Arbitrarily high-$T_{\rm c}$ lattices Davidson Noby Joseph

Reference

Phys. Rev. E
Families of planar lattices with arbitrarily high $T_{\rm c}$ for the ferromagnetic Ising model

Davidson Noby Joseph, Connor M. Walsh, and Igor Boettcher

2026

Abstract arXiv Journal

We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, $T_{\rm c}$, for the classical nearest-neighbor 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 $T_{\rm c}/J \sim A \ln q_{\rm max}-2 \ln \ln q_{\rm max}$ with a universal prefactor $A=2/\ln 2$. We introduce a function $T_{\rm c}^*(q_{\rm max})$ that we conjecture to be an upper bound on the critical temperature of any periodic tessellation 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.
@article{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}, month = {}, primaryclass = {cond-mat.stat-mech}, publisher = {American Physical Society}, url = {https://link.aps.org/doi/10.1103/y4zd-dffy}, doi = {https://doi.org/10.1103/y4zd-dffy} }

We start with the recently found exact bounds for $T_{\rm c}$ for the ferromagnetic Ising model. The work showcases the critical temperatures of over 200 lattices and introduces the Compass-Rose lattice – a planar periodic lattice with the highest known critical temperature for this model. The inspiration to construct the Compass-Rose came from the following observations:

[1] The Triangular lattice has the highest $T_{\rm c}$ among the Archimedean lattices
[2] The Laves-Star lattice has the highest $T_{\rm c}$ among the Archimedean and Laves lattices combined
[3] The Laves-Star lattice is constructed from the Triangular lattice by placing a vertex at the center of each triangle and connecting it to the vertices of the triangle.

We introduce the notion of iterative triangulation, the process of constructing new lattices from lattices which are triangulations by the procedure above. Applied on the Triangular lattice, it looks like the following:

We show that under iterative triangulation, the critical temperature grows without bound. Consequently, any starting lattice (called the base lattice) can be used to construct and a plethora of new lattices belonging to its corresponding family each with a higher critical temperature than its predecessor. Given the critical temperature of the base lattice, we derive an exact expression for the critical temperature(s) of any members of its family in terms of a functional recursion equation. Written in terms of the maximum coordination number $q_{\rm max}$ – the maximum number of edges leaving a site on the lattice – we uncover the asymptotic scaling law under iterative triangulation

\[T_{\rm c}/J \sim A \ln q_{\rm max}- 2 \ln \ln q_{\rm max},\]

where $A=2/\ln 2$ is a universal constant. Additionally, we construct a unique continuous extension for the critical temperatures for any family valid for $q_{\rm max}>0$ instead of integer $q_{\rm max}$. Particularly, the unique continuous extension for the Triangular lattice $T^*(q_{\rm max})$ which lies below the exact bound, serves as an upper bound for all the lattices considered in this work. We conjecture that it might be the tightest upper bound for any planar Euclidean lattice with maximum coordination number $q_{\rm max}>6$.