Kels (J. Phys. A: Math. Theor. 50 (2017) 495202) extended Baxter's Z-invariance for exactly solved two-dimensional lattice models, allowing rapidity lines that form directed closed paths and thereby defining integrable edge-interaction models on planar graphs whose edges connect subsets of next-nearest-neighbour vertices of Z3\mathbb{Z}^3; in the quasi-classical limit this yields an action functional for classical discrete Laplace equations satisfying a closure relation for two-point Lagrangians. The paper left four directions open. Two of these directions have since been completed by the author himself: the extension of extended Z-invariance to vertex and interaction-round-a-face models (J. Stat. Phys. 176 (2019) 1375) and the quasi-classical derivation of the H-type ABS equations from an asymmetric star-triangle relation with complex-valued Boltzmann weights (Lett. Math. Phys. 110 (2020) 1477). The surviving open core consists of two linked questions: (i) whether a systematic regularisation can make the extended Z-invariance well defined for continuous-spin models, whose deformed partition functions currently contain divergent factors ∫Rdx0 S(x0)\int_{\mathbb{R}}dx_0\,S(x_0) and formal δ(0)\delta(0) terms; and (ii) whether three-point Lagrangian systems - the Lagrangian-multiform (pluri-Lagrangian) structure of the ABS quad equations with its closure relation - can arise as the quasi-classical limit of a lattice model satisfying the Yang-Baxter equation, for example on a triangular lattice. Both parts concern the completion of the same correspondence between Yang-Baxter integrability and classical variational structure under the extended deformations.

Background

Two-dimensional lattice models of statistical mechanics assign spins to vertices of a lattice and Boltzmann weights to edges; the partition function is the sum (or integral) over interior spins of the product of all edge weights. A model is exactly solvable when its weights satisfy the Yang-Baxter equation (YBE), a functional identity guaranteeing commuting transfer matrices. For edge-interaction (spin) models the YBE takes the form of the star-triangle relation (STR): an identity in which the integral over an internal spin of three edge weights meeting at a star equals a product of three weights on the complementary triangle, up to a normalisation factor. Baxter showed that the STR, together with simple inversion relations for the weights, implies Z-invariance: the partition function is invariant, up to elementary factors, under continuous deformations of the rapidity graph - the system of directed rapidity lines carrying the spectral parameters that cross the lattice edges - so that an integrable model can be transferred from the square lattice to general planar graphs (Baxter 1978; Baxter 1986). In Baxter's formulation the rapidity lines must never form directed closed paths.

The source paper (Kels, J. Phys. A: Math. Theor. 50 (2017) 495202) extended this framework in two ways. First, it allowed new types of rapidity lines that do form directed closed paths and used the resulting local deformations (an elementary square is replaced by configurations of up to five squares lying in adjacent lattice planes) to define integrable edge-interaction models on general planar graphs whose edges connect subsets of next-nearest-neighbour vertices of the three-dimensional cubic lattice Z3\mathbb{Z}^3, with the partition function invariant up to factors supplied by the STR and the inversion relations. Second, it studied the quasi-classical limit: a parameter ℏ\hbar is introduced so that each Boltzmann weight expands as exp⁡(ℏ−1Λ(xi,xj)+O(1))\exp(\hbar^{-1}\Lambda(x_i,x_j)+O(1)), and the leading exponential of the partition function becomes an action functional S=∑Λ\mathcal{S}=\sum\Lambda - a sum of two-point Lagrangians on diagonals of elementary squares - whose saddle points are classical discrete Laplace equations (difference equations generalising the lattice Laplace equation). This action is invariant under the same cubic deformations, and the classical star-triangle relation was shown to be equivalent to a local closure relation, i.e. the vanishing of the sum of Lagrangians around a deformation.

This variational picture parallels the Lagrangian multiform (or pluri-Lagrangian) formalism introduced by Lobb and Nijhoff and developed by Bobenko and Suris for the Adler-Bobenko-Suris (ABS) equations - the classification of two-dimensional quad equations that are consistent around a cube, which is the standard integrability notion for equations on quad-graphs. There the elementary object is a three-point corner Lagrangian L(x,xi,xj)\mathcal{L}(x,x_i,x_j) depending on one vertex and two lattice directions, and integrability is encoded in a closure relation: the sum of the corner Lagrangians over the faces of an elementary cube vanishes. Before and alongside the source paper, Bazhanov, Kels and Sergeev (J. Phys. A 49 (2016) 464001) showed that quasi-classical expansions of continuous-spin solutions of the STR reproduce the Q-type ABS equations, with the H-type cases completed later by Kels (Lett. Math. Phys. 110 (2020) 1477); in all of these expansions, however, the Lagrangians that appear are two-point functions of two spins and a rapidity. The source paper contrasted this with the ABS systems of three-point Lagrangians and asked whether those, too, can arise as a quasi-classical limit of a Yang-Baxter lattice model.

Among the models solved by the STR are continuous-spin models, in which spins take real values or values on a circle and spin sums become integrals: the Zamolodchikov fishing-net model (arising from planar quantum-field-theory diagrams), the Faddeev-Volkov-type models, and the Bazhanov-Sergeev master solution built from the elliptic gamma function, which contains most known discrete- and continuous-spin solutions as limiting cases. When the extended deformations of the source paper are applied to such models, divergent factors appear: the factor ∑x0S(x0)\sum_{x_0}S(x_0) attached to an internal spin becomes ∫Rdx0 S(x0)\int_{\mathbb{R}}dx_0\,S(x_0), which diverges for models whose spins run over the whole real line, and the Kronecker deltas δx,x\delta_{x,x} produced by the inversion relation become formal Dirac deltas δ(0)\delta(0) for all continuous-spin models. The author observed that these divergent constants are independent of spins and rapidities, so observables - typically derivatives of log⁡Z\log Z - are unaffected, and that the δ(0)\delta(0) terms can be avoided by giving up the second inversion relation at the cost of restricting the allowed deformations; nevertheless the deformed partition functions themselves remain ill-defined without a systematic regularisation.

The source paper left four directions open: the continuous-spin regularisation just described; the extension of extended Z-invariance to models satisfying other forms of the YBE - vertex models, whose Boltzmann weights sit at the crossings of rapidity lines (as in the six-vertex model), and interaction-round-a-face (IRF) models, whose weights depend on the spins surrounding a face; the question of a quasi-classical realisation of three-point Lagrangian systems; and the treatment of the ABS H-type equations, whose star-triangle relations are of an asymmetric form with generically complex-valued Boltzmann weights requiring complex saddle-point analysis. The second and fourth of these were later resolved by the author; the first and third constitute the present problem.

Problem Statement

The problem is to complete the extended Z-invariance and quasi-classical closure correspondence of Kels (J. Phys. A: Math. Theor. 50 (2017) 495202) in the two settings it does not currently reach. The two parts serve one overarching objective - extending the correspondence between Yang-Baxter integrability and classical variational structure under the extended deformations to its remaining cases: the continuous-spin case on the statistical-mechanical side, and the three-point-Lagrangian case on the variational side. They are stated at the source's own generality; the source's other two open directions (the vertex/IRF extension and the H-type complex-weight cases) were resolved by Kels in 2019 and 2020 and are not part of this problem.

(1) Continuous-spin extended Z-invariance. Determine whether there is a systematic regularisation of the divergent factors that the extended deformations generate for continuous-spin edge-interaction models - the factor ∫Rdx0 S(x0)\int_{\mathbb{R}}dx_0\,S(x_0) arising from internal spins for models whose spins take values on the whole real line (such as the Zamolodchikov fishing-net model), and the formal δ(0)\delta(0) factors generated by the inversion relation for all continuous-spin models - under which the deformed partition functions on planar graphs with edges connecting subsets of next-nearest-neighbour vertices of Z3\mathbb{Z}^3 are well defined and the extended Z-invariance holds for the regularised models up to explicitly controlled factors, including for the deformations that require the second inversion relation; or establish that no such regularisation exists. A regularisation counts as systematic only if it is a single scheme, defined uniformly for the class of models and for all the extended deformation moves under consideration rather than chosen ad hoc per model or per move, that is consistent with the discrete-spin case (to which it reduces in the appropriate limit); and 'up to explicitly controlled factors' means that the invariance is an equality up to factors written out explicitly from the star-triangle and inversion relations, rather than only shown to be finite or bounded.

(2) Three-point Lagrangian systems as quasi-classical limits. Determine whether a system of three-point Lagrangians with its closure relation - the Lobb-Nijhoff/Bobenko-Suris Lagrangian-multiform (pluri-Lagrangian) structure of the ABS quad equations - arises as the quasi-classical limit of a lattice model satisfying the Yang-Baxter equation, in any of its standard forms (edge-interaction/star-triangle, vertex, or interaction-round-a-face) and on any lattice geometry (the source suggests, without requiring, a triangular lattice). Concretely: construct such a lattice model whose quasi-classical expansion of the partition function yields an action functional built from three-point Lagrangians whose closure relation follows from the underlying Yang-Baxter equation, in the same way that the source derived the two-point closure relation for the square-lattice discrete Laplace systems; or establish that such a realisation is impossible, identifying the obstruction.

References

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