Ising model on nonorientable surfaces: Exact solution for the Möbius strip and the Klein bottle

Closed-form expressions are obtained for the partition function of the Ising model on an $\mathcal{M}\times \mathcal{N}$ simple-quartic lattice embedded on a Möbius strip and a Klein bottle. The solutions all lead to the same bulk free energy, but for finite $\mathcal{M}$ and $\mathcal{N}$ the expressions are different depending on whether the strip width $\mathcal{M}$ is odd or even. Finite-size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions. Our results are consistent with the conformal field prediction of a central charge $c=1/2,$ provided that the twisted Möbius boundary condition is regarded as a free or fixed boundary.