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

This paper constructs families of periodic planar lattices, specifically Apollonian lattices, that achieve arbitrarily high critical temperatures for the ferromagnetic Ising model by demonstrating that $T_{\rm c}$ scales logarithmically with the maximal coordination number and conjecturing this family to be optimal for such systems.

The Gist

Imagine you are building a giant, flat city made of tiny magnets. In this city, every magnet wants to point in the same direction as its neighbors. This is the Ising model, a famous way physicists describe how materials become magnetic.

The big question this paper asks is: How hot can this city get before the magnets lose their order and start pointing in random directions?

In physics, this temperature is called the Critical Temperature ($T_c$). If the city gets hotter than $T_c$, the "magnetic spell" breaks, and the material stops being a magnet. The authors wanted to find a way to build cities (lattices) that can survive being incredibly hot while still staying magnetic.

Here is the story of how they did it, using simple analogies:

1. The Problem: The "Crowded Room" Limit

Usually, the hotter the room, the harder it is for people to agree on a direction. In magnetism, the "agreement" is the magnetic order.

• The Old Rule: Scientists knew that to make a magnet survive high heat, you needed to pack the magnets very tightly together (high "coordination number," or how many neighbors each magnet has).
• The Limit: There was a theoretical "speed limit" on how hot these magnets could get. It was thought that even with the tightest packing, the temperature couldn't go beyond a certain linear line.

2. The Solution: The "Triangle Expansion" Game

The authors discovered a clever trick to break this speed limit. They used a method they call Iterative Triangulation.

Imagine you have a floor covered in triangles.

1. Step 1: Take one triangle. Put a new magnet right in the dead center.
2. Step 2: Connect this new center magnet to the three corners of the triangle.
3. Step 3: Now, that one big triangle has been split into three smaller triangles.
4. Step 4: Repeat this forever. Every time you do it, you add more magnets in the middle of every new triangle.

The Magic Effect:
Every time you do this, the magnets on the original corners get more and more neighbors.

• In the first step, a corner magnet might have 6 neighbors.
• In the next step, it has 12.
• Then 24, then 48, and so on.

The authors found that by doing this "infinite expansion," they could create families of magnets that stay ordered at temperatures that are arbitrarily high. You can make them as hot as you want, provided you keep adding layers of triangles.

3. The "Apollonian" Champions

They tested this on many different starting shapes, but one family stood out as the absolute champion. They called these Apollonian Lattices.

Think of these as the "Olympic Gold Medalists" of magnetic cities.

• They start with a simple triangular grid (like a honeycomb, but made of triangles).
• They apply the "center-point" trick over and over.
• The Result: These lattices stay magnetic at higher temperatures than any other flat pattern the authors could find.

The paper claims that if you look at the relationship between how crowded the magnets are (neighbors) and how hot they can get, the Apollonian lattices hit the absolute ceiling. They define a new "ceiling line" (called $T^*_c$) that no other flat magnet city can cross.

4. The "Growth Rate" Surprise

Here is the most interesting part about the math:

• Scientists previously thought that if you doubled the number of neighbors, the heat limit would go up by a straight, linear amount (like climbing a steady ramp).
• The authors found that with their triangle trick, the heat limit actually grows much slower, like a logarithmic curve.
• The Analogy: Imagine you are filling a bucket with water.
  • The old theory said: "If you double the size of the hose, you fill the bucket twice as fast."
  • The new discovery says: "If you double the size of the hose, you fill the bucket faster, but the speed of filling slows down as the bucket gets bigger."
  • Even though the growth is slower than the theoretical maximum, the authors proved you can still reach any temperature you want if you keep adding layers.

5. Why This Matters (According to the Paper)

The paper suggests two main places where this could be useful:

1. Coherent Ising Machines: These are experimental computers that use light (lasers) to solve complex math problems by acting like magnets. If engineers can build these machines using the "Apollonian" patterns, the machines might work at much higher temperatures, making them easier to build and run.
2. Topoelectrical Circuits: These are circuits that act like magnets but use electricity. The authors suggest these specific patterns could be built in the lab to test these theories.

Summary

The paper is about inventing a new way to arrange magnets on a flat surface. By repeatedly adding a "center piece" to every triangle, they created a family of patterns that can withstand extreme heat better than any other known flat pattern. They proved that while there is a theoretical limit to how hot these magnets can get, their specific "Apollonian" design gets closer to that limit than anything else, offering a blueprint for building super-stable magnetic systems for future computers.

Technical Summary

Technical Summary: Families of Planar Lattices with Arbitrarily High $T_c$ for the Ferromagnetic Ising Model

Problem Statement
The ferromagnetic Ising model on two-dimensional planar lattices exhibits a phase transition at a critical temperature $T_c$. While the critical behavior of correlation functions is universal (dependent on dimension but not lattice structure), $T_c$ itself is a non-universal quantity determined by the lattice geometry and the interaction strength $J$. Recent exact bounds established for periodic tilings of the plane indicate that $T_c$ is finite and asymptotically bounded by the maximal coordination number $q_{\text{max}}$ (the largest number of nearest neighbors for any site). Specifically, for $q_{\text{max}} \ge 6$, the bound is $T_c/J \le (2/\pi)q_{\text{max}}$. However, existing examples of lattices with high $q_{\text{max}}$ (e.g., Laves-Star, Compass-Rose) suggest that actual $T_c$ values scale more slowly than this linear bound. The central problem addressed is whether families of lattices can be constructed to achieve arbitrarily high $T_c$ and how these values scale with $q_{\text{max}}$, specifically seeking to determine if a tighter, optimal upper bound exists for planar tessellations in Euclidean space.

Methodology
The authors employ a geometric construction method termed iterative triangulation. Starting from an arbitrary base triangulation $\Lambda$ (a lattice composed solely of triangles), the procedure involves:

1. Placing a new vertex at the center of every triangular face.
2. Connecting this new vertex to the three vertices of the original triangle.
3. Repeating this process $n$ times to generate a family of lattices $\Lambda_n$.

To analyze the thermodynamic properties of these generated families, the authors utilize the star-triangle identity (also known as the Yang-Baxter equation in this context) to perform a decimation of the spins introduced at the centers of the triangles. This allows for the exact mapping of the partition function $Z_n(t)$ of the triangulated lattice $\Lambda_n$ to the partition function $Z_0(h(t))$ of the base lattice, where $t = \tanh(\beta J)$ is the temperature weight and $h(t)$ is a specific recursive function derived from the identity.

The critical weight $t_c$ for the $n$-th iteration is determined by the functional relation $h(t_c^{(n)}) = t_c^{(0)}$, or equivalently $t_c^{(n)} = g(t_c^{(0)})$, where $g = h^{-1}$. The critical temperature is then recovered via $T_c/J = 1/\text{artanh}(t_c)$. The authors analyze the asymptotic behavior of the sequence $t_c^{(n)}$ as $n \to \infty$ using the Stolz–Cesàro theorem to derive the scaling of $T_c$ with respect to $q_{\text{max}}$.

Key Contributions and Results

1. Construction of High-$T_c$ Families: The paper demonstrates that iterative triangulation generates infinite families of lattices where $q_{\text{max}}$ doubles with each iteration ($q_{\text{max}}^{(n)} = 2^n q_{\text{max}}^{(0)}$), while the average coordination number remains fixed at $\bar{q}=6$.
2. Exact Recursive Relations: Explicit expressions are derived for the partition function, free energy per site, and critical temperature of the resulting lattices in terms of the base lattice properties. For example, the critical weight of the Laves-Star lattice (derived from the Triangular lattice) is exactly $T_c/J = 5.007$, and the Compass-Rose lattice (next iteration) is $T_c/J = 6.492$.
3. Asymptotic Scaling Law: For large $n$ (and thus large $q_{\text{max}}$), the critical temperature scales as:
   $$\frac{T_c}{J} \sim A \ln q_{\text{max}} - 2 \ln \ln q_{\text{max}} - K_\Lambda$$
   where $A = 2/\ln 2 \approx 2.89$ is a universal prefactor independent of the base lattice. The term $K_\Lambda$ is a lattice-dependent constant determined by the critical weight of the base lattice.
4. Optimality of Apollonian Lattices: Among the twelve base lattices studied (including Triangular, Laves-CaVO, and various SrCuBO variants), the Triangular lattice yields the smallest $K_\Lambda$ ($K_\Delta \approx 1.024$). Consequently, the family derived from it, termed the Apollonian lattices, achieves the highest $T_c$ for a given $q_{\text{max}}$.
5. Conjectured Upper Bound $T_c^*(q_{\text{max}})$: The authors define a continuous function $T_c^*(q_{\text{max}})$ based on the Apollonian family. They conjecture that this function represents the tightest possible upper bound for the critical temperature of any periodic planar tessellation in Euclidean space with $q_{\text{max}} \ge 6$. This bound is strictly lower than the previously known linear bound $(2/\pi)q_{\text{max}}$ for large $q_{\text{max}}$.
6. Numerical Verification: The authors computed exact $T_c$ values for twelve families of lattices. None of the calculated values exceeded the curve defined by $T_c^*(q_{\text{max}})$, supporting the conjecture.

Significance and Claims
The paper claims to resolve the question of whether $T_c$ can be made arbitrarily large for 2D Ising systems, confirming that it can, but only logarithmically with respect to the maximal coordination number, rather than linearly. The significance lies in:

• Theoretical Optimality: Providing a concrete candidate for the optimal lattice structure (Apollonian) that maximizes $T_c$ for a given connectivity constraint in Euclidean space.
• Exact Solvability: Demonstrating that iterative triangulation preserves the exact solvability of the Ising model, allowing for the derivation of thermodynamic quantities for complex, non-uniform lattices from simple base cases.
• Experimental Relevance: The authors note that these lattices are relevant for theoretical questions of optimality in network systems and may be realized in future experimental platforms such as Coherent Ising Machines (CIMs) or topolectrical circuits, where tunable couplings and specific graph topologies can be engineered.

The authors remain modest regarding the conjecture, stating that the existence of a single counter-example (a planar lattice with $T_c > T_c^*(q_{\text{max}})$) would disprove the claim, establishing this as an open question for future investigation. They also note that while the scaling law holds for hyperbolic lattices, the specific bound $T_c^*$ applies only to Euclidean space.