We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, T_\rm c, for the classical ferromagnetic Ising model. Our approach is motivated by recently found exact bounds, which imply that large values of T_\rm c require large values of the maximal coordination number of the lattice, q_\rm max. We create such lattices through iterative triangulation and derive explicit expressions for their T_\rm c. Furthermore, we show that T_\rm c for these families scales asymptotically as T_\rm c/J ∼A \ln q_\rm max with a universal prefactor A=2/\ln 2. We introduce a function T_\rm c^*(q_\rm max) that we conjecture to be optimal for all periodic tessellations of the plane. We show that the family of so-called Apollonian lattices, which are derived from the Triangular lattice through iterative triangulation, saturates this bound. The lattices discussed in this work are relevant for theoretical questions of optimality in network systems and may be realized experimentally in Coherent Ising Machines or topoelectric circuits in the future.
@misc{joseph2026familiesplanarlatticesarbitrarily,title={Families of planar lattices with arbitrarily high $$T_{\rm c}$$ for the ferromagnetic Ising model},author={Joseph, Davidson Noby and Walsh, Connor M. and Boettcher, Igor},year={2026},eprint={2605.10017},archiveprefix={arXiv},primaryclass={cond-mat.stat-mech},url={https://arxiv.org/abs/2605.10017},}
Under Review
Exact critical-temperature bounds for two-dimensional Ising models
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},}
Accepted
A simple quantum dot: numerical and variational solutions
Connor
Walsh, Ian
MacPherson, Davidson
Joseph, and
4 more authors
We describe a simple quantum dot that consists of two crossed troughs. As such there is no potential well; nonetheless this geometry gives rise to a bound state, centred around the point at which these troughs cross one another. In this paper we review existing numerical methods to solve this problem, and highlight one which we feel is particularly elegant and, in this case, provides the most accurate solution to the problem. The bound state is well-contained on the scale of the trough width, and yields a bound state energy of 0.659606 in units of the minimum continuum state energy. This method also motivates a simple variational solution which yields the lowest energy known to date (0.6812 in the same units) to arise out of an analytical variational solution.
@article{walsh2025simplequantumdotnumerical,title={A simple quantum dot: numerical and variational solutions},author={Walsh, Connor and MacPherson, Ian and Joseph, Davidson and Kabra, Suyash and Toor, Ripanjeet Singh and Protter, Mason and Marsiglio, Frank},year={2026},volume={},issue={},pages={},numpages={},eprint={2511.16053},primaryclass={cond-mat.mes-hall},url={https://arxiv.org/abs/2511.16053},journal={Am. J. Phys},month=mar,publisher={American Association of Physics Teachers},}
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},}
