Conformal field theories and critical exponents of the exact scattering solutions for the two-dimensional random-bond q-state Potts model

Delfino and Tartaglia (J. Stat. Mech. 2017) obtained the exact scale-invariant scattering solutions of two-dimensional quantum field theories with replicated permutational symmetry $S_q\times S_n$; in the zero-replica limit these describe the renormalization-group fixed points of the q-state Potts model with quenched bond disorder, and every solution with non-zero disorder contains sectors that do not depend on q, making some critical exponents potentially superuniversal (independent of $q$). The missing piece is the identification of the conformal field theories to which these scattering solutions correspond: without it, scaling dimensions and hence critical exponents cannot be extracted from the exact data. The problem is to determine this conformal data together with the operator assignments for lattice observables, and thereby the magnetic exponent $\eta$ at the strongly disordered Nishimori-like and zero-temperature fixed points of the $\pm J$ model — settling whether $\eta$ is superuniversal, as a duality hypothesis suggests, whether that duality hypothesis holds in the replicated continuum limit, and whether the two fixed points correspond to the same or to distinct scattering solutions. Later work made the thermal exponents at the $q=2$ Nishimori point exact ($\nu_1=3/2$, $\nu_2=4$) but leaves the spin sector and the conformal identification untouched.

Background

The q-state Potts model places on each site of a lattice a variable ('color') taking $q$ values, with an energy that favors equal colors on neighboring sites; its global symmetry is the permutation group $S_q$ of the colors. In two dimensions the pure ferromagnet has a second-order phase transition for $q\le 4$ and a first-order one for $q>4$. In the random-bond version the nearest-neighbor couplings are independent random variables fixed once and for all ('quenched' disorder), and the average over them is taken on the free energy; theoretically this is implemented by the replica trick, which couples $n$ identical copies of the system and then sends $n\to 0$. Disorder is relevant, by the Harris criterion, whenever the pure specific heat diverges, and it rigorously rounds first-order transitions into continuous ones (Aizenman-Wehr); a rigorous inequality of Chayes et al. requires the correlation length exponent of a disordered critical point in $d$ dimensions to satisfy $\nu\ge 2/d$, i.e. $\nu\ge 1$ in two dimensions.

The two-dimensional random-bond Potts model has been the main playground of a long-standing puzzle. Simulations beginning with Chen, Ferrenberg and Landau (1992) found that disorder softens the first-order transition of the $q=8$ model into a continuous one with exponents close to the Ising values ('strong universality'), while transfer-matrix studies (Cardy and Jacobsen, 1997) found exponents depending on $q$, consistent with the perturbative replica expansion around $q=2$ (Ludwig, 1990; Dotsenko, Picco and Pujol, 1995). For $q=2$ with a mixture of ferro- and antiferromagnetic ($\pm J$) bonds, the phase diagram contains, on the boundary of the ferromagnetic phase, a multicritical 'Nishimori point' lying on a line where a gauge symmetry makes the internal energy exactly known (Nishimori, 1981); it is a prototype strong-disorder fixed point, with numerical exponents $\nu\approx 1.5$ and $\eta\approx 0.18$ (Merz and Chalker, 2002; Picco, Honecker and Pujol, 2006; Hasenbusch et al., 2008), and a zero-temperature endpoint of the same boundary. For $q=3$ a similar 'Nishimori-like' fixed point is observed numerically (Sørensen, Gingras and Huse, 1998), although the gauge symmetry, and hence the exact line, exist only at $q=2$.

The exact approach underlying this problem rests on the fact that a critical point of a two-dimensional statistical system corresponds to a massless, scale-invariant scattering theory of right- and left-moving particles, whose two-particle amplitudes are fixed by exact unitarity and crossing equations. For pure models this framework classified the critical lines of the Potts model (Delfino and Tartaglia, Phys. Rev. E 96, 042137 (2017)) and, whenever the underlying conformal field theory (CFT) could be recognized, delivered the critical exponents through the scaling dimensions: a CFT is a field theory invariant under the infinite-dimensional conformal algebra of two dimensions, and the scaling dimension $X_\varepsilon$ of the energy-density field gives $\nu = 1/(2-X_\varepsilon)$ while the spin dimension $X_\sigma$ gives $\eta = 2X_\sigma$. Delfino (Phys. Rev. Lett. 118, 250601 (2017)) showed that the replica method can be implemented exactly in this framework: the particles carry color and replica indices, the amplitudes obey $S_q\times S_n$-symmetric unitarity equations, with $S_n$ the permutation group of the replicas, and the $n\to 0$ limit describes quenched disorder. The solution space (Delfino and Tartaglia, J. Stat. Mech. 2017) contains a line of fixed points interpolating from weak to strong disorder as $q$ grows — the random ferromagnet, whose $S_q$-invariant sector becomes exactly $q$-independent at $n=0$, providing the mechanism by which $\nu$ can keep the Ising value 1 (saturating the Chayes bound) while $\eta = 2\beta/\nu$ varies with $q$ — and two strongly disordered solutions defined for all $q$: one completely $q$-independent, naturally describing the zero-temperature percolation fixed point of the bond-diluted ferromagnet, and one $q$-independent only in an $S_q$-charged sector which, under a duality hypothesis extending Kramers-Wannier self-duality (the exact equivalence of the ferromagnetic and paramagnetic phases of the pure model) to the replicated continuum theory, contains the dual spin field: this is the main candidate for the Nishimori-like fixed point of the $\pm J$ model and leaves room for superuniversality (symmetry-independence) of $\eta$.

The solutions are exact, but they do not by themselves produce exponents: for pure models, extracting scaling dimensions from scattering required recognizing the underlying CFT, and for quenched disorder this identification is missing. What is now exactly known at the $q=2$ Nishimori point are only the thermal-sector exponents: Delfino (J. Stat. Mech. 2025) showed that regularity of the exactly known internal energy along the Nishimori line forces discrete values of the two relevant even scaling dimensions, giving $\nu_1 = 3/2$ and $\nu_2 = 4$ (renormalization eigenvalues $y_1 = 2/3$, $y_2 = 1/4$, crossover exponent $8/3$) in two dimensions — values that the best numerics confirm — but this argument cannot reach the spin sector, does not apply at $q>2$, and does not touch the zero-temperature fixed point. The magnetic exponent $\eta$ at the strongly disordered fixed points therefore remains undetermined by theory, and the conformal field theories of the random fixed points — which the source paper observes must possess features distinguishing them from all CFTs known so far — remain unidentified.

Problem Statement

For the two-dimensional q-state Potts model with quenched bond disorder, the renormalization-group fixed points are known exactly as scale-invariant scattering solutions of the replicated $S_q\times S_n$ field theory in the $n\to 0$ limit (Delfino, PRL 2017; Delfino-Tartaglia, JSTAT 2017), including the line of solutions describing the random ferromagnet, the completely $q$-independent strongly disordered solution, and the strongly disordered solution that is $q$-independent in an $S_q$-charged sector. The problem is to determine the critical behavior carried by these solutions — specifically:

1. Identify the conformal field theory (spectrum of scaling dimensions with symmetry quantum numbers, operator content and fusion structure, central charge) corresponding to each scattering solution, and with it the operator assignments that tell which scaling fields describe the lattice spin and energy observables in the quenched ($n\to 0$) limit, including how replica-mixed operators enter.

2. Derive the magnetic exponent $\eta$ (equivalently the spin scaling dimension $X_\sigma$, with $\eta=2X_\sigma$ and $\beta/\nu=X_\sigma$) at the strongly disordered fixed points — the Nishimori-like fixed point of the $\pm J$ model for $q\ge 2$ and the zero-temperature fixed point — and settle whether $\eta$ is superuniversal (independent of $q$), as follows if the Kramers-Wannier-type duality (the exact self-duality relating the ferromagnetic and paramagnetic phases of the pure model) extends to the replicated continuum limit; establishing or refuting that duality hypothesis is part of the problem. A q-dependent $\nu$ together with a q-independent $\eta$ would select the charged-sector solution; equal exponents at the two fixed points would bear on whether they share a solution.

3. Determine whether the Nishimori-like fixed point and the zero-temperature fixed point of the $\pm J$ model correspond to the same or to distinct scattering solutions, and whether they share exponents — noting the source paper's caveat that a single scattering solution can describe critical lines with different exponents.

A complete answer supplies the conformal data for the named solutions, derives the exponent values, and settles the superuniversality questions; it must be consistent with the constraints now known exactly — the $q\to 2$ weak-disorder limit matching the perturbative replica exponents (Ludwig; Dotsenko-Picco-Pujol), random percolation exponents for the completely $q$-independent solution at $q=1$, the value $\nu=1$ on the random ferromagnetic line (saturating the Chayes bound $\nu\ge 1$), and the exact thermal exponents $\nu_1=3/2$, $\nu_2=4$ at the $q=2$ Nishimori point — and with the numerical estimates ($\eta=0.180(1)$ at the $q=2$ Nishimori point; $\eta\approx 0.17$-$0.22$ at $q=3$; the conflicting zero-temperature estimates $\eta\approx 0.13$-$0.18$).

References

1. G. Delfino and E. Tartaglia, 'On superuniversality in the q-state Potts model with quenched disorder', J. Stat. Mech. Theory Exp. (2017) 123303, DOI: 10.1088/1742-5468/aa9bad, arXiv:1709.00364, https://arxiv.org/abs/1709.00364
2. G. Delfino, 'Exact results for quenched bond randomness at criticality', Phys. Rev. Lett. 118 (2017) 250601, DOI: 10.1103/PhysRevLett.118.250601, arXiv:1701.01816, https://arxiv.org/abs/1701.01816
3. G. Delfino and E. Tartaglia, 'Classifying Potts critical lines', Phys. Rev. E 96 (2017) 042137, DOI: 10.1103/PhysRevE.96.042137, https://doi.org/10.1103/PhysRevE.96.042137
4. G. Delfino, 'Critical exponents at the Nishimori point', J. Stat. Mech. Theory Exp. (2025) 043203, DOI: 10.1088/1742-5468/adc4ce, arXiv:2412.07383, https://arxiv.org/abs/2412.07383
5. G. Delfino, 'Particles, conformal invariance and criticality in pure and disordered systems', Eur. Phys. J. B 94 (2021) 65, DOI: 10.1140/epjb/s10051-021-00076-0, arXiv:2010.12275, https://arxiv.org/abs/2010.12275
6. G. Delfino and N. Lamsen, 'On the phase diagram of the random bond q-state Potts model', Eur. Phys. J. B 92 (2019) 278, DOI: 10.1140/epjb/e2019-100451-6, arXiv:1909.04445, https://arxiv.org/abs/1909.04445
7. T. Chen, E. Guo, W. Zhang, P. Zhang and Y. Deng, 'Tensor network Monte Carlo simulations for the two-dimensional random-bond Ising model', Phys. Rev. B 111 (2025) 094201, DOI: 10.1103/PhysRevB.111.094201, arXiv:2409.06538, https://arxiv.org/abs/2409.06538
8. G. Delfino and N. Lamsen, 'Exact results for the O(N) model with quenched disorder', JHEP 04 (2018) 077, DOI: 10.1007/JHEP04(2018)077, arXiv:1802.07151, https://arxiv.org/abs/1802.07151
9. Y. Makoudi and G. Delfino, 'Criticality in the disordered N-color Ashkin-Teller model', arXiv:2605.29810 (2026), https://arxiv.org/abs/2605.29810
10. G. Delfino, 'Nonuniversality in random criticality', J. Stat. Mech. Theory Exp. (2025) 013211, DOI: 10.1088/1742-5468/ada695, arXiv:2408.04548, https://arxiv.org/abs/2408.04548
11. G. Delfino, 'Exact results for spin glass criticality', J. Stat. Mech. Theory Exp. (2025) 063204, DOI: 10.1088/1742-5468/ade135, https://doi.org/10.1088/1742-5468/ade135
12. G. Delfino, 'On the nature of the spin glass transition', J. Stat. Mech. Theory Exp. (2026) 043201, DOI: 10.1088/1742-5468/ae5a20, arXiv:2511.20163, https://arxiv.org/abs/2511.20163
13. N. G. Fytas, J. Zierenberg, P. E. Theodorakis, M. Weigel, W. Janke and A. Malakis, 'Universality from disorder in the random-bond Blume-Capel model', Phys. Rev. E 97 (2018) 040102(R), DOI: 10.1103/PhysRevE.97.040102, arXiv:1712.07703, https://arxiv.org/abs/1712.07703
14. V. Anfray and C. Chatelain, 'Numerical evidence of superuniversality of the two-dimensional and three-dimensional random quantum Potts models', Phys. Rev. B 103 (2021) 174207, DOI: 10.1103/PhysRevB.103.174207, arXiv:2104.02490, https://arxiv.org/abs/2104.02490
15. E. Vatansever, Z. D. Vatansever, P. E. Theodorakis and N. G. Fytas, 'Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field', Phys. Rev. E 102 (2020) 062138, DOI: 10.1103/PhysRevE.102.062138, https://doi.org/10.1103/PhysRevE.102.062138
16. E. S. Sørensen, M. J. P. Gingras and D. A. Huse, 'Non-trivial fixed-point structure of the two-dimensional ±J 3-state Potts ferromagnet/spin glass', Europhys. Lett. 44 (1998) 504, DOI: 10.1209/epl/i1998-00502-1, https://doi.org/10.1209/epl/i1998-00502-1
17. F. Merz and J. T. Chalker, 'Two-dimensional random-bond Ising model, free fermions, and the network model', Phys. Rev. B 65 (2002) 054425, DOI: 10.1103/PhysRevB.65.054425, https://doi.org/10.1103/PhysRevB.65.054425
18. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto and E. Vicari, 'Multicritical Nishimori point in the phase diagram of the ±J Ising model on a square lattice', Phys. Rev. E 77 (2008) 051115, DOI: 10.1103/PhysRevE.77.051115, https://doi.org/10.1103/PhysRevE.77.051115
19. M. Picco, A. Honecker and P. Pujol, 'Strong disorder fixed points in the two-dimensional random-bond Ising model', J. Stat. Mech. (2006) P09006, DOI: 10.1088/1742-5468/2006/09/P09006, https://doi.org/10.1088/1742-5468/2006/09/P09006
20. S. Chen, A. M. Ferrenberg and D. P. Landau, 'Randomness-induced second-order transition in the two-dimensional eight-state Potts model: A Monte Carlo study', Phys. Rev. Lett. 69 (1992) 1213, DOI: 10.1103/PhysRevLett.69.1213, https://doi.org/10.1103/PhysRevLett.69.1213
21. J. Cardy and J. L. Jacobsen, 'Critical behavior of random-bond Potts models', Phys. Rev. Lett. 79 (1997) 4063, DOI: 10.1103/PhysRevLett.79.4063, https://doi.org/10.1103/PhysRevLett.79.4063
22. J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, 'Finite-size scaling and correlation lengths for disordered systems', Phys. Rev. Lett. 57 (1986) 2999, DOI: 10.1103/PhysRevLett.57.2999, https://doi.org/10.1103/PhysRevLett.57.2999
23. H. Nishimori, 'Internal energy, specific heat and correlation function of the bond-random Ising model', Prog. Theor. Phys. 66 (1981) 1169, DOI: 10.1143/PTP.66.1169, https://doi.org/10.1143/PTP.66.1169
24. A. W. W. Ludwig, 'Infinite hierarchies of exponents in a diluted ferromagnet and their interpretation', Nucl. Phys. B 330 (1990) 639, DOI: 10.1016/0550-3213(90)90126-X, https://doi.org/10.1016/0550-3213(90)90126-X
25. V. Dotsenko, M. Picco and P. Pujol, 'Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models', Nucl. Phys. B 455 (1995) 701, DOI: 10.1016/0550-3213(95)00534-Y, https://doi.org/10.1016/0550-3213(95)00534-Y
26. G. Delfino, W. Selke and A. Squarcini, 'Structure of interfaces at phase coexistence. Theory and numerics', J. Stat. Mech. (2018) 053203, DOI: 10.1088/1742-5468/aabbe2, arXiv:1803.04759, https://arxiv.org/abs/1803.04759
27. N. G. Fytas, P. E. Theodorakis and A. Malakis, 'Interfacial adsorption in two-dimensional pure and random-bond Potts models', Phys. Rev. E 95 (2017) 032126, DOI: 10.1103/PhysRevE.95.032126, https://doi.org/10.1103/PhysRevE.95.032126
28. M. Aizenman and J. Wehr, 'Rounding of first-order phase transitions in systems with quenched disorder', Phys. Rev. Lett. 62 (1989) 2503, DOI: 10.1103/PhysRevLett.62.2503, https://doi.org/10.1103/PhysRevLett.62.2503