Complex CFTs of the two-dimensional Potts model for $Q>4$ and complex $Q$: proving the lattice analytic-continuation conjectures

The two-dimensional $Q$-state Potts model has a first-order transition for $Q>4$. Jacobsen and Wiese (2024) constructed a triangular-lattice loop-model realization in which $Q$ is a continuous, indeed complex, parameter: along a self-dual manifold the critical and tricritical fixed points collide at $Q=4$ and re-emerge as a pair of complex fixed points for $Q>4$ and complex $Q$, whose finite-size spectra match the analytic continuation of the exact $Q\le4$ conformal data. A 44-page companion (2026) formalizes this as two conjectures: the continuum limit at the complex fixed point is a complex conformal field theory (CFT), and it is the prescribed analytic continuation of the non-unitary loop-model CFT for $Q\le4$. These are verified numerically in depth (central charge, critical exponents, Kac-module structure, three-point structure constants, defect sectors with $j=2$ and $j=4$ defect lines), with independent numerical confirmation in a non-Hermitian quantum 5-state Potts chain and in boundary-criticality studies. The open problem is the analytic, non-numerical establishment of these conjectures: a first-principles proof that the complex fixed points have continuum limits that are complex CFTs, and that all conformal data (spectrum, null-state/module structure, and OPE coefficients, including defect sectors at general defect number and phase) are given by the analytic continuation of the exact $Q\le4$ results. A disproof exhibiting a sector in which the identification fails in the $L\to\infty$ limit is an admissible alternative answer.

Background

The $Q$-state Potts model assigns each lattice site a spin $\sigma_i\in\{1,\dots,Q\}$ and favors neighboring spins being equal. In two dimensions with ferromagnetic couplings, the thermal transition is continuous for $0\le Q\le4$ and described by a conformal field theory (CFT), but first order for $Q>4$, where exact conformal symmetry is lost (just above $Q=4$ the transition is only weakly first order, so approximately conformal behavior survives over a wide but finite range of scales). In the Fortuin-Kasteleyn (FK) representation the model becomes a sum over bond configurations whose cluster boundaries form completely packed loops; each closed loop carries a weight $\sqrt{Q}$, so the loop reformulation makes $Q$ a continuous parameter (the Temperley-Lieb loop weight), and the transfer matrix acts on spaces of link patterns known as standard modules $\mathcal{W}_{j,\tilde z}$ of the periodic (affine) Temperley-Lieb algebra, where $j$ counts defect lines, i.e. FK cluster boundaries winding around the periodic direction, and $\tilde z$ is a phase factor for those lines. For $Q\le4$ the continuum limit is a non-unitary CFT with central charge $c\le1$ and a discrete spectrum, known from the Coulomb-gas (CG) construction (a solution method representing the scaling limit by a free boson with a background charge) and a decade-long conformal-bootstrap program: its spaces of states are representations of the Virasoro algebra (the symmetry algebra of two-dimensional conformal transformations) together with the global permutation symmetry $S_Q$, organized into Kac modules (families of primary operators with null states, i.e. vanishing descendant states at fixed levels), and its three-point functions are given by a DOZZ-type formula — the exact closed-form three-point formula of Liouville theory at central charge $c\le1$ — a subject on which rigorous probability results now exist for real $Q\le4$.

Gorbenko, Rychkov and Zan proposed the complex-CFT mechanism: when a pair of renormalization-group (RG) fixed points collides, as the critical and tricritical Potts fixed points do at $Q=4$, the merged points leave the real coupling axis and persist as a pair of complex-conjugate fixed points at complex coupling, described by complex CFTs, non-unitary conformal theories whose data (central charge, operator dimensions, operator-product-expansion coefficients) are complex. Observables of the weakly first-order transition and of walking RG flows are then computable by conformal perturbation theory around the complex CFTs. For the two-dimensional Potts model with $Q>4$ these authors predicted the complex-CFT spectrum by analytically continuing the exact $Q\le4$ results, but a microscopic lattice model actually sitting at the complex fixed points was lacking.

Jacobsen and Wiese supplied such a realization: a triangular-lattice Potts model with competing two- and three-spin interactions (Wu-Lin type), equivalently a loop model whose transfer matrix is built from periodic Temperley-Lieb generators. Along a self-dual manifold parametrized by a coupling $z$, the model has, for $Q>4$ and also complex $Q$, a pair of complex fixed points $z_c$; finite-size scaling of transfer-matrix eigenvalues yields the central charge and scaling dimensions, which match the analytic continuation of the exact $Q\le4$ results for $Q=5,10$ and complex values such as $Q=5+2i$ and $8+i$. Their 2026 companion paper formalizes the claim as two precise statements. Conjecture 1: for any $Q$ in the critical domain of the complex plane, the loop model at its self-dual critical point $z_c$ has a continuum limit that is a complex CFT. Conjecture 2: this complex CFT is the analytic continuation, in a prescribed branch of the CG parametrization, of the non-unitary CFT of the critical loop models with $Q\le4$. The companion verifies both numerically in depth, including three-point structure constants (compared with independent results from a non-Hermitian quantum 5-state Potts chain and with the conjectured exact loop-model three-point formula) and non-diagonal operators in the sectors with $j=2$ and $j=4$ defect lines.

What no source provides is an analytic derivation. The probabilistic machinery that recently put the real-$Q$ loop-model structure constants on rigorous footing (conformal loop ensembles, Schramm-Loewner evolution, Liouville quantum gravity) depends on positive Boltzmann weights and does not extend to complex couplings; the CG and bootstrap arguments yield formulas, not constructions; and the transfer-matrix evidence, however extensive, remains numerical. Establishing, or refuting, the analytic-continuation conjectures for the Potts complex CFTs is the open problem.

Problem Statement

Prove, or disprove, the analytic-continuation conjectures for the two-dimensional $Q$-state Potts model with $Q>4$ and complex $Q$, as precisely formulated by Jacobsen and Wiese:

(i) Existence (Conjecture 1 of arXiv:2606.18125). For every $Q$ in the critical domain (the region of the complex-$Q$ plane inside the separatrix $\operatorname{Re}h_{3,1}=1$, in which one of the two fixed points is stable, as defined in arXiv:2606.18125, Sec. II.6 and Fig. 4; this excludes real $Q>4$, whose two fixed points are both unstable), the triangular-lattice FK loop model with two- and three-spin interactions, at its self-dual critical point $z_c(Q)$, has a continuum limit that is a complex conformal field theory.

(ii) Identification (Conjecture 2 of arXiv:2606.18125). For any $Q\in\mathbb{C}$, that complex CFT is the analytic continuation, in the branch prescribed by the Coulomb-gas parametrization, of the non-unitary CFT of the critical loop models with $Q\le4$: its complete conformal data, including the central charge, the full operator spectrum with its Kac-module (null-state) organization and $S_Q$-symmetric module decomposition, and the three-point structure constants and OPE coefficients, coincide with the analytic continuation of the exact $Q\le4$ results, including the exact three-point formula conjectured for real $Q\le4$ (arXiv:2510.04701).

(iii) Defect sectors. The identification extends to sectors with defect lines: the spectra and module decompositions of the standard modules $\mathcal{W}_{j,\tilde z}$ with $j\neq0$ and arbitrary phase $\tilde z$ in the complex regime follow from the same analytic continuation (the existing numerical treatment covers only $j=2$ and $j=4$ at trivial phase).

A complete answer derives these statements from first principles, for example by a controlled construction of the scaling limit at complex weights, an exact conformal-bootstrap solution, or an equivalent analytic argument; numerical transfer-matrix agreement, however extensive, is supporting evidence and not a solution. A valid disproof exhibits a sector, observable, or parameter region in which the lattice data at the complex fixed point demonstrably differ from the prescribed analytic continuation in the $L\to\infty$ limit (a finite-size anomaly alone does not suffice); a disproof of part (i) may instead show that for some $Q$ in the critical domain the self-dual model has no scaling limit at $z_c(Q)$, or a scaling limit that is not a complex conformal field theory. Refuting any single one of (i)-(iii) resolves the task.

References

1. J. L. Jacobsen and K. J. Wiese, 'Lattice Realization of Complex Conformal Field Theories: Two-Dimensional Potts Model with Q>4 States' (arXiv title: 'Lattice realization of complex CFTs: Two-dimensional Potts model with Q>4 states'), Phys. Rev. Lett. 133 (2024) 077101, DOI: 10.1103/PhysRevLett.133.077101, https://arxiv.org/abs/2402.10732 (source paper)
2. J. L. Jacobsen and K. J. Wiese, 'Making complex CFTs real: The two-dimensional Potts model for Q>4 and complex Q', arXiv:2606.18125 (2026), https://arxiv.org/abs/2606.18125 (companion paper, 44 pages; states Conjectures 1-2)
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