Marginal quenched bond disorder in two-dimensional Ising systems: strong versus weak universality, crossover scales, and the universality of disorder-induced scaling

The two-dimensional random-bond Ising model sits exactly at the Harris marginality boundary — the pure model's specific-heat critical exponent is exactly zero — and whether weak quenched disorder in the bond strengths changes its critical behavior has been debated since 1983. In the strong-universality scenario the exact pure Ising exponents survive with universal double-logarithmic corrections, for example a specific heat growing as $\ln[b\ln(cL)]$ on lattices of linear size $L$; in the weak-universality scenario the system instead follows simple power laws with exponents varying continuously with disorder strength. A 2008 Wang–Landau study by Fytas, Malakis and Hadjiagapiou favored the double-logarithmic form for strong disorder and sizes up to $L=200$, but reported difficulties in reaching a definite discrimination between the two scenarios at weak disorder. The same study reported a strongly saturating specific heat in the random-bond superantiferromagnetic model with competing interactions, which it contrasted with other two-dimensional random-bond systems, and it closed by announcing an ongoing similar study of bond disorder in two-dimensional systems whose clean limit undergoes a first-order transition. The subsequent literature — renormalization-group analyses, conformal perturbation theory, and simulations up to $L=2240$ — strongly supports the strong-universality picture, but a definitive discrimination of the two scenarios for the marginal random-bond Ising model remains unresolved, as do the related questions of the universality of specific-heat saturation in competing-interaction models and of a quantitative crossover theory of disorder-rounded first-order two-dimensional transitions.

Background

The ferromagnetic Ising model on the square lattice is the paradigm of an exactly solved critical point: its specific heat diverges logarithmically at the critical temperature, so its specific-heat critical exponent is exactly $\alpha_{\mathrm{p}}=0$. The Harris criterion states that weak quenched (frozen-in, spatially fixed) randomness in the interaction strengths changes the critical behavior only if the pure system has $\alpha_{\mathrm{p}}>0$ (disorder relevant); if $\alpha_{\mathrm{p}}<0$, disorder is irrelevant; and $\alpha_{\mathrm{p}}=0$ is the marginal case about which the criterion is silent. The two-dimensional (2D) random-bond Ising model, in which each nearest-neighbor coupling is drawn independently from a distribution such as two positive values $rJ$ and $J$ with equal probability, sits exactly at this boundary, and the nature of its critical behavior has been controversial since the early 1980s.

Two competing scenarios define the debate. The strong-universality (logarithmic-corrections) scenario, derived in the 1980s by Dotsenko and Dotsenko (1983), Shankar (1987) and Ludwig (1987) using renormalization-group and conformal-field-theory methods, holds that the pure fixed point remains stable with a marginally irrelevant disorder coupling: all critical exponents keep their exact pure values ($\nu=1$, $\eta=1/4$, $\beta/\nu=1/8$, $\gamma/\nu=7/4$), but universal double-logarithmic corrections appear — for example the specific heat at criticality grows as $C\sim\ln[b\ln(cL)]$ on lattices of linear size $L$, and the temperature derivative of a dimensionless ratio $R$ (such as a Binder cumulant, the fourth-order cumulant of the magnetization used in finite-size scaling analyses, or the ratio $\xi/L$ of correlation length to system size) scales as $\mathrm{d}R/\mathrm{d}T\sim L(\ln L)^{-1/2}$. The weak-universality scenario, developed heuristically from early numerical data and defended in particular by Kim and Patrascioiu (1994), holds instead that observables follow simple power laws whose exponents drift continuously with disorder strength while certain exponent ratios stay fixed at their pure values; in this scenario the specific heat of the disordered system saturates at a finite value. Any resolution must respect the rigorous inequality of Chayes et al. (1986), $d\nu\ge 2$, for the correlation-length exponent at disordered critical points. The practical difficulty in deciding between the scenarios is intrinsic: a logarithm is nearly indistinguishable from a very small power over accessible system sizes, because the effective exponents drift only as $1/\ln L$, so all numerical discriminations are statements about crossovers as much as about asymptotics.

The source paper (Fytas, Malakis and Hadjiagapiou 2008) approached this problem with entropic simulations based on the Wang–Landau algorithm (a flat-histogram Monte Carlo method that estimates the density of states directly; Wang and Landau 2001), contrasting two random-bond Ising systems. The first is the simple 2D Ising model — the marginal case. The second is the square superantiferromagnetic (SAF) model, an Ising model with competing nearest- and next-nearest-neighbor antiferromagnetic couplings whose pure transition is second order with exponents close to those of the 2D three-state Potts model (a generalized Ising model with three spin states per site), i.e. $\alpha_{\mathrm{p}}>0$: the non-marginal case in which disorder is Harris-relevant and must drive the system to a new, disorder-dominated universality class. For the random-bond SAF model the authors verified weak universality (fixed ratios $\beta/\nu=1/8$, $\gamma/\nu=7/4$ with $\nu>1$ satisfying hyperscaling and the Chayes inequality) and found a strongly saturating specific heat — qualitatively different from the diverging specific heat of the random-bond three-state Potts ferromagnet — a behavior they attributed to the competing nature of the interactions. For the marginal random-bond simple Ising model, at weak disorder ($r=3/5$) and strong disorder ($r=1/7$), their finite-size scaling analysis up to $L=200$ favored the double-logarithmic form, with the correlation-length exponent consistent with the pure value $\nu=1$ and critical temperatures matching the exactly known dual values; but the authors state that at weak disorder and smaller sizes they could not conclusively discriminate between the two alternatives, while for the strong-disorder case they present the stability of the double-logarithmic fits as evidence in favor of the logarithmic-correction scenario. The paper closes by announcing the same machinery for 2D systems with first-order transitions, where the Aizenman–Wehr theorem guarantees that quenched disorder rounds a first-order transition in two dimensions, and Cardy's renormalization-group analysis of the disordered Ashkin-Teller model (a model of two coupled Ising systems whose coupling can drive a first-order transition, degenerating at special points into independent Ising models) predicts that the emerging continuous transition lies in the clean 2D Ising universality class with the same type of logarithmic corrections.

Problem Statement

Resolve the open question left by the source study of Fytas, Malakis and Hadjiagapiou (2008): for the two-dimensional random-bond Ising model at Harris marginality — each nearest-neighbor coupling drawn independently from the binary distribution of two positive values $rJ$ and $J$ with equal probability — decide definitively between

• the strong-universality scenario: the exact pure Ising exponents ($\nu=1$, $\beta/\nu=1/8$, $\gamma/\nu=7/4$) with universal double-logarithmic corrections, such as $C\sim\ln[b\ln(cL)]$ for the specific heat at criticality on lattices of linear size $L$ and $\mathrm{d}R/\mathrm{d}T\sim L(\ln L)^{-1/2}$ for the temperature derivative of a dimensionless ratio $R$; and
• the weak-universality scenario: simple power laws with exponents varying continuously with disorder strength.

The source study favored the double-logarithmic form but reported that a definite discrimination between the two alternatives could not be reached at weak disorder and smaller sizes. A resolution must make the discrimination definite — the evidence must exclude one of the two scenarios rather than merely favor it, and the resolution must state the evidence standard (analytic or numerical) it relies on and what that standard does not settle — and must characterize the regime — the finite-size range and the disorder strengths of the binary bond distribution — over which it is established.

Closely related questions remain open in the same literature but are not part of the required target of this statement: whether the strong specific-heat saturation found in the random-bond superantiferromagnetic (competing-interaction) model is generic across two-dimensional Ising-like systems, and the universality class, logarithmic corrections, and quantitative crossover (breakup) scales of the continuous transition emerging when quenched bond randomness is added to two-dimensional systems whose clean limit undergoes a first-order transition.

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