The dimer and Ising models on Klein bottles

Abstract

We study the dimer and Ising models on a planar weighted graph with periodicantiperiodic boundary conditions, i.e. a graph Γ in the Klein bottle K . Let Γmn denote the graph obtained by pasting m rows and n columns of copies of Γ, which embeds in K for n odd and in the torus T 2 for n even. We compute the dimer partition function Zmn of Γmn for n odd, in terms of the well-known characteristic polynomial P of Γ12 ⊂ T 2 together with a new characteristic polynomial R of Γ ⊂ K .Using this result together with work of Kenyon, Sun and Wilson [31], we show that in the bipartite case, this partition function has the asymptotic expansionwhere f0 is the bulk free energy for Γ12 ⊂ T 2 and fsc an explicit finite-size correction term.The remarkable feature of this later term is its universality: it does not depend on the graph Γ, but only on the zeros of P on the unit torus and on an explicit (purely imaginary) shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of P .We then show that this asymptotic expansion holds for the Ising partition function as well, with fsc taking a particularly simple form: it vanishes in the subcritical regime, is equal to log(2) in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale [2].

Graph coverings and twisted operators

“…It turns out that they split as products of representations of degree 1 and 2, yielding a closed formula for Z dimer (Γ mn , x) in terms of determinants of A τ Γ , with τ of degree 1 and 2. This result is at the core of the study of the dimer model on Klein bottles of the first-named author [9].…”

“…Applying Corollary 3.9 to the case of a graph embedded in the torus yields an immediate proof of the classical fact that the dimer characteristic polynomial behaves multiplicatively under so-called enlargement of the fundamental domain [24,Theorem 3.3]. However, applying our results to the study of the dimer model on graphs embedded in the Klein bottle leads to new powerful results, that are harnessed in the parallel article [9].…”

“…This brings into what we believe to be the correct framework several well-known results in combinatorics, statistical mechanics, and L-function theory, but also provides new ones. In order to preserve its nontechnical nature, we focus in the present article on relatively direct consequences, leaving the more elaborate implications to subsequent papers [9,10].…”

Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

“…There are a number of interesting problems which could be studied if this limitation could be overcome. In the last few years, similar expansions for periodic graphs imbedded in the torus [KSW16] or Klein bottle [Cim20] based on the characteristic polynomial method have found a constant term which is universal (i.e. independent of many of the details of exactly which graph is chosen); it seems reasonable to expect that the same is true for some class of planar graphs.…”