Exact critical-temperature bounds for two-dimensional Ising models

TL;DR

The paper derives exact, universal upper bounds on the critical temperature $T_{\rm c}$ for the classical ferromagnetic Ising model on any 2D periodic lattice, showing $t_{\rm c,min}= \tan(\pi/(2 q_{\max}))$ with $t=\tanh(\beta J)$ and $T_{\rm c,max}/J= 2/\log\Bigl( \tfrac{1+t_{\rm c,min}}{1-t_{\rm c,min}} \Bigr)$. The core method uses the Feynman–Kac–Ward (Kac–Ward) formalism, recasting the partition function in terms of Kac–Ward matrices $W(\boldsymbol{k})$, whose structure factorizes into blocks tied to local coordination numbers, yielding a universal bound via the largest eigenvalue of $W_0^{\dagger}W_0$. A key result is that the largest eigenvalue of $W_0^{\dagger}W_0$ depends only on the maximal coordination number $q_{\max}$, and is computed from reference matrices $\Phi_0^{(q)}$ with eigenvalue $\lambda_{\max}(q)=\cot(\pi/(2q))$, giving $u_{\max}=\lambda_{\max}(q_{\max})^2$ and hence $t_{\rm c,min}=1/\lambda_{\max}(q_{\max})=\tan(\pi/(2 q_{\max}))$. The authors validate the bound across hundreds of lattices, saturating for Honeycomb, Square, and Triangular lattices, and further demonstrate Tc engineering by constructing a 24-coordinated lattice with record high $T_{\rm c}/J=6.49190$, approaching the bound for large $q_{\max}$. The work thus provides a simple graph-theoretic criterion for high-temperature order in 2D Ising systems and a framework to analyze a broad class of periodic lattices via exact Kac–Ward techniques, with potential extensions to non-Euclidean tessellations.

Abstract

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 a 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.

Figures (31)

• Figure 1: Critical temperature of the ferromagnetic Ising model, $T_{\rm c}$, on various 2D periodic tessellations versus their maximal coordination number, $q_{\rm max}$. The solid line is the exact upper bound from Eq. (\ref{['intro3']}). For each $q_{\rm max}$, we show a subunit of the tessellation with the largest value of $T_{\rm c}/J$ among the lattices considered here. The Honeycomb, Square, and Triangular lattices saturate their bounds, while all other lattices considered here lie strictly below. The plot encompasses all $k$-uniform lattices with $k\leq 3$ and their dual lattices, with $k=1$ corresponding to the Archimedean and Laves lattices. Purple dots constitute a selection of lattices with $q_{\rm max}=7,9,10,11$. Lieb-like lattices are the sequence of lattices where each edge of a Square lattice is replaced by a chain of $2^n$ two-coordinated sites, with $0\leq n\leq 50$, yielding arbitrarily low $T_{\rm c}/J \sim 1/n$ as $n$ increases. The one-dimensional chain with $q_{\rm max}=2$ and $T_{\rm c}=0$ is included for completeness. The plot comprises data from 236 lattices.
• Figure 2: Starting from the Triangular lattice (one tile shown in a, $q_{\rm max}=6$), the Laves-Star lattice (b, $q_{\rm max}=12$) is obtained by adding a 3-coordinated site into each triangle. Iterating this step once again (c), we obtain the Compass-Rose lattice (d) with $q_{\rm max}=24$, the name inspired by the 24 classical wind directions. Its critical temperature $T_{\rm c}/J=6.49190$ computed in the SM SI is, to the best of our knowledge, the largest value ever reported for the ferromagnetic Ising model on a periodic tessellation of the plane. All three lattices have a high $T_{\rm c}$ for their value of $q_{\rm max}$. They also have average coordination number $\bar{q}=6$, which is the maximal possible value for 2D periodic tessellations SI.
• Figure S1: The eleven Archimedean lattices are the 1-uniform periodic tessellations of the plane by regular polgons. Names follow common conventions in the literature. Note that the Ruby lattice is also referred to as Bounce lattice. The Honeycomb lattice is topologically equivalent to the Brickwork lattice discussed below. Similarly, the SrCuBO lattice is topologically equivalent to the lattice SrCuBO$^*$ shown here, which appears in the quantum spin model by Shastry and Sutherland SRIRAMSHASTRY19811069.
• Figure S2: The Laves lattices are the dual lattices to the Archimedean lattices. They use only one type of tile, since the Archimedean lattices only use one type of vertex. We exclude the dual lattices of the Honeycomb, Square, and Triangular lattices, since they are again Archimedean lattices. Since the Star (and SHD) and CaVO lattices employ regular 12- and 8-gons, it follows that the Laves-Star (and Laves-SHD) and Laves-CaVO lattices have vertices with coordination numbers 12 and 8, respectively. The Laves-SrCuBO lattice is topologically equivalent to the Laves-SrCuBO$^*$ lattices shown here. The Laves-Star lattice is also known as Asanoha or hemp-leaf lattice.
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