The dimer and Ising models on Klein bottles
Authors: Cimasoni David 
Published: April 2023

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph

\Gamma

in the Klein bottle

\mathcal{K}

. Let

\Gamma_{mn}

denote the graph obtained by pasting

m

rows and

n

columns of copies of

\Gamma

, which embeds in

\mathcal{K}

for

n

odd and in the torus

\mathbb{T}^2

for

n

even. We compute the dimer partition function

Z_{mn}

of

\Gamma_{mn}

for

n

odd in terms of the well-known characteristic polynomial

P

of

\Gamma_{12}\subset\mathbb{T}^2

together with a new characteristic polynomial

R

of

\Gamma\subset\mathcal{K}

. 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

\log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1)

for

m

,

n

tending to infinity and

m/n

bounded below and above, where

\mathbf{f}_0

is the bulk free energy for

\Gamma_{12}\subset\mathbb{T}^2

and

\mathsf{fsc}

is an explicit finite-size correction term. The remarkable feature of this later term is its
universality
: it does not depend on the graph

\Gamma

, but only on the zeros of

P

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

P

. We then show that this asymptotic expansion holds for the Ising partition function as well, with

\mathsf{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.We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph

\Gamma

in the Klein bottle

\mathcal{K}

. Let

\Gamma_{mn}

denote the graph obtained by pasting

m

rows and

n

columns of copies of

\Gamma

, which embeds in

\mathcal{K}

for

n

odd and in the torus

\mathbb{T}^2

for

n

even. We compute the dimer partition function

Z_{mn}

of

\Gamma_{mn}

for

n

odd in terms of the well-known characteristic polynomial

P

of

\Gamma_{12}\subset\mathbb{T}^2

together with a new characteristic polynomial

R

of

\Gamma\subset\mathcal{K}

. 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

\log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1)

for

m

,

n

tending to infinity and

m/n

bounded below and above, where

\mathbf{f}_0

is the bulk free energy for

\Gamma_{12}\subset\mathbb{T}^2

and

\mathsf{fsc}

is an explicit finite-size correction term. The remarkable feature of this later term is its
universality
: it does not depend on the graph

\Gamma

, but only on the zeros of

P

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

P

. We then show that this asymptotic expansion holds for the Ising partition function as well, with

\mathsf{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.