Kac–Ward Solution of the 2D Classical and 1D Quantum Ising Models

We give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants and (ii) the one-dimensional quantum Ising model. We use the method of Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni-Duminil-Copin-Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model.