The $\mathrm{SLE}(4,\sqrt{g}-1,\sqrt{g}-1)$ conjecture for Ashkin–Teller parafermionic interfaces and its Schramm-formula test

Ikhlef and Rajabpour (2011) constructed a discrete-holomorphic parafermion (a fractional-spin lattice observable satisfying discrete Cauchy–Riemann equations) for the square-lattice Ashkin–Teller (AT) model along its entire critical line and conjectured that the interface created by inserting this parafermion at two boundary points converges in the scaling limit to $\mathrm{SLE}(4,\sqrt{g}-1,\sqrt{g}-1)$, where $g=4(\pi-\lambda)/\pi$ is the coupling of the associated Gaussian (solid-on-solid height) field and $\lambda$ parametrizes the critical line. Proving or refuting this conjecture is open. The authors also identify the missing tool: no closed-form Schramm (left-passage) formula — the probability that the curve passes to the left of a given point — exists for $\mathrm{SLE}(\kappa,\rho_1,\rho_2)$ with force points, although $\kappa=4$ should be tractable through the coupling with the Gaussian free field. The problem is to settle the continuum law of the critical AT parafermionic interface — by rigorous proof or by refutation; deriving the force-point Schramm formula and comparing it with lattice measurements can corroborate or falsify a candidate law, but such evidence cannot by itself establish the scaling limit. The authors name a supporting program: Monte-Carlo and martingale measurements of interface statistics, investigation of further discrete-holomorphic parafermions that may exist in the AT model, and extension of the boundary conformal field theory analysis from the decoupled point $\lambda=\pi/2$ to the full critical line.

Background

The Ashkin–Teller (AT) model places two Ising-like spins $\sigma_j=\pm1$ and $\tau_j=\pm1$ on each site $j$ of the square lattice, with Boltzmann weight per edge $\exp[\beta(\sigma_i\sigma_j+\tau_i\tau_j)+\alpha\,\sigma_i\sigma_j\tau_i\tau_j]$, so that it interpolates between two independent Ising models ($\alpha=0$) and the four-state Potts model ($\alpha=\beta$). Unlike the Ising model, it has a whole critical (self-dual) line: along it the continuum theory is a conformal field theory of central charge $c=1$ (a compactified free boson, in $\mathbb{Z}_2$-orbifold form, i.e. the boson quotiented by the identification $\varphi\equiv-\varphi$) whose critical exponents vary continuously. A convenient parametrization uses an angle $\lambda$ running from the four-state Potts point ($\lambda=0$) through the Fateev–Zamolodchikov $\mathbb{Z}_4$-parafermion point ($\lambda=\pi/4$; a critical point whose order parameters include fractional-spin parafermions) and the decoupled point of two independent Ising models ($\lambda=\pi/2$) to the XY endpoint ($\lambda=3\pi/4$).

Schramm–Loewner evolution (SLE) describes scaling limits of critical interfaces: a random curve in a domain is encoded by a Loewner evolution driven by a one-dimensional Brownian motion with diffusivity $\kappa$. $\mathrm{SLE}(\kappa,\rho_1,\rho_2)$ is the variant with force points: extra marked boundary points enter the driving dynamics through additional drift terms proportional to $\rho_i/(W_t-V_t)$, where $W_t$ is the driving Brownian motion and $V_t$ the force-point position, so that the curve feels boundary degrees of freedom. For Ising, percolation and FK-Ising (the Fortuin–Kasteleyn random-cluster representation of the Ising model), convergence of critical interfaces to SLE has been proved rigorously via discrete-holomorphic observables (lattice observables satisfying discrete Cauchy–Riemann equations); for the coupled AT model no such result exists.

Ikhlef and Rajabpour mapped suitable AT interfaces to the $O(n=1)$ loop model (a model of non-crossing lattice loops, each carrying weight $n=1$; the mapping goes through the six-vertex, i.e. ice-type vertex, representation of the AT model) and thereby exhibited a discrete-holomorphic parafermion $\psi$ of fractional spin $s=1-\lambda/\pi$, valid along the whole critical line. In continuum language $\psi\sim e^{i\theta}\times\sigma\times\mu_{\tau'}$, where $e^{i\theta}$ accumulates the winding angle of the path, $\sigma$ is a spin operator, and $\mu_{\tau'}$ creates a domain wall of the variable $\tau'=\sigma\tau$. Inserting $\psi$ at two boundary points $a,b$ forces an interface $\gamma$ from $a$ to $b$ (boundary conditions BC1: $\sigma$ free on the boundary, $\tau'=+1$ on one boundary arc and $-1$ on the other; a variant BC2 lets $\gamma$ end anywhere on a prescribed boundary arc). The six-vertex model can equivalently be written as a solid-on-solid (SOS) height model, i.e. an integer-valued height field on the faces of the lattice, which renormalizes to a Gaussian free field (a massless Gaussian scalar field) with action $\frac{g}{4\pi}\int|\nabla\varphi|^2d^2x$ and coupling $g=4(\pi-\lambda)/\pi$. Schramm and Sheffield proved that contour lines of the Gaussian free field with boundary heights $-a$ on one arc and $b$ on the other are exactly $\mathrm{SLE}(4,a/\varphi_*-1,b/\varphi_*-1)$ for a universal height $\varphi_*$. Comparing the height gap $\varphi_1=\varphi_2=\pi/2$ inserted by the parafermion with $\varphi_*=\pi/\sqrt{4g}$ yields the conjecture: the continuum limit of $\gamma$ is $\mathrm{SLE}(4,\sqrt{g}-1,\sqrt{g}-1)$, with $\sqrt{g}-1$ varying along the line — it equals $0$ at the XY endpoint, $\sqrt{2}-1$ at the decoupled point, and $1$ at the four-state Potts endpoint. A standard observable for testing an SLE identification is the left-passage probability — the probability that the curve passes to the left of a given interior point — known in closed form (Schramm's formula) for chordal $\mathrm{SLE}(\kappa)$ (Schramm, 2001) but not for $\mathrm{SLE}(\kappa,\rho_1,\rho_2)$.

Problem Statement

Determine the continuum probability law of the critical Ashkin-Teller parafermionic interface — the curve $\gamma$ created by inserting the discrete-holomorphic parafermion at two boundary points, under boundary conditions BC1 ($\sigma$ free, $\tau'=+1$ on one boundary arc, $-1$ on the other) or BC2 ($\gamma$ free to end on a prescribed boundary arc) — and in particular prove or refute the conjecture of Ikhlef and Rajabpour that, for each point of the critical line parametrized by $\lambda$ (equivalently by the Gaussian coupling $g=4(\pi-\lambda)/\pi$), the scaling limit of $\gamma$ is $\mathrm{SLE}(4,\sqrt{g}-1,\sqrt{g}-1)$.

The single scientific objective is establishing (or refuting) this identification. The authors name the following supporting open directions, all left open by the source:

1. Derive the closed-form Schramm (left-passage) formula for $\mathrm{SLE}(\kappa,\rho_1,\rho_2)$ with force points — at least in the case $\kappa=4$, where the Gaussian-free-field coupling of SLE$_4$ (imaginary geometry, the framework realizing SLE curves as level lines of the field) should make the computation tractable. This is the missing quantitative prediction (e.g. refined left-right hitting probabilities) against which the conjecture can be tested.

2. Perform the corresponding lattice measurements — Monte-Carlo simulation and/or transfer-matrix computation of the left-passage probability and other interface statistics and martingale observables (quantities whose conditional expectation remains constant as the interface is progressively revealed) of the critical AT model, as was done for Ising — and compare them with the predicted $\mathrm{SLE}(4,\sqrt{g}-1,\sqrt{g}-1)$ law at several points of the critical line, so as to test the conjecture.

3. Investigate whether other discrete-holomorphic parafermions exist in the AT model along the critical line (the known one arises from the $O(n=1)$ mapping; the source states there may be others), together with the boundary conditions and local operators they correspond to.

4. Extend the boundary conformal field theory analysis — the null-vector equations, i.e. the differential equations satisfied by correlation functions of degenerate fields, and precise identification of the boundary operators, established at the decoupled point $\lambda=\pi/2$ through the $\mathbb{Z}_2$-orbifold theory of Oshikawa and Affleck — to the full critical line, relating the AT boundary fields to the SLE-type martingales and placing the height-gap argument on firm footing.

Directions 1 and 2 supply the quantitative prediction and the lattice measurements against which the conjecture can be tested, but agreement of such measurements with the predicted law cannot by itself settle the scaling limit; 3 and 4 supply the observable toolkit and the continuum identification on which a rigorous proof would rest.

References

1. Y. Ikhlef and M. A. Rajabpour, 'Discrete holomorphic parafermions in the Ashkin-Teller model and SLE', J. Phys. A 44 (2011) 042001. DOI: 10.1088/1751-8113/44/4/042001; arXiv:1009.3374. https://arxiv.org/abs/1009.3374
2. Y. Ikhlef and M. A. Rajabpour, 'Spin interfaces in the Ashkin-Teller model and SLE', J. Stat. Mech. (2012) P01012. DOI: 10.1088/1742-5468/2012/01/P01012; arXiv:1111.3197. https://arxiv.org/abs/1111.3197
3. M. Picco and R. Santachiara, 'Critical interfaces of the Ashkin-Teller model at the parafermionic point', J. Stat. Mech. (2010) P07027. DOI: 10.1088/1742-5468/2010/07/P07027; arXiv:1005.0493. https://arxiv.org/abs/1005.0493
4. M. Picco and R. Santachiara, 'Critical interfaces and duality in the Ashkin Teller model', Phys. Rev. E 83 (2011) 061124. DOI: 10.1103/PhysRevE.83.061124; arXiv:1011.1159. https://arxiv.org/abs/1011.1159
5. M. Caselle, S. Lottini and M. A. Rajabpour, 'Critical domain walls in the Ashkin-Teller model', J. Stat. Mech. (2011) P02039. DOI: 10.1088/1742-5468/2011/02/P02039; arXiv:0907.5094. https://arxiv.org/abs/0907.5094
6. D. B. Wilson, 'XOR-Ising Loops and the Gaussian Free Field', arXiv:1102.3782 (2011). https://arxiv.org/abs/1102.3782
7. M. N. Najafi, 'Left Passage Probability of SLE(kappa,rho)', arXiv:1205.6851 (2012). https://arxiv.org/abs/1205.6851
8. Y. Fukusumi, M. Picco and R. Santachiara, 'Spin interfaces and crossing probabilities of spin clusters in parafermionic models', arXiv:2006.09850 (2020). https://arxiv.org/abs/2006.09850
9. M. Tanhayi-Ahari and S. Rouhani, 'Discrete Holomorphic Parafermions in the Eight Vertex Model', arXiv:1209.4253 (2012). https://arxiv.org/abs/1209.4253
10. Hong-Bin Chen and Jiaming Xia, 'Multi-point functions of a full-plane two-state fuzzy 4-Potts model', arXiv:2605.31374 (2026). https://arxiv.org/abs/2605.31374
11. Tomás Alcalde López and Avelio Sepúlveda, 'Excursion decomposition of the XOR-Ising model', arXiv:2602.06011 (2026). https://arxiv.org/abs/2602.06011
12. Juhan Aru and Titus Lupu, 'Renormalised two-point functions of CLE_4 gaskets', arXiv:2604.15146 (2026). https://arxiv.org/abs/2604.15146
13. O. Schramm and S. Sheffield, 'Contour lines of the two-dimensional discrete Gaussian free field', Acta Mathematica 202 (2009) 21-137. DOI: 10.1007/s11511-009-0034-y; arXiv:math/0605337. https://arxiv.org/abs/math/0605337
14. C. Hagendorf, D. Bernard and M. Bauer, 'The Gaussian free field and SLE(4) on doubly connected domains', J. Stat. Phys. 140 (2010) 1-26. DOI: 10.1007/s10955-010-9980-1; arXiv:1001.4501. https://arxiv.org/abs/1001.4501
15. J. Lenells and F. Viklund, 'Schramm's formula and the Green's function for multiple SLE', J. Stat. Phys. 176 (2019) 873-931. DOI: 10.1007/s10955-019-02325-0; arXiv:1701.03698. https://arxiv.org/abs/1701.03698
16. M. Khristoforov, M. Skopenkov and S. Smirnov, 'A generalization of Cardy's and Schramm's formulae', Commun. Math. Phys. 406, 105 (2025). DOI: 10.1007/s00220-025-05255-z; arXiv:2305.12368. https://arxiv.org/abs/2305.12368
17. O. Schramm, 'A percolation formula', Electron. Commun. Probab. 6 (2001). DOI: 10.1214/ecp.v6-1041. https://doi.org/10.1214/ecp.v6-1041
18. M. Oshikawa and I. Affleck, 'Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line', Nucl. Phys. B 495 (1997) 533-582. DOI: 10.1016/S0550-3213(97)00219-8; arXiv:cond-mat/9612187. https://arxiv.org/abs/cond-mat/9612187