Exact critical temperature bounds for ferromagnetic two-dimensional Ising models

The Ising model, a cornerstone of statistical physics, has been thoroughly investigated for over a century. Despite remarkable progress, an exact solution remains elusive to this day for three dimensions and higher, with various numerical methods like Metropolis-Hastings traditionally used to explore the model. In two-dimensions, the Feynman-Kac-Ward method provides an analytic framework to solve the model exactly for periodic planar lattices in zero field. We extend this method to any periodic tiling in two-dimensions and derive explicit expressions for the corresponding free energy densities. Using our generalized formalism, we obtain an exact analytic upper bound on the critical temperatures that depends solely on the maximum coordination number of the underlying lattice. We verify our results across all eleven Archimedean lattices and their duals, commonly known as the Laves lattices. The framework provides an analytic treatment to evaluate thermodynamic quantities of interest like the free energy density and specific heat, offering a new analytic tool complementary to computational approaches that is useful for benchmarking.