On crystallization in the plane for pair potentials with an arbitrary norm

This paper establishes that two-dimensional crystallization occurs for the Heitmann-Radin sticky disk potential under any arbitrary norm, proving that minimizers are affine transforms of triangular or square lattices determined by the norm's kissing number, while also constructing explicit $p$-norm families for lattice crystallization and numerically identifying unexpected phase transitions in minimizers for Lennard-Jones and Epstein zeta potentials.

The Gist

Imagine you are a city planner trying to build the most efficient city possible. You have a huge number of identical houses (particles) and a set of rules about how they interact with each other. Some rules say, "Stay close, but not too close," while others say, "If you get too close, you explode; if you get too far, you drift apart."

Your goal is to arrange these houses so that the total "stress" or "energy" of the city is as low as possible. In physics, this is called crystallization: figuring out why atoms naturally snap into perfect, repeating patterns (crystals) rather than forming messy piles.

This paper explores a fascinating twist on that problem: What happens if the "rules of space" themselves are weird?

1. The Weird Ruler (The Arbitrary Norm)

In our normal world, distance is measured with a standard ruler (Euclidean distance). If you walk 3 steps east and 4 steps north, you are 5 steps away from where you started (thanks to the Pythagorean theorem).

But in this paper, the authors imagine a world where the "ruler" is different. They use arbitrary norms.

• The "Taxi Driver" Ruler ($L_1$): Imagine you can only walk along city streets (grid lines). To get from A to B, you can't cut diagonally through buildings. You have to walk the sum of the horizontal and vertical distances.
• The "King's Move" Ruler ($L_\infty$): Imagine a chess King. You can move one square in any direction (including diagonally). The distance is determined by the longest single step you take, not the total path.

The authors ask: If we change the ruler, does the "perfect city" (the crystal) change shape?

2. The Sticky Disks (The Simplest Case)

First, they look at a very simple rule called the Heitmann-Radin "Sticky Disk" potential.

• The Rule: If two houses are exactly 1 unit apart, they stick together happily (energy = -1). If they are closer than 1, they repel violently (energy = infinity). If they are farther apart, they don't care (energy = 0).

The Discovery:
In our normal world (standard ruler), the best way to pack these sticky disks is in a honeycomb (triangular) pattern. Every house has 6 neighbors.

But when the authors changed the ruler:

• If the ruler is "Taxi-style" ($L_1$) or "King-style" ($L_\infty$): The best pattern changes! The houses arrange themselves in a square grid. Now, every house has 8 neighbors that are "equally close" in this weird geometry.
• If the ruler is anything else (mostly): The pattern stays honeycomb (triangular).

The Analogy: Think of the "kissing number." In a normal circle, you can fit 6 coins around a central coin. In a square world (Taxi/King rules), you can fit 8 coins around a central one because the "circle" looks like a square or diamond. The crystal shape simply adapts to fit the most neighbors possible under the new rules.

The Big Takeaway: You can force atoms to form a square crystal or a triangular crystal just by changing the geometry of space, without changing the atoms themselves. This explains how anisotropy (directional differences) naturally emerges in crystals.

3. The Real-World Mess (Lennard-Jones Potential)

Next, they looked at a more realistic, complex rule called the Lennard-Jones potential. This is the rule used to model real atoms (like in water or gold).

• The Rule: Atoms attract each other if they are a bit far apart (like magnets), but repel fiercely if they get too close (like trying to push two north poles together).

In our normal world, we know the triangular honeycomb is usually the winner. But what happens with the weird rulers?

The Surprise:
The authors ran computer simulations to see what the "perfect city" looks like for different rulers. They found a phase transition (a sudden switch in the winning shape) that nobody expected.

• For some rulers: The triangular honeycomb wins.
• For others: The square grid wins.
• For a weird middle ground: The city doesn't look like a perfect square or a perfect honeycomb at all! It looks like a squashed, tilted, or distorted grid.

It's as if you told the city planner, "Build the most efficient city," and for some strange rulers, the best answer wasn't a square or a hexagon, but a weird, stretched-out shape that only exists in that specific mathematical universe.

4. Why Does This Matter?

• Understanding Materials: This helps scientists understand why some materials form strange, non-standard crystal structures. It suggests that the "shape" of the space (or the forces acting on the atoms) dictates the structure, not just the atoms themselves.
• Designing New Materials: If we can engineer materials where the "effective distance" between atoms is manipulated (perhaps by pressure or external fields), we might be able to force them to form specific, useful shapes (like perfect squares) that don't happen naturally.
• Mathematical Beauty: It shows that even in simple 2D worlds, the answer to "what is the best shape?" is surprisingly complex and depends entirely on how you measure distance.

Summary in One Sentence

This paper proves that if you change the way you measure distance in a 2D world, the perfect arrangement of atoms will shift from a honeycomb to a square grid, or even to strange, distorted shapes, revealing that the "shape" of space is just as important as the atoms themselves in building crystals.

Technical Summary

1. Problem Statement

The paper investigates the crystallization phenomenon in two dimensions, specifically focusing on the emergence of periodic lattice structures as minimizers of interaction energies. The study generalizes previous results by considering:

• Arbitrary Norms: Instead of the standard Euclidean norm ($\|\cdot\|_2$), the authors analyze pair potentials of the form $V(\|x - y\|)$, where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$.
• Two Potential Types:
  1. Heitmann-Radin Sticky Disk Potential ($V_{HR}$): A hard-core potential with a single attractive well at distance 1. The goal is finite crystallization (proving that for a finite number of particles $N$, the ground state is a patch of a specific lattice).
  2. Lennard-Jones Potential ($V_{LJ}$) and Epstein Zeta Functions: Attractive-repulsive potentials ($r^{-12} - 2r^{-6}$ and $r^{-s}$). The goal here is thermodynamic limit crystallization (minimizing energy per point among unit-density lattices).

The central question is how the geometry of the norm (specifically its unit sphere) influences the optimal lattice structure (triangular vs. square vs. anisotropic) and whether anisotropy can emerge naturally from the norm itself rather than being artificially imposed.

2. Methodology

A. Geometric Classification of Norms

The authors utilize a classification of norms based on their kissing number (the maximum number of non-overlapping unit balls that can touch a central unit ball), denoted as $K_{\|\cdot\|}$.

• Class $\mathcal{N}_8$: Norms where the unit sphere $S_{\|\cdot\|}$ is a parallelogram (e.g., $\ell_1$ and $\ell_\infty$ norms). Here, $K_{\|\cdot\|} = 8$.
• Class $\mathcal{N}_6$: Norms where the unit sphere is strictly convex (e.g., $\ell_p$ for $1 < p < \infty$). Here, $K_{\|\cdot\|} = 6$.

B. Finite Crystallization (Sticky Disk)

For the Heitmann-Radin potential, minimizing energy is equivalent to maximizing the number of pairs of points at the minimal distance (distance 1).

• Key Tool: The authors rely on Brass's combinatorial geometry results (generalizing Harborth's work) which determine the maximum number of unit distances in a set of $N$ points for a given norm.
• Construction: They construct specific configurations:
  • $H_N$: Hexagonal patches on the triangular lattice $\mathcal{A}_2$ (optimal for $\mathcal{N}_6$).
  • $Z_N$: Octagonal patches on the square lattice $\mathbb{Z}^2$ (optimal for $\mathcal{N}_8$).
• Transformation: They use linear maps to transform these canonical configurations to any arbitrary lattice $L$ within the respective norm class.

C. Lattice Minimization (Lennard-Jones & Epstein Zeta)

For the continuous potentials, the problem is reduced to minimizing energy per point over the space of unit-density lattices $\mathcal{L}_2(1)$.

• Reduction: Using the homogeneity of the Epstein zeta function $\zeta_{L, \|\cdot\|}(s)$, they reduce the minimization over all dilated lattices to a minimization over unit-density lattices.
• Numerical Investigation: Since rigorous proofs for Lennard-Jones minimizers with arbitrary norms are difficult, the authors employ numerical simulations (Nelder-Mead algorithm and contour plots) on the half-fundamental domain $D$ of the lattice space. They parametrize lattices by $(x, y)$ and compute energies for various $p$-norms ($\|\cdot\|_p$).

3. Key Contributions and Results

A. Finite Crystallization with Arbitrary Norms

The paper provides a complete classification of minimizers for the sticky disk potential based on the norm's kissing number:

• Theorem 3.2:
  • If $\|\cdot\| \in \mathcal{N}_6$ (strictly convex unit sphere), the minimizer is a patch of a lattice $L$ that is an affine transform of the triangular lattice ($\mathcal{A}_2$). The minimal energy is $-\lfloor 3N - \sqrt{12N-3} \rfloor$.
  • If $\|\cdot\| \in \mathcal{N}_8$ (parallelogram unit sphere), the minimizer is a patch of a lattice $L$ that is an affine transform of the square lattice ($\mathbb{Z}^2$). The minimal energy is $-\lfloor 4N - \sqrt{28N-12} \rfloor$.
• Anisotropy Construction: The authors explicitly construct an infinite family of norms $\{N_{p,L}\}$ such that a specific given lattice $L$ becomes the unique minimizer. This demonstrates how to engineer anisotropy in crystallization purely through the choice of the interaction norm.
• Application to $p$-norms:
  • For $p=1$ and $p=\infty$, the minimizer is a patch of the square lattice (affine transformed).
  • For $1 < p < \infty$, the minimizer is a patch of the triangular lattice (affine transformed).
• Soft Potential on $\mathbb{Z}^2$: They solve the minimization of the soft potential $V_{BPD}$ constrained to points on $\mathbb{Z}^2$, showing it yields the same energy as the sticky disk potential with the $\ell_\infty$ norm.

B. Lennard-Jones and Epstein Zeta Minimization

The numerical study reveals unexpected phase transitions in the optimal lattice structure as the parameter $p$ of the norm $\|\cdot\|_p$ varies:

• Epstein Zeta Function ($\zeta(s)$):
  • For $p \in [1, 2]$, the minimizer is the triangular lattice.
  • For $p \in (p_1, p_2)$ (where $p_1 \approx 2.45, p_2 \approx 5.45$), the minimizer is the square lattice.
  • For $p > p_2$, the minimizer transitions to a non-standard, anisotropic lattice (parametrized by $(1/2, y_p)$).
• Lennard-Jones Energy:
  • The phase diagram is even more complex. While $p=1$ favors the square lattice and $p=2$ favors the triangular lattice, intermediate values of $p$ lead to a sequence of transitions involving anisotropic lattices before returning to the square lattice for large $p$.
• Significance of Phase Transitions: Unlike the Euclidean case where the triangular lattice is optimal for both sticky disk and Lennard-Jones potentials, the arbitrary norm case shows that the Lennard-Jones potential does not always select the same lattice as the sticky disk potential. The attractive-repulsive nature of $V_{LJ}$ interacts with the anisotropy of the norm to produce new, non-triangular, non-square ground states.

4. Significance

1. Generalization of Crystallization Theory: The work extends the classical Heitmann-Radin results from the Euclidean setting to arbitrary norms, proving that crystallization occurs for any norm but the resulting lattice structure depends fundamentally on the geometry of the norm's unit sphere (specifically its kissing number).
2. Control of Anisotropy: The paper provides a rigorous mathematical framework to "design" the ground state of a crystal. By choosing a specific norm (or constructing a norm $N_{p,L}$), one can force the system to crystallize on a desired lattice $L$, offering a theoretical tool for material design.
3. Discovery of New Phase Transitions: The numerical results for Lennard-Jones potentials reveal a rich phase diagram where the optimal lattice changes from triangular to square to anisotropic and back as the norm parameter $p$ varies. This challenges the intuition that the triangular lattice is universally optimal in 2D for short-range interactions.
4. Bridging Discrete and Continuous: The paper connects discrete finite-particle problems (finite crystallization) with thermodynamic limit problems (lattice energy minimization), showing how the "kissing number" dictates the finite case, while the interplay of attraction/repulsion and norm geometry dictates the infinite case.

In summary, this paper establishes that the geometry of the interaction norm is a primary driver of crystallization patterns, capable of inducing anisotropy and complex phase transitions that are not present in standard isotropic models.