Kramers-Wannier critical-temperature equations beyond planarity: Tutte-polynomial duality for Ising and Potts models on non-planar lattices

The exact critical temperatures of Ising and Potts models on planar lattices follow from the Kramers-Wannier duality, which Kazakov (2024) re-derived from the Tutte-polynomial duality $T_G(x,y)=T_{G^*}(y,x)$ of a planar graph and its planar dual. That work explicitly poses the follow-up problem treated here: adapt the derivation to non-planar graphs, using the established generalizations of the Tutte polynomial - the signed Bollobas-Riordan polynomial of graphs on surfaces, Krushkal's surface polynomial, and the Tutte polynomials of Hopf algebras of graphs - so as to obtain equations that determine the critical temperatures of $n$-state Potts (including Ising) models on lattices that lie outside planar Tutte duality - among them periodic lattices embedded in surfaces of positive genus and lattices whose underlying infinite graph is non-planar - generalizing the planar equation $(x(t_c)-1)(x(t_c^*)-1)=n$. A secondary question from the same source asks how the Biggs high-temperature expansion compares with the $*$-exponential (Baker-Campbell-Hausdorff-type) convolution formulas of the Hopf algebra of graphs. At present, exact partition functions and determinant-level duality relations are known for Ising models on surface graphs, and the surface Tutte polynomials satisfy clean duality theorems, but no critical-temperature equation of Kramers-Wannier type is known for any genuinely non-planar lattice, and no adaptation of the Tutte-duality technique to non-planar graphs has been published.

Background

The $n$-state Potts model on a finite graph $G=(V,E)$ assigns to each vertex $v$ a spin $\sigma(v)\in\mathbb{Z}_n$ and gives each edge $e=vw$ a weight that depends only on whether the endpoint spins agree: $i(0)=\alpha$ if $\sigma(v)=\sigma(w)$ and $i(a)=\beta$ for $a\neq 0$ otherwise; the partition function is $Z_n(G)=\sum_{\sigma}\prod_{e\in E}i(\sigma(v)-\sigma(w))$ (Biggs's interaction-model formalism). The Ising model is the case $n=2$ with $\alpha=e^{J/kT}$ and $\beta=e^{-J/kT}$. A classical result of the random-cluster representation (the Fortuin-Kasteleyn rewriting of the Potts partition function as a sum over edge subsets; see Biggs's Interaction Models and Sokal's survey) states that $Z_n(G)$ is, up to explicit prefactors, the Tutte polynomial $T_G(x,y)$ - the two-variable graph invariant defined recursively by deletion and contraction of edges - evaluated at $((x+n-1)/(x-1),x)$ with $x=\alpha/\beta$, a point on the Ising hyperbola $(x-1)(y-1)=n$. If $G$ is planar (drawable in the plane without edge crossings), it has a planar dual graph $G^*$, and the Tutte polynomials satisfy the duality $T_G(x,y)=T_{G^*}(y,x)$. This combinatorial fact underlies the Kramers-Wannier duality of 1941: on two mutually dual planar lattices the high-temperature expansion of one model is the low-temperature expansion of the other, so their critical points - the temperatures $t_c$ at which the free energy per site $f(t)=\lim_{k\to\infty}\ln Z_n(G_k)/v(G_k)$ for finite periodic pieces $G_k$ is non-analytic - are related by $(x(t_c)-1)(x(t_c^*)-1)=n$, where $x(t)=\alpha(t)/\beta(t)$. On the self-dual square lattice this gives $(x(t_c)-1)^2=n$, hence the exact Ising critical temperature $e^{2J/kT_c}=1+\sqrt{2}$; combined with the star-triangle ($Y-\Delta$) transformation (a local replacement of three edges meeting at a vertex by a triangle, or conversely, that preserves the partition function) one also obtains the triangular- and honeycomb-lattice critical points, and Wu's review covers the general $q$-state theory. In the limit $n\to 1$ the same equations give the exact bond-percolation thresholds $p_c+p_c^*=1$ of dual planar lattices. Every exact critical point known for two-dimensional lattice spin systems rests on this planar duality mechanism. Kazakov (2024) re-derived this circle of results directly from the Tutte duality within the Biggs formalism - the Kramers-Wannier equation for dual regular planar lattices, the self-dual equation, the triangular-lattice critical-point system via star-triangle, and the high-temperature (even-subgraph) expansion (an expansion over subgraphs in which every vertex meets an even number of selected edges), derived via the contraction-deletion relation - under the explicit working hypothesis that $f(t)$ has a unique critical point; the paper builds on the author's earlier work with Bychkov and Talalaev on functional relations for anisotropic Potts models. The open problem posed there concerns the established generalizations of the Tutte polynomial to embedded graphs: a ribbon graph is a graph together with a cyclic order of half-edges at each vertex, equivalently a band decomposition of an orientable surface; the Bollobas-Riordan polynomial of a ribbon graph refines the Tutte polynomial with a third variable tracking genus (the number of handles of the surface) and boundary components, and Chmutov proved a generalized duality for its signed version; Krushkal introduced a polynomial of graphs embedded in arbitrary surfaces with a topological duality theorem generalizing $T_G(x,y)=T_{G^*}(y,x)$; and Krajewski, Moffatt and Tanasa unified these polynomials as canonical Tutte polynomials of Hopf algebras of graphs (a graded algebra of graphs whose coproduct implements simultaneous deletion and contraction of edge subsets), where the Tutte polynomial arises as a convolution exponential $\exp^*$, the natural setting of Baker-Campbell-Hausdorff-type expansions. The rigorous status around the question is as follows. For planar graphs the picture is complete: Cimasoni and Duminil-Copin proved that for any planar doubly periodic Ising graph the critical inverse temperature is exactly the self-dual solution, characterized as a linear equation in $\tanh(\beta J_e)$ via the Kac-Ward method (the determinant method indexed by directed edges that solves the planar Ising model). For graphs embedded in orientable surfaces of genus $g$, Cimasoni's generalized Kac-Ward formula expresses the Ising partition function exactly as an alternating sum of $2^{2g}$ Kac-Ward determinants, a generalized Kramers-Wannier duality holds as explicit determinant equalities between a surface graph and its surface dual, and duality of Kac-Ward determinants extends to arbitrary weighted surface graphs. On the combinatorial side, Chmutov's, Krushkal's and the Hopf-algebraic duality theorems are well developed. None of these results, however, produces an equation determining a critical temperature of a non-planar lattice: they are finite-graph partition-function or polynomial identities. For genuinely non-planar lattices - lattices whose underlying infinite graph is not planar, such as two-dimensional lattices with crossing bonds (e.g. equivalent-neighbor lattices) and all three-dimensional lattices - no exact critical-temperature equations are known; thresholds are computed numerically, for instance by the critical-polynomial method (finite-lattice polynomials whose limiting roots approximate critical surfaces), extended to non-planar lattices as a numerical extrapolation tool. Classical $\mathbb{Z}_n$ spin-gauge duality does hold on arbitrary cell complexes but maps spin systems to gauge systems and yields no closed critical equations for spin systems (Savit's review).

Problem Statement

Following the future-work statement of Kazakov (2024), the problem is to determine whether the derivation of the Kramers-Wannier duality from the Tutte duality - a technique formulated for planar graphs - can be applied, verbatim or with modifications, to non-planar graphs, using the established generalizations of the Tutte polynomial: the signed Bollobas-Riordan polynomial for graphs on surfaces, the polynomial of Krushkal, or the Tutte polynomials of Hopf algebras of graphs. The scientific objective is to obtain equations determining the critical temperatures of $n$-state Potts models (including the Ising model, $n=2$) on lattices that lie outside planar Tutte duality - both lattices whose underlying infinite graph is not embeddable in the plane (such as two-dimensional lattices with crossing bonds, or three-dimensional lattices) and periodic lattices carried by a closed surface of positive genus, whose non-planarity resides in the chosen embedding rather than in the abstract infinite graph (a positive-genus surface lattice lifts to a planar infinite graph) - in the same way as the planar equation $(x(t_c)-1)(x(t_c^*)-1)=n$ determines the critical points of mutually dual planar lattices (with the bond-percolation limit $n\to 1$ giving $p_c+p_c^*=1$). A complete answer to the main objective should: (i) establish how the duality relations of the relevant generalized Tutte polynomial constrain the Potts/Ising partition function of a lattice embedded in a surface of genus $g\ge 1$, where the genus $g$ of the chosen embedding, not the planarity of the abstract graph, encodes the departure from planarity, including the bookkeeping of the homology (winding) data that the embedding introduces; (ii) derive the explicit critical-temperature equation(s) generalizing the Kramers-Wannier relation to this setting, stating precisely the class of lattices and embeddings to which they apply; (iii) show - rigorously, or under the same explicit critical-point-uniqueness hypothesis used in the planar derivation - that the solutions of these equations are the critical points of at least one lattice family outside planar duality - a positive-genus surface lattice, or a lattice whose underlying infinite graph is not embeddable in the plane - whose critical temperature is not already obtainable from the planar self-dual equation and its star-triangle relatives, with the genus-zero (planar) limit reproducing the classical Kramers-Wannier equations; and (iv) if exact equations are unattainable in full generality, characterize the non-planar lattice families for which they can be obtained and identify the obstruction for the remaining lattices. A complete answer to the main objective is either (a) the explicit critical-temperature equation(s) of (ii) for a stated class of lattices together with the demonstration required in (iii), or (b) the characterization and obstruction of (iv) should exact equations prove unattainable in full generality. A secondary component of the same program, posed in the same source and not a condition of a resolution, is to compare the Biggs high-temperature (even-subgraph) expansion - derived there from the contraction-deletion relation - with the $*$-exponential and convolution formulas (the Baker-Campbell-Hausdorff-type analog) of the Hopf algebra of graphs of Krajewski-Moffatt-Tanasa, clarifying whether that framework supplies the mechanism by which Tutte-type duality could act beyond planarity.

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