The three-dimensional (3D) three-state Potts transition is believed from decades of lattice simulations to be weakly first order, yet a fuzzy-sphere exact-diagonalization/DMRG study recently uncovered an approximately conformal energy spectrum with the subleading S3S_3-singlet scaling dimension Δϵ′\Delta_{\epsilon'} drifting around the relevance threshold 33, which the authors interpret as pseudo-criticality driven by a nearby fixed point just outside the studied Hermitian parameter space. Two linked questions remain open: (i) is ϵ′\epsilon' relevant (Δϵ′<3\Delta_{\epsilon'}<3), marginal, or irrelevant in the thermodynamic limit, and (ii) which fixed points generate the pseudo-criticality — a pair of complex-conjugate nonunitary fixed points (a complex conformal field theory) reachable only in a non-Hermitian extension of coupling space, or a unitary multicritical (tricritical) fixed point reachable by tuning an additional Hermitian singlet coupling? Resolving both facets would settle the renormalization-group mechanism of this canonical weakly first-order transition.

Background

The QQ-state Potts model generalizes the Ising model: each site carries a spin with QQ possible values, and ferromagnetic nearest-neighbor interactions favor alignment. For Q=3Q=3 the global symmetry is S3S_3, the permutation group of the three spin states. In two dimensions the model is exactly solvable: its transition is continuous for Q≤4Q\le 4 and first order for Q>4Q>4, with the marginal case Qc(D=2)=4Q_c(D=2)=4 understood as the collision and annihilation of the critical and tricritical renormalization-group (RG) fixed points; a tricritical point is the multicritical point at which a line of continuous transitions terminates. In three dimensions (equivalently (2+1)(2+1)-dimensional quantum systems) no exact solution exists. Monte Carlo and tensor-network studies since 1979 consistently find a weakly first-order transition for the 3D 3-state Potts model, and estimates of the largest QQ admitting a continuous transition cluster around Qc(3)≈2.1Q_c(3)\approx 2.1–2.72.7, but the mechanism — the RG structure behind the apparent scaling behavior at a first-order transition — was not identified.

A first-order transition has finite correlation length and thus no exact scale invariance; the puzzle is that weakly first-order transitions can show approximate scaling over a wide range. Gorbenko, Rychkov, and Zan proposed that when two real fixed points collide and annihilate, a pair of complex-conjugate fixed points survives in complexified coupling space, described by a complex conformal field theory (CFT): a CFT whose scaling dimensions and operator product expansion (OPE) coefficients — the constants in the expansion of a product of two local operators as a sum of local operators — are complex. RG flow passing between these complex fixed points is exponentially slow ('walking'), producing pseudo-critical scaling almost indistinguishable from a continuous transition. This picture is verified in the 2D Potts model for Q>4Q>4, where complex fixed points have been identified analytically, numerically via non-Hermitian lattice models, and most recently through analytic continuation of exact loop-model data. Whether an analogous complex CFT governs any genuinely three-dimensional transition is unknown.

The numerical conformal bootstrap — deriving consistency bounds on CFT data from the crossing symmetry of four-point functions, i.e. the requirement that correlators computed in different pairings of the four operators agree — has been applied to the 3-state Potts model in general spacetime dimension dd by Chester and Su using the 'navigator function' method (a bootstrap technique that measures the distance from a trial point in theory space to the nearest consistent CFT): they find the critical and tricritical S3S_3 fixed points approach each other as dd increases and merge near d≈2.5d\approx 2.5–2.62.6, so that no unitary S3S_3-symmetric CFT is expected at d=3d=3, consistent with the complex-CFT scenario but not locating any complex fixed point.

The fuzzy-sphere regularization provides a microscopic route to 3D CFT data: interacting fermions confined to the lowest Landau level on a sphere pierced by a monopole flux realize a (2+1)-D quantum Hamiltonian whose energy levels at criticality organize into conformal towers via the state-operator correspondence, the identification of energy eigenstates on the sphere with local scaling operators of the CFT (demonstrated quantitatively for the 3D Ising transition). Applying this to Q=3Q=3, Yang, Yue, Tang, Han, Zhu, and Chen (Phys. Rev. B 112, 024436 (2025)) constructed an S3S_3-symmetric three-flavor fuzzy-sphere model with a direct paramagnet–ferromagnet transition line, studied by exact diagonalization and density matrix renormalization group (DMRG, a tensor-network method for one-dimensional-like entanglement structure) up to No≤14N_o\le 14 Landau orbitals. At the transition the spectrum shows a nearly integer-spaced tower structure (approximate conformal symmetry), and the scaling dimension of the subleading S3S_3-singlet primary operator ϵ′\epsilon' — a primary is the head of a conformal tower; in a 3D CFT an operator is relevant if its scaling dimension is below the spacetime dimension 33 — flows with system size and drifts around the marginal value ∼3\sim 3, so its thermodynamic-limit status cannot be decided. The authors argue the transition is pseudo-critical, caused either by a unitary fixed point outside their two-parameter Hermitian space, reachable by tuning an additional coupling gg (a multicritical/tricritical scenario), or by a complex-conjugate pair of nonunitary fixed points in a non-Hermitian extension that might locate at couplings gϵ=0g_\epsilon=0, gϵ′=±ia(Q−Qc(D))g_{\epsilon'}=\pm i\sqrt{a(Q-Q_c(D))} if the RG equation takes the same form as in the 2D cases (the complex-CFT scenario). Yang, Yue, Tang, Han, Zhu, and Chen identify locating the putative fixed point(s) as the challenging point of this study and propose, as follow-up directions, larger-system fuzzy-sphere calculations, a non-Hermitian version of the model, a dedicated 3D S3S_3 conformal bootstrap, and lattice simulations with competing nearest-neighbor ferromagnetic (J1J_{1}) and further-neighbor antiferromagnetic (J2J_{2}) couplings.

Problem Statement

Determine the renormalization-group structure that governs the pseudo-critical, weakly first-order transition of the three-dimensional three-state Potts universality class, expressed as thermodynamic-limit conformal data. The single underlying objective — explaining this transition's approximate conformality in RG terms — decomposes into two facets:

(1) The scaling dimension of the subleading S3S_3-singlet primary ϵ′\epsilon' in the thermodynamic limit: establish whether Δϵ′<3\Delta_{\epsilon'}<3 (relevant), =3=3 (marginal), or >3>3 (irrelevant), where the comparison is made on the real part of Δϵ′\Delta_{\epsilon'} if the governing fixed point is complex, replacing the current fit-sensitive finite-size values that drift around ∼3\sim 3 for No≤14N_o\le 14 orbitals with a controlled extrapolation, and supply the extrapolated conformal data of the low-lying S3S_3-symmetry primaries with quantified uncertainties.

(2) The identity and location of the fixed point(s) responsible for the pseudo-critical behavior: decide which of the two scenarios holds — the scenario of a pair of complex-conjugate nonunitary fixed points in a non-Hermitian extension of the coupling space (a 3D complex CFT), which might locate along the RG trajectory at gϵ=0g_\epsilon=0, gϵ′=±ia(Q−Qc(D))g_{\epsilon'}=\pm i\sqrt{a(Q-Q_c(D))} if the RG equation takes the same form as in the 2D cases, and the scenario of a unitary multicritical (tricritical) S3S_3 fixed point reachable inside an enlarged Hermitian parameter space by tuning an additional singlet coupling gg. An answer should locate the relevant fixed point(s) in the appropriate parameter space and report their conformal data (complex or real, as the case may be).

Facet (2) explains the pseudo-criticality that facet (1) quantifies.

References

  1. Shuai Yang, Yan-Guang Yue, Yin Tang, Chao Han, W. Zhu, Yan Chen, 'Microscopic study of the three-dimensional Potts phase transition via fuzzy sphere regularization', Phys. Rev. B 112, 024436 (2025), DOI: 10.1103/x1qn-x6xb, arXiv:2501.14320, https://arxiv.org/abs/2501.14320
  2. Shai M. Chester, Ning Su, 'Upper critical dimension of the 3-state Potts model', Phys. Rev. D 111, L121701 (2025), DOI: 10.1103/vqss-l6f9, arXiv:2210.09091, https://arxiv.org/abs/2210.09091
  3. Victor Gorbenko, Slava Rychkov, Bernardo Zan, 'Walking, Weak first-order transitions, and Complex CFTs', J. High Energ. Phys. 2018, 108 (2018), DOI: 10.1007/JHEP10(2018)108, arXiv:1807.11512, https://arxiv.org/abs/1807.11512
  4. Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, Yin-Chen He, 'Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization', Phys. Rev. X 13, 021009 (2023), DOI: 10.1103/PhysRevX.13.021009, arXiv:2210.13482, https://arxiv.org/abs/2210.13482
  5. Yin Tang, Han Ma, Qicheng Tang, Yin-Chen He, W. Zhu, 'Reclaiming the Lost Conformality in a Non-Hermitian Quantum 5-State Potts Model', Phys. Rev. Lett. 133, 076504 (2024), DOI: 10.1103/PhysRevLett.133.076504, arXiv:2403.00852, https://arxiv.org/abs/2403.00852
  6. Yin-Chen He, W. Zhu, 'A Fuzzy Sphere Journey in Critical Phenomena', Annu. Rev. Condens. Matter Phys. 17, 1–25 (2026), DOI: 10.1146/annurev-conmatphys-031424-020256, arXiv:2607.01310, https://arxiv.org/abs/2607.01310
  7. Jin-Xiang Hao, Zheng Zhu, Yang Qi, 'Multi-target density matrix renormalization group for 3D CFTs on the fuzzy sphere', arXiv:2601.18648 (2026), https://arxiv.org/abs/2601.18648
  8. Ruihua Fan, Junkai Dong, Ashvin Vishwanath, 'Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere', arXiv:2505.06342 (2025), https://arxiv.org/abs/2505.06342
  9. Jesper Lykke Jacobsen, Kay Joerg Wiese, 'Making complex CFTs real: The two-dimensional Potts model for Q>4Q>4 and complex QQ', arXiv:2606.18125 (2026), https://arxiv.org/abs/2606.18125