Exact critical-temperature bounds for the Ising model Davidson Noby Joseph

Reference

Phys. Rev. E
Exact critical-temperature bounds for two-dimensional Ising models

Davidson Noby Joseph, and Igor Boettcher

Phys. Rev. E 113, 064113 – Published Jun 2026

Abstract arXiv Journal

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.
@article{svhn-yfv1, title = {Exact critical-temperature bounds for two-dimensional Ising models}, author = {Joseph, Davidson Noby and Boettcher, Igor}, journal = {Phys. Rev. E}, volume = {113}, issue = {6}, pages = {064113}, numpages = {23}, year = {2026}, month = jun, publisher = {American Physical Society}, doi = {10.1103/svhn-yfv1}, url = {https://link.aps.org/doi/10.1103/svhn-yfv1}, }

The Ising model is one of the simplest classical models for ferromagnetism that is able to capture a sudden phase transition under a change of temperature. Although simple, its lack of sophistication is by no means an indicator of its crucial role in the development of many-body physics in the last century. The classical ferromagnetic nearest-neighbor Ising model consists of spins $s_i=\pm 1$ on sites $i$ of a lattice (or graph), that interact with a coupling constant $J>0$ described by the Hamiltonian

\[H(s_1,s_2,\dots)= -J\sum_{\langle i,j\rangle } s_is_j.\]

Here the sum is over all the nearest-neighbors (undirected edges) $\langle i,j\rangle$ of the graph. The coupling constant is chosen to be positive in order to facilitate alignment, i.e, conformity is the rule of the game. Placing the system in a thermal bath at temperature $T$, we describe the system using the partition function

\[Z= \sum_{\{s_1,s_2,\dots\}}e^{-\frac{H(s_1,s_2,\dots)}{k_{\rm B}T}},\]

where the sum is over all spin configurations. The system tries to conform under thermal fluctuations; for large $T/J$, the system is paramagnetic and alignment isn’t feasible. However, for small $T/J$, the system is ferromagnetic and prefers to align. Between these two phases, there exists a so-called critical temperature $T_{\rm c}/J$ that delineates both phases. One can observe this transition happen in the following lattice (here $s_i=+1$ is represented by green whereas yellow represents $s_i=-1$ and the transition is shown by the onset of collective ‘blinking’.)

The critical temperature for the above periodic lattice is $T_{\rm c}/J\approx 1.8$. Lattice with higher $T_{\rm c}/J$ stabilize these aligned (ordered) more easily that its counterparts with low critical temperature and thus, high-$T_{\rm c}$ lattices are sought out for. However, although critical exponents are universal for this model, the critical temperature heavily depends on the lattice and must be estimated using computational methods like Markov Chain Monte Carlo (MCMC) or tedious analytic methods like the Feynman-Kac-Ward method. Is there a simple way to estimate the critical temperature given a lattice in order to search for lattices with high $T_{\rm c}$?

We answer this question by showing the existence of an exact bound for the critical temperature of any planar Euclidean lattice in terms of its maximum coordination number $\boldsymbol{q_{\rm max}}$ – the maximum number of edges that leave a site. The bound is expressed in terms of the inequality

\[\tanh\left(\frac{J}{T_{\rm c}}\right)\geq \tan\left(\frac{\pi}{2 q_{\rm max}}\right).\]

Unlike the loose mean-field bound, our bound is tight and saturates for the Square, Triangular and Honeycomb lattice. The proof uses a new representation of the Kac-Ward matrix which is used to bound the analytic $T_{\rm c}$. Additionally, we construct the Leib-n lattices which have arbitrarily low $T_{\rm c}$. We verify our bound by testing over 200 Euclidean lattices.