Discrete-spin solutions of the star-triangle relation from root-of-unity limits beyond the $\mathbb{Z}_N$-symmetric family

The root-of-unity limit of the lens elliptic gamma function solution of the star-triangle relation (Kels and Yamazaki, 2018) converts an integrable lattice model with a continuous spin and a $\mathbb{Z}_r$-valued discrete spin into a model with purely discrete $\mathbb{Z}_{rN}$-valued spins, reproducing the Kashiwara-Miwa, chiral Potts, and Fateev-Zamolodchikov models, with the saddle-point equations identified with three-leg forms of the ABS quad equations $Q4$ and $Q3_{(\delta=0)}$. The open problem is to extend this construction beyond the known $\mathbb{Z}_N$-symmetric family: find non-singular solutions of the $Q3_{(\delta=1)}$ three-leg equation yielding a hyperbolic discrete-spin star-triangle relation with broken $\mathbb{Z}_N$ symmetry (no cases known); determine, via a suitable parametrization of $Q4$, whether elliptic discrete-spin solutions other than the Kashiwara-Miwa model exist; work out the root-of-unity limit of hyperbolic solutions whose spin integrals run over the whole real line, possibly producing models with unbounded integer-valued spins; and work out the root-of-unity limit of multi-spin solutions, including the simplest $r=1$ case. All of these directions remain open as of 2026; partial progress exists only on adjacent fronts (a hyperbolic-level lens algebra in 2025, and independent integer-spin star-triangle solutions in 2023).

Background

An Ising-type (edge-interaction) lattice model of statistical mechanics is specified by Boltzmann weights $W(\sigma_i,\sigma_j)$ and $V(\sigma_i,\sigma_j)$ attached to the edges of a triangular lattice; the weights give the amplitude of each local spin configuration. The model is exactly solvable if the weights satisfy the star-triangle relation (STR), which is the form the Yang-Baxter equation (the fundamental consistency condition for exactly solvable two-dimensional models) takes for such models: summing the product of three $W$-weights over the spin at the central site equals a product of three $V$-weights,

$\sum_{\sigma_0} W_{12}(\sigma_1,\sigma_0)\,W_{23}(\sigma_2,\sigma_0)\,W_{31}(\sigma_3,\sigma_0) \;=\; \mathcal{R}\,V_{13}(\sigma_1,\sigma_3)\,V_{32}(\sigma_2,\sigma_3)\,V_{21}(\sigma_1,\sigma_2).$

The STR guarantees commuting transfer matrices and infinitely many conserved quantities. Solutions with discrete spins $\sigma\in\mathbb{Z}_N$ are rare: the classical families are the Kashiwara-Miwa model (elliptic level) and the chiral Potts model and its self-dual degeneration, the Fateev-Zamolodchikov model (trigonometric level). Their weights depend on rapidity variables constrained to lie on an algebraic curve — of genus greater than one for $N>2$ in the chiral Potts case — rather than on differences of rapidities.

A separate route to STR solutions comes from hypergeometric integrals: the univariate elliptic beta integral of Spiridonov is equivalent to the 'master solution' of the STR, an integral model with continuous spins whose weights are built from the elliptic gamma function (an infinite double product generalizing the q-Pochhammer symbol). Kels (2015) introduced the lens elliptic gamma function, a combination of two ordinary elliptic gamma functions depending on an integer $m\in\{0,\dots,r-1\}$, which yields STR solutions with a continuous spin together with a discrete $\mathbb{Z}_r$ spin; through the gauge/YBE correspondence these arise as supersymmetric lens indices of four-dimensional quiver gauge theories on $S^1\times S^3/\mathbb{Z}_r$.

Kels and Yamazaki (2018) analyzed the singular limit in which one elliptic nome (a modulus parameter of the elliptic functions) is sent to a root of unity, $q\to e^{\pi i/(rN)}$ with $N$ a new positive integer, so that $q$ becomes a primitive $2rN$-th root of unity. The weights diverge, but an asymptotic expansion of the logarithm of the STR yields, at leading order, saddle-point equations for the continuous spin components, and at subleading order a new STR for a model with purely discrete spins valued in $\mathbb{Z}_{rN}$: the original $\mathbb{Z}_r$ spin and the periodicity introduced by the root of unity fuse into a single $\mathbb{Z}_{rN}$ spin, essentially independently of $r$. The saddle-point equations turn out to be three-leg forms (a rewriting of a lattice equation on an elementary square as a product of two-point 'leg' factors) of quad equations from the Adler-Bobenko-Suris (ABS) classification of 3D-consistent equations — lattice equations on a square that admit consistent extensions to a cube: $Q4$ at the elliptic level and $Q3_{(\delta=0)}$ at the trigonometric level. The construction is thus a path-integral quantization of these classical discrete integrable equations. Particular solutions of the saddle-point equations recover the known discrete-spin STR solutions: Kashiwara-Miwa, chiral Potts, and Fateev-Zamolodchikov. For the trigonometric case Kels and Yamazaki also proved a uniqueness statement: if the rapidity curve is required to be a rational relation $Y=F(X)$ between the two parametrizing variables, the saddle-point constraint forces $F(X)=X/(CX-1)$, which is precisely the chiral Potts curve.

Everything produced by this mechanism so far has weights invariant under simultaneous $\mathbb{Z}_{rN}$ translations of the spins, reducing to the three classical families. In this task the label '$\mathbb{Z}_N$ symmetry' denotes the invariance of the discrete-spin Boltzmann weights under a simultaneous shift of all spins — the standard spin-shift symmetry of the chiral Potts and Kashiwara-Miwa weights, of which the simultaneous $\mathbb{Z}_{rN}$ translations noted above are the general-$r$ form; a target model with 'broken $\mathbb{Z}_N$ symmetry' is one whose Boltzmann weights are not invariant under any nontrivial such simultaneous shift. The source paper states explicitly that a hyperbolic discrete-spin STR with broken $\mathbb{Z}_N$ symmetry has no known cases; that the known solutions of the $\delta=0$ three-leg equation fail for the $\delta=1$ member of $Q3$ (either no solution or singular weights); that the elliptic-level question — whether discrete-spin elliptic STRs other than Kashiwara-Miwa exist, via a suitable $Q4$ parametrization — is untouched; that root-of-unity limits of hyperbolic solutions, whose spin integrals run over the whole real line and may therefore discretize to unbounded integer spins valued in $\mathbb{Z}$ rather than $\mathbb{Z}_{rN}$, are unworked; and that even the root-of-unity limit of multi-spin (star-star type) solutions is unknown, including in the simplest $r=1$ case.

Problem Statement

The problem is to extend the root-of-unity discretization mechanism of Kels and Yamazaki beyond the known $\mathbb{Z}_N$-symmetric family of discrete-spin star-triangle relations, and to determine which new discrete-spin integrable lattice models it can produce. The concrete instances identified by the source are:

(1) Hyperbolic level. Find a solution of the three-leg (saddle-point) equation of the ABS equation $Q3_{(\delta=1)}$ — for example by generalizing the uniqueness analysis that forced the chiral Potts curve in the $\delta=0$ case — that leads to non-singular Boltzmann weights for a discrete-spin solution of the star-triangle relation with broken $\mathbb{Z}_N$ symmetry. No such solution is known.

(2) Elliptic level. Find a parametrization of $Q4$ suitable for the same construction, and thereby determine whether there exist elliptic solutions of the star-triangle relation with discrete integer-valued spins other than the Kashiwara-Miwa model.

(3) Unbounded spins. For hyperbolic solutions of the Yang-Baxter equation, whose spin integrations run over the whole real line, work out the root-of-unity limit; determine whether it produces discrete-spin models with spins valued in $\mathbb{Z}$ (unbounded).

(4) Multi-spin. Work out the root-of-unity limit of multi-spin (star-star type) solutions of the Yang-Baxter equation, including the simplest $r=1$ case, which is also not yet known.

Resolving any one of the four instances constitutes a full solution; resolving all four is not required. A valid answer to any instance must deliver an explicit model — Boltzmann weight formulas together with the spin domain, the rapidity-curve parametrization, and all parameter ranges — satisfying the relevant star-triangle or star-star relation exactly, with non-singular weights on the stated domains; alternatively, a rigorous non-existence or uniqueness result for the corresponding class (for example, a proof that the $\delta=1$ three-leg equation admits no non-singular solution of the required type, or that the Kashiwara-Miwa model is the only elliptic discrete-spin star-triangle solution).

References

1. A. P. Kels and M. Yamazaki, 'Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity', J. Stat. Mech. (2018) 023108, arXiv:1709.07148, https://arxiv.org/abs/1709.07148, DOI:10.1088/1742-5468/aaa8fd
2. V. V. Bazhanov and S. M. Sergeev, 'A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations', Adv. Theor. Math. Phys. 16 (2012) 65-95, arXiv:1006.0651, https://arxiv.org/abs/1006.0651, DOI:10.4310/ATMP.2012.v16.n1.a3
3. V. V. Bazhanov, A. P. Kels and S. M. Sergeev, 'Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs', J. Phys. A 49 (2016) 464001, arXiv:1602.07076, https://arxiv.org/abs/1602.07076, DOI:10.1088/1751-8113/49/46/464001
4. A. P. Kels, 'New solutions of the star-triangle relation with discrete and continuous spin variables', J. Phys. A 48 (2015) 435201, arXiv:1504.07074, https://arxiv.org/abs/1504.07074, DOI:10.1088/1751-8113/48/43/435201
5. V. E. Adler, A. I. Bobenko and Yu. B. Suris, 'Classification of Integrable Equations on Quad-Graphs. The Consistency Approach', Commun. Math. Phys. 233 (2003) 513, DOI:10.1007/s00220-002-0762-8, https://doi.org/10.1007/s00220-002-0762-8
6. M. Kashiwara and T. Miwa, 'A class of elliptic solutions to the star-triangle relation', Nucl. Phys. B 275 (1986) 121, DOI:10.1016/0550-3213(86)90591-2, https://doi.org/10.1016/0550-3213(86)90591-2
7. V. A. Fateev and A. B. Zamolodchikov, 'Self-dual solutions of the star-triangle relations in ZN-models', Phys. Lett. A 92 (1982) 37, DOI:10.1016/0375-9601(82)90736-8, https://doi.org/10.1016/0375-9601(82)90736-8
8. H. Au-Yang, B. M. McCoy, J. H. H. Perk, S. Tang and M.-L. Yan, 'Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus > 1', Phys. Lett. A 123 (1987) 219, DOI:10.1016/0375-9601(87)90065-X, https://doi.org/10.1016/0375-9601(87)90065-X
9. R. J. Baxter, J. H. H. Perk and H. Au-Yang, 'New solutions of the star triangle relations for the chiral Potts model', Phys. Lett. A 128 (1988) 138, DOI:10.1016/0375-9601(88)90896-1, https://doi.org/10.1016/0375-9601(88)90896-1
10. E. K. Sklyanin, 'Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras', Funct. Anal. Appl. 17 (1983) 273, DOI:10.1007/BF01076718, https://doi.org/10.1007/BF01076718
11. A. P. Kels, 'Integrable quad equations derived from the quantum Yang-Baxter equation', Lett. Math. Phys. 110 (2020) 1477-1557, arXiv:1803.03219, https://arxiv.org/abs/1803.03219, DOI:10.1007/s11005-020-01255-3
12. A. P. Kels, 'Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations', J. Stat. Phys. 176 (2019) 1375-1408, arXiv:1812.10893, https://arxiv.org/abs/1812.10893, DOI:10.1007/s10955-019-02346-9
13. A. P. Kels, 'Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations', Proc. R. Soc. A 481 (2025) 20250386, arXiv:2412.21096, https://arxiv.org/abs/2412.21096, DOI:10.1098/rspa.2025.0386
14. I. Gahramanov and S. Jafarzade, 'Comments on the multi-spin solution to the Yang-Baxter equation and basic hypergeometric sum/integral identity', Mod. Phys. Lett. A 34 (2019) 1950140, arXiv:1710.09106, https://arxiv.org/abs/1710.09106, DOI:10.1142/S0217732319501402
15. M. Yamazaki, 'Integrability as duality: The Gauge/YBE correspondence', Phys. Rep. 859 (2020) 1, arXiv:1808.04374, https://arxiv.org/abs/1808.04374, DOI:10.1016/j.physrep.2020.01.006
16. Yağmur Bülbül, Ilmar Gahramanov, Ali Mert Yetkin and Reyhan Yumuşak, 'Lens Hyperbolic Modular Double', arXiv:2511.06400, https://arxiv.org/abs/2511.06400
17. V. V. Bazhanov and S. M. Sergeev, 'A distant descendant of the six-vertex model', arXiv:2310.08427, https://arxiv.org/abs/2310.08427
18. Sergey Sergeev, 'On algebraic structures underlying the rational Kashiwara-Miwa-type models', arXiv:2508.12537, https://arxiv.org/abs/2508.12537
19. A. Arabi Ardehali and S. Murthy, 'The 4d superconformal index near roots of unity and 3d Chern-Simons theory', JHEP 2021 (2021) 207, arXiv:2104.02051, https://arxiv.org/abs/2104.02051, DOI:10.1007/JHEP10(2021)207
20. Meer Ashwinkumar, Jun-ichi Sakamoto and Masahito Yamazaki, 'Dualities and discretizations of integrable quantum field theories from 4d Chern-Simons theory', Adv. Theor. Math. Phys. 29 (2025) 1509-1694, arXiv:2309.14412, https://arxiv.org/abs/2309.14412, DOI:10.4310/ATMP.251118001759
21. Seyed Faroogh Moosavian, Masahito Yamazaki and Yehao Zhou, 'Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models', arXiv:2502.17545, https://arxiv.org/abs/2502.17545
22. V. P. Spiridonov, 'Rarefied elliptic hypergeometric functions', Adv. Math. 331 (2018) 830-873, arXiv:1609.00715, https://arxiv.org/abs/1609.00715
23. V. P. Spiridonov, 'The rarefied elliptic Bailey lemma and the Yang-Baxter equation', J. Phys. A 52 (2019) 355201, arXiv:1904.12046, https://arxiv.org/abs/1904.12046, DOI:10.1088/1751-8121/ab3358
24. V. P. Spiridonov, 'Elliptic beta integrals and solvable models of statistical mechanics', Contemp. Math. 563 (2012) 181-211, arXiv:1011.3798, https://arxiv.org/abs/1011.3798, DOI:10.1090/conm/563/11170
25. I. Gahramanov, 'Integrability from supersymmetric duality: a short review', Anal. Math. Phys. 16 (2026) 98, arXiv:2201.00351, https://arxiv.org/abs/2201.00351, DOI:10.1007/s13324-026-01224-4