Paper detail

The dimer and Ising models on Klein bottles

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $Γ$ in the Klein bottle $K$. Let $Γ_{mn}$ 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 $\mathbb{T}^2$ for $n$ even. We compute the dimer partition function $Z_{mn}$ of $Γ_{mn}$ for $n$ odd, in terms of the well-known characteristic polynomial $P$ of $Γ_{12}\subset\mathbb{T}^2$ together with a new characteristic polynomial $R$ of $Γ\subset K$. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion $\log Z_{mn}=mn f_0/2 +\mathrm{fsc}+o(1)$, for $m, n$ tending to infinity and $m/n$ bounded below and above, where $f_0$ is the bulk free energy for $Γ_{12}\subset\mathbb{T}^2$ and $\mathrm{fsc}$ 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 $\mathrm{fsc}$ 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.