The rapidities of the NN-state chiral Potts model (CPM) live on a genus N3−2N2+1N^3-2N^2+1 algebraic curve Wk′,kW_{k',k} depending on temperature-like parameters (k′,k)(k',k) with k′2+k2=1k'^2+k^2=1. Roan (2013) showed that all rapidity curves of the NN-state CPM together form a degree-2N2N Fermat hypersurface in P3\mathbb{P}^3 fibered over the parameter space Λ≃P1\Lambda\simeq\mathbb{P}^1, whose automorphism group is the modular-extended group G~N\widetilde{G}_N of order 96N396N^3 with quotient G~N/GN≃PSL2(Z4)≅S4\widetilde{G}_N/G_N\simeq\mathrm{PSL}_2(\mathbb{Z}_4)\cong S_4 acting on the parameter κ′=k′+ik\kappa'=k'+ik through 24 Möbius transformations. The element TT of this quotient is the rapidity identification of the Kramers–Wannier-type duality that Roan (2009) proved for the general NN-state CPM, so the cyclic subgroup ⟨T⟩\langle T\rangle yields known equivalences. The open problem, stated explicitly in the source ('the possibility of other equivalent theories in CPM is currently under investigation'), is to determine which of the 24 modular transformations identify chiral Potts models that are the same statistical-mechanical theory — in the precise sense of transfer-matrix spectral equality under the appropriate relabeling of ZN\mathbb{Z}_N-charge and boundary sectors — thereby classifying the equivalence classes of NN-state CPMs and the exact identities for eigenvalues, eigenvectors, and order parameters that each confirmed equivalence implies. Beyond the cyclic subgroup ⟨T⟩\langle T\rangle generated by the known duality, no resolution is known in the literature to date.

Background

The two-dimensional NN-state chiral Potts model (CPM) is an exactly solvable lattice model of ZN\mathbb{Z}_N-valued spins on a square lattice, generalizing the Ising model (N=2N=2). Its solvability rests on the star-triangle relation (the Yang–Baxter integrability equation for this model), which forces the Boltzmann weights to depend on 'rapidities' — auxiliary variables p=[a,b,c,d]p=[a,b,c,d] lying on an algebraic curve Wk′,k:kaN+k′cN=dN,kbN+k′dN=cN,W_{k',k}:\quad ka^N+k'c^N=d^N,\qquad kb^N+k'd^N=c^N, of genus N3−2N2+1N^3-2N^2+1 (genus 1, an elliptic curve, when N=2N=2; genus 10 when N=3N=3). Here k′,kk',k are temperature-like parameters constrained by k′2+k2=1k'^2+k^2=1, with k′k' small at low temperature. Unlike most solvable models, the CPM lacks the rapidity-difference property (the weights cannot be written as functions of a difference of rapidities on the curve), which is why even the free energy and the order parameter — the latter proved by Baxter (2005) — required special functional-relation methods. For N≥3N\ge 3 a single curve Wk′,kW_{k',k} has a finite automorphism group GNG_N of order 4N34N^3 (Roan, 2005).

Roan (2013) enlarged the picture from one curve to the whole family: the collection of all rapidity curves, including degenerate fibers at κ′∈{0,∞,±1,±i}\kappa'\in\{0,\infty,\pm1,\pm i\}, forms a degree-2N2N Fermat hypersurface (a hypersurface defined by one homogeneous equation whose monomials are pure 2N2N-th powers) in complex projective 3-space, fibered over the parameter space Λ≃P1\Lambda\simeq\mathbb{P}^1 with coordinate κ′=k′+ik\kappa'=k'+ik. For N=2N=2 this is the Fermat quartic K3 surface (a simply connected complex surface with trivial canonical bundle) carrying the classical elliptic fibration. Theorem 3.1 of that paper identifies the automorphism group of this fibration as a modular extension G~N\widetilde{G}_N of GNG_N, of order 96N396N^3, whose quotient G~N/GN≃PSL2(Z4)≅S4\widetilde{G}_N/G_N\simeq\mathrm{PSL}_2(\mathbb{Z}_4)\cong S_4 — the projective special linear group of 2×22\times 2 matrices over Z/4Z\mathbb{Z}/4\mathbb{Z}, of order 24 — acts on the parameter κ′\kappa' by exactly the 24 Möbius transformations that preserve the degenerate set; the explicit list of the induced transformations of (k′,k)(k',k) is display (3.10) of the source. Two generators are T:(k′,k)↦(1/k′, ik/k′)T:(k',k)\mapsto(1/k',\,ik/k') and S:(k′,k)↦(k,k′)S:(k',k)\mapsto(k,k').

The physical anchor is duality. Kramers and Wannier (1941) showed that the planar Ising model at low temperature equals, as a statistical theory, the Ising model at its dual high temperature; Baxter (1989) exhibited an 'inverse' superintegrable chiral Potts model hinting at an analogue. Roan (2009) proved the general-NN Kramers–Wannier-type duality: for the finite inhomogeneous CPM, the transfer matrix (the matrix whose iterates generate the partition function) over the k′k'-curve carrying ZN\mathbb{Z}_N-charge QQ with boundary condition rr has exactly the same eigenvalue spectrum as the transfer matrix over the k′−1k'^{-1}-curve with charge and boundary condition interchanged, (Q∗,r∗)=(r,Q)(Q^*,r^*)=(r,Q), the dual Boltzmann weights being related by a discrete Fourier transform; at N=2N=2 this reduces to the classical Ising duality tanh⁡J∗=e−2J\tanh J^*=e^{-2J}. Roan (2013) then identified this duality with the modular generator TT acting on the rapidity family. What the geometric framework does not settle is the converse direction: it guarantees only that two parameter values could define the same theory when their rapidity curves are isomorphic under G~N\widetilde{G}_N — curve isomorphism is a pre-condition, not a proof of physical equivalence, since the Boltzmann weights and the sector labels must also match. Which of the 24 transformations actually identify equivalent statistical-mechanical theories is the question the source explicitly leaves 'currently under investigation'.

Problem Statement

Determine which NN-state chiral Potts models define the same statistical-mechanical theory, within the modular rapidity-symmetry framework of Roan (2013). Concretely: for each of the 24 transformations M∈G~N/GN≃PSL2(Z4)≅S4M\in\widetilde{G}_N/G_N\simeq\mathrm{PSL}_2(\mathbb{Z}_4)\cong S_4 acting on the temperature-like parameter κ′=k′+ik∈Λ≃P1\kappa'=k'+ik\in\Lambda\simeq\mathbb{P}^1 (the list (3.10) of the source; for instance T:(k′,k)↦(1/k′, ik/k′)T:(k',k)\mapsto(1/k',\,ik/k') and S:(k′,k)↦(k,k′)S:(k',k)\mapsto(k,k')), decide whether the CPM at κ′\kappa' and the CPM at M(κ′)M(\kappa') are equivalent theories. Equivalence is meant in the sense of the correspondence proved for the Kramers–Wannier element TT by Roan (2009): an explicit correspondence of rapidities and Boltzmann weights under which the transfer matrices of the two models on every finite inhomogeneous lattice have coinciding eigenvalue spectra after the appropriate relabeling of the ZN\mathbb{Z}_N-charge and boundary-condition sectors, together with the induced matching of eigenvectors; the further matching of the thermodynamic-limit free energy and of observables such as the order parameter unpacks the source's requirement that the two models 'provide the same theory in statistical mechanics'. The required outcome is the complete classification, for every N≥2N\ge2 (presented, where necessary, as an explicit stratification in NN): the subset of the 24 candidates that yield genuine equivalences — the cyclic subgroup ⟨T⟩\langle T\rangle generated by the known duality is contained in it, and the question is whether any element outside ⟨T⟩\langle T\rangle, such as SS, also does — proved at the level of rigor of Roan (2009) Theorem 3.1 for the positive cases, with rigorously computed distinguishing invariants for the negative cases; the induced partition of the parameter space Λ\Lambda into equivalence classes of NN-state CPMs; and the exact identities for eigenvalues, eigenvectors, and order parameters implied by each confirmed equivalence. These deliverables share one objective — the equivalence classification itself: the spectral and observable identities are the content of each equivalence, and the partition of Λ\Lambda is its restatement, not independent targets.

References

  1. Shi-shyr Roan, Fermat Surface and Group Theory in Symmetry of Rapidity Family in Chiral Potts Model, arXiv:1301.5260 (2013), https://arxiv.org/abs/1301.5260 (primary source)
  2. Shi-shyr Roan, Duality and symmetry in chiral Potts model, Journal of Statistical Mechanics: Theory and Experiment (2009) P08012, arXiv:0905.1924, DOI 10.1088/1742-5468/2009/08/P08012, https://arxiv.org/abs/0905.1924
  3. Shi-shyr Roan, Chiral Potts rapidity curve descended from six-vertex model and symmetry group of rapidities, Journal of Physics A: Mathematical and General 38 (2005) 7483–7499, arXiv:cond-mat/0410011, DOI 10.1088/0305-4470/38/34/003, https://arxiv.org/abs/cond-mat/0410011
  4. H. A. Kramers and G. H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Physical Review 60 (1941) 252–262, DOI 10.1103/PhysRev.60.252, https://doi.org/10.1103/PhysRev.60.252
  5. R. J. Baxter, Superintegrable chiral Potts model: Thermodynamic properties, an 'inverse' model, and a simple associated Hamiltonian, Journal of Statistical Physics 57 (1989) 1–39, DOI 10.1007/BF01023632, https://doi.org/10.1007/BF01023632
  6. R. J. Baxter, The Order Parameter of the Chiral Potts Model, Journal of Statistical Physics 120 (2005) 1–36, DOI 10.1007/s10955-005-5534-3, https://doi.org/10.1007/s10955-005-5534-3
  7. Helen Au-Yang and Jacques H. H. Perk, Spontaneous magnetization of the integrable chiral Potts model, Journal of Physics A: Mathematical and Theoretical 44 (2011) 445005, arXiv:1003.4805, DOI 10.1088/1751-8113/44/44/445005, https://arxiv.org/abs/1003.4805
  8. Helen Au-Yang and Jacques H. H. Perk, About 30 Years of Integrable Chiral Potts Model, Quantum Groups at Roots of Unity and Cyclic Hypergeometric Functions, arXiv:1601.01014 (2016), https://arxiv.org/abs/1601.01014
  9. M. P. Bellon, J.-M. Maillard, G. Rollet and C.-M. Viallet, Deformations of dynamics associated to the chiral Potts model, International Journal of Modern Physics B 6 (1992) 3575–3584, DOI 10.1142/S021797929200164x, https://doi.org/10.1142/S021797929200164x
  10. Seyed Faroogh Moosavian, Masahito Yamazaki and Yehao Zhou, Hyperbolic Monopoles, (Semi-)Holomorphic Chern–Simons Theories, and Generalized Chiral Potts Models, arXiv:2502.17545 (2025), https://arxiv.org/abs/2502.17545
  11. Zhao Zhang, The Yang-Baxter Equation for the Chiral Potts Model and Integrable Parafermions, arXiv:2606.11146 (2026), https://arxiv.org/abs/2606.11146