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Critical modular lattices in the Gaussian core model

Arian Joharian, Frank Vallentin, Marc Christian Zimmermannn

TL;DR

This work analyzes the local behavior of Gaussian core model energy for modular lattices by leveraging spherical designs and modular forms. It derives gradient and Hessian formulas for $\mathcal{E}(\alpha,L)$, showing that shells forming 2- or 4-designs greatly simplify the analysis and sometimes force uniform Hessian spectra, enabling precise stability conclusions. The paper provides explicit bounds for Eisenstein and cusp form coefficients, and uses these to study notable lattices such as the 3-modular Coxeter-Todd lattice $K_{12}$ and the 2-modular Barnes-Wall lattice $BW_{16}$, proving they are not locally universally optimal but can be local maxima for suitable $\alpha$. It further demonstrates that even extremal lattices can be non critical and outlines a program for extending the approach to other extremal modular lattices, given additional cusp form data and coefficient bounds. The results advance understanding of energy landscapes in the Gaussian core model and highlight the deep link between lattice design properties and optimality in high dimensional sphere packings.

Abstract

We discuss the local analysis of Gaussian potential energy of modular lattices. We show for instance that the $3$-modular $12$-dimensional Coxeter-Todd lattice and the $2$-modular $16$-dimensional Barnes-Wall lattice, which both provide excellent sphere packings, are not, even locally, universally optimal (in the sense of Cohn and Kumar).

Critical modular lattices in the Gaussian core model

TL;DR

This work analyzes the local behavior of Gaussian core model energy for modular lattices by leveraging spherical designs and modular forms. It derives gradient and Hessian formulas for , showing that shells forming 2- or 4-designs greatly simplify the analysis and sometimes force uniform Hessian spectra, enabling precise stability conclusions. The paper provides explicit bounds for Eisenstein and cusp form coefficients, and uses these to study notable lattices such as the 3-modular Coxeter-Todd lattice and the 2-modular Barnes-Wall lattice , proving they are not locally universally optimal but can be local maxima for suitable . It further demonstrates that even extremal lattices can be non critical and outlines a program for extending the approach to other extremal modular lattices, given additional cusp form data and coefficient bounds. The results advance understanding of energy landscapes in the Gaussian core model and highlight the deep link between lattice design properties and optimality in high dimensional sphere packings.

Abstract

We discuss the local analysis of Gaussian potential energy of modular lattices. We show for instance that the -modular -dimensional Coxeter-Todd lattice and the -modular -dimensional Barnes-Wall lattice, which both provide excellent sphere packings, are not, even locally, universally optimal (in the sense of Cohn and Kumar).

Paper Structure

This paper contains 31 sections, 13 theorems, 152 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The Gaussian core model
  3. Modular lattices in the Gaussian core model
  4. Structure of the paper and main results
  5. Preliminaries
  6. Spherical designs
  7. Theta series as modular forms
  8. Strongly modular lattices
  9. Extremal modular lattices and spherical designs
  10. Strategy
  11. The gradient in the presence of $2$-designs
  12. The Hessian in the presence of $4$-designs
  13. The Hessian in the presence of $2$-designs
  14. Asymptotic behavior and properties of spaces of cusp forms
  15. Explicit bounds on the coefficients of modular forms
  16. ...and 16 more sections

Key Result

Theorem 1.2

The $3$-modular $12$-dimensional Coxeter-Todd lattice $K_{12}$ and the $2$-modular $16$-dimensional Barnes-Wall lattice $\operatorname{BW}_{16}$ are not locally universally optimal.

Figures (1)

  • Figure 1: Eigenvalues of the Hessian for $L_{16}$

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Lemma 2.1 in Heimendahl2023
  • Lemma 3.1
  • Theorem 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • ...and 12 more