Critical frontier and disorder-strength dependence of the two-dimensional random-bond triangular Ising ferromagnet and the complementary random-bond three-state Potts case

For the two-dimensional random-bond triangular Ising ferromagnet (RBTrIM) — Ising spins on a triangular lattice with nearest-neighbor couplings drawn from a symmetric binary ferromagnetic distribution of strength $r=J_2/J_1$ — the only detailed finite-size-scaling study (Fytas and Malakis, 2010) examined a single disorder strength, $r=1/3$, at lattice sizes up to $L=200$, and found evidence for the strong-universality scenario: pure-Ising critical exponents with a double-logarithmically diverging specific heat. The open problem is the systematic completion of that program: map the critical frontier $T_c(r)$ across the disorder parameter space including stronger disorder, test at each $r$ the strong-universality scenario ($\nu=1$, $\beta=1/8$, $\gamma=7/4$; $C^*\sim C_1+C_2\ln\ln L$) against the weak-universality alternative (disorder-dependent exponents, saturating specific heat), consolidate the delicate discrimination between double-logarithmic and small-power-law forms at larger sizes with quantified control of entropic-sampling systematic errors, and carry out the complementary equilibrium study of the ferromagnetic three-state Potts model with bond randomness on the triangular lattice, where disorder is relevant rather than marginal. Later literature has effectively settled the strong-versus-weak universality controversy for the two-dimensional random-bond Ising class as a whole in favor of strong universality. That class-level verdict does not settle the model-specific quantitative consolidation on the triangular lattice, which remains open: the mapping of $T_c(r)$ and, at each disorder strength, the finite-size verification of the pure-Ising exponent values and of the $C^*\sim C_1+C_2\ln\ln L$ form. The weak-universality scenario thus remains a live hypothesis to be tested at each $r$, not one pre-empted by the class-level result.

Background

The Ising model — spins taking values $\pm1$ on a lattice, coupled by nearest-neighbor ferromagnetic bonds — is the paradigm of continuous phase transitions. On the two-dimensional triangular lattice its pure critical behavior is exactly known: the transition occurs at $k_BT_c/J=4/\ln3$ and the critical exponents are $\nu=1$ (correlation length), $\beta=1/8$ (magnetization) and $\gamma=7/4$ (susceptibility), while the specific heat diverges only logarithmically; the exact finite-size form of this logarithmic anomaly was computed by Ferdinand and Fisher.

A canonical question of disordered-systems physics is what happens to such criticality when the bonds become quenched random variables (frozen in, not thermally equilibrated), here drawn independently from a symmetric binary ferromagnetic distribution $P(J)=\frac{1}{2}[\delta(J-J_1)+\delta(J-J_2)]$ with $(J_1+J_2)/2=1$; the ratio $r=J_2/J_1<1$ measures the disorder strength. The Harris criterion states that weak quenched disorder is relevant if the specific-heat exponent of the pure system satisfies $\alpha>0$ and irrelevant if $\alpha<0$. For the two-dimensional Ising model $\alpha=0$: disorder is marginal, and its effect is unusually subtle. Since the early 1980s two competing scenarios have been debated. In the strong-universality (logarithmic-corrections) scenario, established by field-theoretic renormalization-group and conformal-field-theory calculations (reviewed by Shalaev; the systematic expansion in $q-2$ is due to Ludwig), random bonds are marginally irrelevant: the leading exponents retain their pure Ising values and the singularities acquire logarithmic corrections, the most striking being a double-logarithmic specific heat, $C\propto\ln\ln(1/t)$ with $t=(T-T_c)/T_c$, which at criticality appears in finite-size scaling (FSS — the analysis of how observables at criticality grow with the linear system size $L$) as $C^*\sim C_1+C_2\ln\ln L$. In the weak-universality scenario the exponents $\beta$, $\gamma$, $\nu$ vary continuously with disorder strength while the ratios $\beta/\nu$ and $\gamma/\nu$ stay at their pure values, and the specific heat saturates at a finite value, with $\nu\geq 2/d$ required by rigorous bounds for disordered systems. An exact solution claimed by Shankar, which would have implied non-Ising exponents, was corrected by Ludwig, leaving strong universality as the field-theoretic prediction. Numerically the two scenarios are notoriously hard to distinguish, since for finite $L$ a $\ln\ln L$ curve and a small power law mimic each other; decades of numerical work on the square lattice (random-site, random-bond and diluted versions), including the transfer-matrix finite-size-scaling analyses of Aarão Reis, de Queiroz and dos Santos and the random-site finite-size-scaling study of Gordillo-Guerrero, Kenna and Ruiz-Lorenzo, progressively favored strong universality, culminating in the high-precision simulations of Hasenbusch, Parisen Toldin, Pelissetto and Vicari, which confirmed the renormalization-group predictions and the universality of the logarithmic corrections for the site-diluted and $\pm J$ random-bond models. On the weak-universality side, the numerical claims of Kim and Patrascioiu for the site-diluted specific heat remained the principal dissident evidence.

Fytas and Malakis (2010) extended the test to the triangular lattice, which had not previously been investigated, using a two-stage Wang–Landau entropic sampling scheme. The Wang–Landau algorithm is a Monte Carlo method that estimates the density of states directly by iteratively flattening the energy histogram while progressively reducing a multiplicative modification factor; it is efficient for disordered systems, where observables must be averaged over many disorder realizations, but its recursion violates detailed balance in the early stages, so its systematic error is difficult to bound in general. At the single disorder strength $r=1/3$ and for $L\leq200$, Fytas and Malakis found $\nu\approx1$, $\beta/\nu\approx1/8$, $\gamma/\nu\approx1.75$, and specific-heat maxima better described by $C^*\sim C_1+C_2\ln\ln L$ than by a small power law — evidence for strong universality in the RBTrIM. They also used the sample-to-sample fluctuations of disordered quantities (which at marginality violate self-averaging, i.e. do not shrink relative to their means as $L$ grows) as an independent diagnostic. Unlike the square lattice, where duality gives the random-bond critical temperature exactly as a function of $r$, the triangular-lattice critical frontier $T_c(r)$ is not known exactly; before their work it had not been mapped at any disorder strength, and their study itself covered only $r=1/3$.

As a complementary case in a different marginality class the authors proposed the three-state Potts model (spins taking three values, with energy favoring equal neighboring states) with bond randomness on the triangular lattice: the pure two-dimensional three-state Potts model has $\alpha_p=1/3>0$, so by the Harris criterion disorder is relevant there, and the flow is expected to be toward a new random fixed point rather than toward the pure Ising point — a qualitatively different test of disorder relevance.

Problem Statement

For the two-dimensional random-bond triangular Ising ferromagnet (RBTrIM) — Ising spins on a triangular lattice with nearest-neighbor couplings independently drawn from $P(J)=\frac{1}{2}[\delta(J-J_1)+\delta(J-J_2)]$ with $(J_1+J_2)/2=1$ and disorder strength $r=J_2/J_1$ — map the critical frontier $T_c(r)$ across the disorder parameter space, from weak disorder ($r\to1$) through strong disorder well below the studied value $r=1/3$, and at each disorder strength test the strong-universality scenario — exponents fixed at the pure Ising values $\nu=1$, $\beta=1/8$, $\gamma=7/4$ with the specific heat diverging double-logarithmically, $[C^*]_{\mathrm{av}}\sim C_1+C_2\ln\ln L$ at criticality, where $C^*$ denotes the specific-heat maximum and $[\cdot]_{\mathrm{av}}$ the average over disorder realizations — against the competing weak-universality prediction, in which $\beta$, $\gamma$, $\nu$ drift with disorder while $\beta/\nu$ and $\gamma/\nu$ stay constant and the specific heat saturates. The discrimination between the double-logarithmic and small-power-law finite-size forms must be consolidated at substantially larger lattice sizes than the $L\leq200$ of the existing study, with the systematic errors of the entropic-sampling (Wang–Landau-type) method quantified rather than assumed negligible. As the complementary disorder-relevant case explicitly proposed by the source, carry out the analogous equilibrium study of the ferromagnetic three-state Potts model with bond randomness on the triangular lattice — where the pure model's positive specific-heat exponent makes disorder relevant — determining its critical frontier and universality class (pure-Potts-like, pure-Ising-like, or a distinct random fixed point) as a function of disorder strength. These four components — the mapping of $T_c(r)$, the per-$r$ test of the strong-universality scenario, the quantified control of the entropic-sampling systematic errors, and the complementary three-state Potts study — are distinct sub-problems, each to be resolved independently; settling any one of them leaves the remaining ones open. The overarching objective is a complete, systematics-controlled characterization of how quenched bond randomness changes triangular-lattice ferromagnetic criticality in the two complementary marginality classes, marginal and relevant.

References

1. N. G. Fytas and A. Malakis, Critical behavior of the pure and random-bond two-dimensional triangular Ising ferromagnet, Phys. Rev. E 81, 041109 (2010), DOI 10.1103/PhysRevE.81.041109, arXiv:1001.0867, https://doi.org/10.1103/PhysRevE.81.041109
2. A. E. Ferdinand and M. E. Fisher, Bounded and inhomogeneous Ising models. I. Specific-heat anomaly of a finite lattice, Phys. Rev. 185, 832 (1969), DOI 10.1103/PhysRev.185.832, https://doi.org/10.1103/PhysRev.185.832
3. A. W. W. Ludwig, Critical behavior of the two-dimensional random q-state Potts model by expansion in (q − 2), Nucl. Phys. B 285, 97 (1987), DOI 10.1016/0550-3213(87)90330-0, https://doi.org/10.1016/0550-3213(87)90330-0
4. R. Shankar, Exact critical behavior of a random bond two-dimensional Ising model, Phys. Rev. Lett. 58, 2466 (1987), DOI 10.1103/PhysRevLett.58.2466, https://doi.org/10.1103/PhysRevLett.58.2466
5. A. W. W. Ludwig, Comment on 'Exact Critical Behavior of a Random-Bond Two-Dimensional Ising Model', Phys. Rev. Lett. 61, 2388 (1988), DOI 10.1103/PhysRevLett.61.2388, https://doi.org/10.1103/PhysRevLett.61.2388
6. B. N. Shalaev, Critical behavior of the two-dimensional Ising model with random bonds, Phys. Rep. 237, 129 (1994), DOI 10.1016/0370-1573(94)90086-8, https://doi.org/10.1016/0370-1573(94)90086-8
7. J.-K. Kim and A. Patrascioiu, Critical behavior of the specific heat in the two dimensional site diluted Ising system, Phys. Rev. Lett. 72, 2785 (1994), DOI 10.1103/PhysRevLett.72.2785, https://doi.org/10.1103/PhysRevLett.72.2785
8. F. D. A. Aarão Reis, S. L. A. de Queiroz and R. R. dos Santos, Universality and logarithmic corrections in two-dimensional random Ising ferromagnets, Phys. Rev. B 56, 6013 (1997), DOI 10.1103/PhysRevB.56.6013, arXiv:cond-mat/9706202, https://doi.org/10.1103/PhysRevB.56.6013
9. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto and E. Vicari, Universal dependence on disorder of two-dimensional randomly diluted and random-bond ±J Ising models, Phys. Rev. E 78, 011110 (2008), DOI 10.1103/PhysRevE.78.011110, https://doi.org/10.1103/PhysRevE.78.011110
10. A. Gordillo-Guerrero, R. Kenna and J. J. Ruiz-Lorenzo, The quenched-disordered Ising model in two and four dimensions, AIP Conf. Proc. 1198, 42 (2009), DOI 10.1063/1.3284424, arXiv:0909.3774, https://doi.org/10.1063/1.3284424
11. N. G. Fytas and P. E. Theodorakis, Universality aspects of the 2d random-bond Ising and 3d Blume-Capel models, Eur. Phys. J. B 86, 30 (2013), DOI 10.1140/epjb/e2012-30705-x, arXiv:2012.02482, https://doi.org/10.1140/epjb/e2012-30705-x
12. Q. Zhu, X. Wan, R. Narayanan, J. A. Hoyos and T. Vojta, Emerging criticality in the disordered three-color Ashkin-Teller model, Phys. Rev. B 91, 224201 (2015), DOI 10.1103/PhysRevB.91.224201, https://doi.org/10.1103/PhysRevB.91.224201
13. V. Dotsenko, Y. Holovatch, M. Dudka and M. Weigel, Self-averaging in the random two-dimensional Ising ferromagnet, Phys. Rev. E 95, 032118 (2017), DOI 10.1103/PhysRevE.95.032118, https://doi.org/10.1103/PhysRevE.95.032118
14. Z. Komargodski and D. Simmons-Duffin, The random-bond Ising model in 2.01 and 3 dimensions, J. Phys. A: Math. Theor. 50, 154001 (2017), DOI 10.1088/1751-8121/aa6087, https://doi.org/10.1088/1751-8121/aa6087
15. R. E. Belardinelli, V. D. Pereyra, R. Dickman and B. J. Lourenço, Intrinsic convergence properties of entropic sampling algorithms, J. Stat. Mech. (2014) P07007, DOI 10.1088/1742-5468/2014/07/P07007, https://doi.org/10.1088/1742-5468/2014/07/P07007
16. R. E. Belardinelli and V. D. Pereyra, Nonconvergence of the Wang-Landau algorithms with multiple random walkers, Phys. Rev. E 93, 053306 (2016), DOI 10.1103/PhysRevE.93.053306, https://doi.org/10.1103/PhysRevE.93.053306
17. M. Picco, Numerical results for the two-dimensional random-bond three-state Potts model, Phys. Rev. B 54, 14930 (1996), DOI 10.1103/PhysRevB.54.14930, https://doi.org/10.1103/PhysRevB.54.14930
18. J. L. Jacobsen and J. Cardy, Critical behaviour of random-bond Potts models: a transfer matrix study, Nucl. Phys. B 515, 701 (1998), DOI 10.1016/S0550-3213(98)00024-8, https://doi.org/10.1016/S0550-3213(98)00024-8
19. C. Chatelain, Numerical study of Schramm–Loewner evolution in the random 3-state Potts model, J. Stat. Mech. (2010) P08004, DOI 10.1088/1742-5468/2010/08/P08004, https://doi.org/10.1088/1742-5468/2010/08/P08004
20. H. P. Ying, B. J. Bian, D. R. Ji, H. J. Luo and L. Schülke, Critical dynamics and universality of the random-bond Potts ferromagnet with tri-distributed quenched disorders, arXiv:cond-mat/0103355 (2001), https://arxiv.org/abs/cond-mat/0103355
21. A. K. Murtazaev and A. B. Babaev, The critical behavior of the two-dimensional three-state Potts model on a triangular lattice with quenched disorder, Mater. Lett. 238, 321 (2019), DOI 10.1016/j.matlet.2018.12.030, https://doi.org/10.1016/j.matlet.2018.12.030