The dimer and Ising models on Klein bottles

  • David Cimasoni

    Université de Genève, Switzerland
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Abstract

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph in the Klein bottle . Let denote the graph obtained by pasting rows and columns of copies of , which embeds in for odd and in the torus for even. We compute the dimer partition function of for odd in terms of the well-known characteristic polynomial of together with a new characteristic polynomial of . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion

for , tending to infinity and bounded below and above, where is the bulk free energy for and is an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph , but only on the zeros of on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of . We then show that this asymptotic expansion holds for the Ising partition function as well, with taking a particularly simple form: it vanishes in the subcritical regime, is equal to 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.We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph in the Klein bottle . Let denote the graph obtained by pasting rows and columns of copies of , which embeds in for odd and in the torus for even. We compute the dimer partition function of for odd in terms of the well-known characteristic polynomial of together with a new characteristic polynomial of . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion

for , tending to infinity and bounded below and above, where is the bulk free energy for and is an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph , but only on the zeros of on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of . We then show that this asymptotic expansion holds for the Ising partition function as well, with taking a particularly simple form: it vanishes in the subcritical regime, is equal to 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.

Cite this article

David Cimasoni, The dimer and Ising models on Klein bottles. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 11 (2024), no. 3, pp. 503–569

DOI 10.4171/AIHPD/166