2+\sigma)}$ crosses over, as $\sigma$ grows, from a nontrivial long-range (LR) universality class with continuously varying exponents to the short-range (SR) Ising class. Angelini, Parisi and Ricci-Tersenghi (2014) discovered that near the Sak crossover decay exponent $\sigma_*=2-\eta_{\rm SR}=7/4$ the critical spin-spin correlation function is not a single power law but $G(x)=(A+B\,x^{-\delta})/x^{\eta}$ with $\delta\approx0.3$-$0.5$, a subdominant term that dominates ordinary finite-size corrections (it forces $\chi\sim L^{2-\eta}(a+bL^{-\delta})$), and they left its analytical origin open while noting that, after the double-power correction, their data remained compatible both with Sak's scenario and with Picco's smooth interpolation ('much larger sizes are needed to exclude one of the two'). The subsequent literature shows: no analytical derivation of the double power law exists (the only mechanism proposal attributes it to the subleading magnetic scaling field, Yao et al. 2024, unverified), and the crossover scenario is now actively contested between mutually contradictory high-precision studies (Horita et al. 2017 supporting Sak; Xiao et al. 2025 with $L=8192$ simulations and Li-Chen-Deng 2026 with a two-loop $(4-\epsilon)$-expansion supporting a sharp change at $\sigma=2$). The problem — the renormalization-group theory of this crossover window — remains open.

Background

Long-range Ising models and their regimes. The ferromagnetic long-range Ising model (LRIM) places Ising spins on a $d$-dimensional lattice with pair couplings that decay algebraically with distance, $J(r)\propto r^{-(d+\sigma)}$; the decay exponent $\sigma$ tunes the interaction range continuously from effectively infinite to effectively nearest-neighbor. At criticality the spin-spin correlation function decays as $G(x)\sim x^{-(d-2+\eta)}$, which defines the anomalous dimension $\eta$. Renormalization-group (RG) theory — the framework relating the critical behavior at different length scales — predicts three regimes: mean-field behavior for $\sigma<d/2$ (below the mean-field boundary $\sigma_{\rm MF}=d/2$, where the system is effectively above its own upper critical dimension $d_c=2\sigma$), a nontrivial long-range (LR) universality class with exponents varying continuously with $\sigma$ for $d/2<\sigma<\sigma_*$, and short-range (SR) universality for $\sigma>\sigma_*$. Fisher, Ma and Nickel (1972) computed $\eta=2-\sigma$ in the LR regime by expanding around the Gaussian (free-field) fixed point, which would place the LR-SR boundary at $\sigma=2$. Sak (1973) showed that the crossover instead occurs at $\sigma_*=2-\eta_{\rm SR}$, where $\eta_{\rm SR}$ is the anomalous dimension of the SR model in the same dimension, so that $\eta=\max(2-\sigma,\eta_{\rm SR})$. We write this boundary as $\sigma_*$ throughout; Angelini et al. denote the same boundary $\sigma_L$ (their 'lower critical exponent', the value of $\sigma$ at which SR behavior is recovered), while other works such as Horita et al. (2017) call the same number the upper critical decay exponent — the boundary itself is identical in all conventions.

The two-dimensional case. In $d=2$ the SR Ising value is exactly $\eta_{\rm SR}=1/4$, so Sak's boundary sits at $\sigma_*=7/4$, where the prescriptions of Fisher et al. and Sak coincide ($\eta=1/4$). The smooth-interpolation and sharp-change scenarios instead keep $\eta>1/4$ in the interior of the window $7/4\leq\sigma<2$ (Picco measured $\eta=0.332(8)$ at $\sigma=1.75$), so the leading exponents of the proposed scenarios separate only inside the contested window $1.6\lesssim\sigma\leq2$ — and there only by amounts of the same order as the bias that subdominant corrections induce in single-power analyses. Numerically this window has been contested for two decades: Luijten and Blöte (2002) supported Sak, reporting logarithmic corrections to scaling at $\sigma=1.75$, though with error bars large enough to remain compatible with Picco's later measurements; Picco (2012) measured $\eta=0.332(8)$ at $\sigma=1.75$, suggesting a smooth interpolation of $\eta(\sigma)$ between the $2-\sigma$ branch and the SR value, an interpretation supported by the RG analysis of Blanchard, Picco and Rajabpour (2013).

The source paper and its open question. Angelini, Parisi and Ricci-Tersenghi (2014) simulated the 2D LRIM with cluster Monte Carlo methods (algorithms that flip correlated blocks of spins rather than single spins, suppressing critical slowing down) up to $L=2048$. At $\sigma=1.75$ they found that the critical correlation function is a sum of two power laws,

$$G(x)=\frac{A+B\,x'(x)^{-\delta}}{x'(x)^{\eta}},$$

with $x'(x)=\sin(\pi x/L)\,L/\pi$ a boundary-corrected distance (equal to $x$ for $x\ll L$, chosen to absorb periodic-boundary effects), $\delta\approx0.3$ from correlation fits with $\eta=1/4$ (about $0.41$ from susceptibility fits; fits using Picco's $\eta_P=0.332$ give $\delta\approx0.43$-$0.5$). This subdominant term $B\,x^{-(\eta+\delta)}$ is much larger than ordinary finite-size effects: the susceptibility obeys $\chi\sim L^{2-\eta}(a+bL^{-\delta})$, and the $L^{-\delta}$ correction overwhelms the usual correction-to-scaling term $L^{-\omega}$ (with $\omega$ the leading correction-to-scaling exponent). Neglecting it biases $\eta$ upward, which they argued explains Picco's deviation. After the two-power correction their data became compatible with Sak's scenario, but Sak's and Picco's scenarios remained statistically compatible ('much larger sizes are needed to exclude one of the two'), and they explicitly identified the analytical origin of the two power laws as an open theoretical challenge.

Theory of the crossover and neighboring results. The LR critical point is described by a nonlocal scale-invariant theory with no local stress tensor (the operator generating coordinate transformations in a local field theory); conformal invariance (invariance under angle-preserving transformations, a stronger symmetry than scale invariance) nevertheless holds in the $\epsilon$-expansion (the expansion in the deviation $4-d=\epsilon$ from the upper critical dimension) via a representation of the LRIM as a defect in an auxiliary higher-dimensional theory (Paulos, Rychkov, van Rees and Zan, 2016). Behan, Rastelli, Rychkov and Zan (2017) constructed a weakly coupled description of the LR-to-SR crossover — the SR fixed point plus an auxiliary field $\chi_{\rm aux}$ whose perturbation realizes Sak's nonlocal operator — keeping the crossover at $\sigma_*=2-\eta_{\rm SR}$ and predicting a marginally irrelevant operator (an operator whose coupling decays only logarithmically under the RG flow) that produces logarithmic corrections exactly at the crossover. Brezin, Parisi and Ricci-Tersenghi (2014) proposed a crossover-function scaling form for the leading part of $G$ near $\sigma_*$, with a universal $1/\log x$ correction at $\sigma=\sigma_*$, warning that in finite systems logarithmic corrections can masquerade as small power-law corrections; the marginal-case logarithmic law was proven rigorously by Chen and Sakai (2019), but only at and above the upper critical dimension, not in the 2D nontrivial regime.

Universality among long-range constructions. Angelini et al. also tested 'superuniversality' — the conjecture that LR exponents depend on $d$ and $\sigma$ only through $\sigma/d$, equivalent to a $(d,\sigma)\leftrightarrow D$ mapping onto an SR model in an effective dimension $D$ — finding it approximate near $\sigma_{\rm MF}$ but not exact near $\sigma_*$, and showing that the Dyson hierarchical model (an iteratively block-constructed 1D long-range model whose exact RG recursion makes it a standard approximation to the LR chain) is not in the same universality class as the lattice LR chain near $\sigma=1$ in $d=1$ — the value above which the one-dimensional chain has no finite-temperature transition, and the paper's $\sigma_L(1)$. They note that superuniversality in the crossover region is compatible only with Sak's scenario, so the crossover question and the mapping question are linked.

Problem Statement

The problem is the renormalization-group theory of the long-range-to-short-range crossover of the two-dimensional ferromagnetic Ising model with algebraic interactions $J(r)\propto r^{-(2+\sigma)}$ in the crossover window $1.6\lesssim\sigma\leq2$ around the Sak exponent $\sigma_*=2-\eta_{\rm SR}=7/4$. It has two inseparably linked parts.

(1) Field-theoretic origin of the double power law. At criticality near $\sigma_*$ the spin-spin correlation function is empirically a sum of two power laws,

$$G(x)=\frac{A+B\,x'(x)^{-\delta}}{x'(x)^{\eta}},$$

with $x'(x)=\sin(\pi x/L)\,L/\pi$ a boundary-corrected distance (equal to $x$ for $x\ll L$) and $\delta\approx0.3$-$0.5$ at $\sigma=1.75$, and the induced finite-size correction $\chi\sim L^{2-\eta}(a+bL^{-\delta})$ dominates the usual correction-to-scaling term $L^{-\omega}$. Provide an analytical derivation — microscopic or field-theoretic RG — of why a distinct subdominant power law $B\,x^{-(\eta+\delta)}$ appears in the critical correlation function near $\sigma_*$, which operator or RG eigenoperator (eigenoperator of the linearized RG transformation) generates it — candidate mechanisms to be confirmed or refuted include the subleading magnetic scaling field $h_2$ (proposed for the two-dimensional long-range XY model by Yao et al., 2024) and the auxiliary field $\chi_{\rm aux}$ of the defect description — what fixes $\delta$ and its $\sigma$- and $d$-dependence, and embed it in an RG description that simultaneously reproduces the standard long-distance RG eigenoperators and the finite-size correction structure.

(2) Universality-class structure of the crossover window. Determine which scenario governs $1.6\lesssim\sigma\leq2$ in $d=2$: (a) Sak's $\eta=\max(2-\sigma,1/4)$ with the universality change at $\sigma_*=7/4$; (b) a sharp universality-class change at $\sigma=2$, with non-SR critical behavior persisting for all $\sigma<2$ (as advocated by Xiao et al., 2025; distinct from the older Fisher-Ma-Nickel scenario $\eta=2-\sigma$ up to $\sigma=2$); or (c) smooth interpolation of $\eta(\sigma)$ between the $2-\sigma$ branch and $1/4$ with no singularity at either $7/4$ or $2$.

Both parts jointly constitute the theory of the LR-to-SR crossover in the 2D LRIM: the double power law of part (1) is precisely what makes the scenario discrimination of part (2) numerically subtle — single-power analyses are biased upward by an amount comparable to the leading-exponent differences being discriminated — and part (2) in turn decides the fate of superuniversality (dependence of exponents only on $\sigma/d$) and of the $(d,\sigma)\leftrightarrow D$ dimensional matching in the crossover region, which Angelini et al. show is compatible only with scenario (a).

References

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