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Unified criteria for crystallization in hard-core lattice systems with applications to polyomino fluids and chiral mixtures

Qidong He

TL;DR

The work develops a unified volume-allocation criterion for crystallization in hard-core lattice systems with infinite interactions, extending Pirogov--Sinai theory via the Mazel--Stuhl--Suhov framework and coarse-graining to a common supercell. By constructing translation-covariant, lower semi-continuous allocations that minimize local volume, the authors reduce global density optimization to local volume considerations and verify a Peierls condition for contours. The approach identifies periodic ground states with perfect configurations and proves high-fugacity crystallization, demonstrating its applicability to polyomino fluids and chiral mixtures using discrete or continuous Voronoi tessellations. Consequences include rigorous crystallization results for polyomino tilings with finite tilings and broader implications for models with multiple crystal structures and rotational degrees of freedom, thereby broadening the landscape of systems amenable to Pirogov--Sinai-type analyses at high density.

Abstract

We present a unified extension of two sets of criteria for high-fugacity crystallization in hard-core lattice systems developed previously by Jauslin, Lebowitz, and the author. Our new criterion is formulated in terms of the existence of a volume allocation rule with desirable properties, in analogy to the scoring function constructed in Hales' proof of the Kepler conjecture. The proof uses a recent systematic extension of Pirogov--Sinai theory to systems with infinite interactions by Mazel--Stuhl--Suhov. Notably, our result applies to a large class of polyomino models with discrete rotational degrees of freedom and chiral mixtures thereof.

Unified criteria for crystallization in hard-core lattice systems with applications to polyomino fluids and chiral mixtures

TL;DR

The work develops a unified volume-allocation criterion for crystallization in hard-core lattice systems with infinite interactions, extending Pirogov--Sinai theory via the Mazel--Stuhl--Suhov framework and coarse-graining to a common supercell. By constructing translation-covariant, lower semi-continuous allocations that minimize local volume, the authors reduce global density optimization to local volume considerations and verify a Peierls condition for contours. The approach identifies periodic ground states with perfect configurations and proves high-fugacity crystallization, demonstrating its applicability to polyomino fluids and chiral mixtures using discrete or continuous Voronoi tessellations. Consequences include rigorous crystallization results for polyomino tilings with finite tilings and broader implications for models with multiple crystal structures and rotational degrees of freedom, thereby broadening the landscape of systems amenable to Pirogov--Sinai-type analyses at high density.

Abstract

We present a unified extension of two sets of criteria for high-fugacity crystallization in hard-core lattice systems developed previously by Jauslin, Lebowitz, and the author. Our new criterion is formulated in terms of the existence of a volume allocation rule with desirable properties, in analogy to the scoring function constructed in Hales' proof of the Kepler conjecture. The proof uses a recent systematic extension of Pirogov--Sinai theory to systems with infinite interactions by Mazel--Stuhl--Suhov. Notably, our result applies to a large class of polyomino models with discrete rotational degrees of freedom and chiral mixtures thereof.
Paper Structure (14 sections, 11 theorems, 31 equations, 2 figures)

This paper contains 14 sections, 11 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Volume allocations and crystallization
  3. The model
  4. The assumptions
  5. Crystallization
  6. Periodic ground states
  7. The Peierls condition
  8. Deduction of Theorem \ref{['thm:main']}
  9. Applications
  10. Construction of volume allocations
  11. Discrete Voronoi tessellation
  12. Voronoi tessellation
  13. Other tessellations
  14. Polyominoes with a finite number of tilings

Key Result

Theorem 2.3

Under Assumptions asm:optimization and asm:supercell, there exists a universal constant $z_0>0$ such that the following holds. Index the perfect configurations by $q$. For all $z\ge z_0$ and every perfect configuration $\overline{X}^q$, the model eqn:formal Hamiltonian admits an $\mathcal{L}^F$-inva

Figures (2)

  • Figure 1: The only crystal (tiling) structures of the chiral $Z$-pentomino on the square lattice up to translations and rotations martin1991polyominoes
  • Figure 2: Other examples of polyominoes that tile $\mathbb{Z}^2$ in multiple non-congruent ways when rotations, or reflections as well, are allowed martin1991polyominoes

Theorems & Definitions (24)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 14 more