We derive exact critical-temperature bounds for the classical ferromagnetic Ising model on two-dimensional periodic tessellations of the plane. For any such tessellation or lattice, the critical temperature is bounded from above by a universal number that is solely determined by the largest coordination number on the lattice. Crucially, these bounds are tight in some cases such as the Honeycomb, Square, and Triangular lattices. We prove the bounds using the Feynman-Kac-Ward formalism, confirm their validity for a selection of over two hundred lattices, and construct a two-dimensional lattice with 24-coordinated sites and record high critical temperature.
@misc{joseph2026exactcriticaltemperatureboundstwodimensional,title={Exact critical-temperature bounds for two-dimensional Ising models},author={Joseph, Davidson Noby and Boettcher, Igor},year={2026},eprint={2601.02502},archiveprefix={arXiv},primaryclass={cond-mat.stat-mech},url={https://arxiv.org/abs/2601.02502},}
2025
Phys. Rev. E
Walking on Archimedean lattices: Insights from Bloch band theory
Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after n steps. Determining the total number of returning walks of a given length n is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As condensed matter physics applications, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices and analytically determine the asymptotics of the return probability. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.
@article{1fvj-91v6,title={Walking on Archimedean lattices: Insights from Bloch band theory},author={Joseph, Davidson Noby and Boettcher, Igor},journal={Phys. Rev. E},volume={112},issue={4},pages={044118},numpages={29},year={2025},month=oct,publisher={American Physical Society},doi={10.1103/1fvj-91v6},url={https://link.aps.org/doi/10.1103/1fvj-91v6},}
