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Can one hear the shape of a crystal?

Haina Wang, Salvatore Torquato

TL;DR

The paper investigates whether inequivalent crystals can share the same theta series by focusing on the minimum multi-particle basis size $n_{\min}(d)$ in dimensions $d=1,2,3$. It develops a precise multi-objective algorithm that combines $D_{g_2}$ and geometric separations to locate inequivalent isospectral crystals, and proves a 1D rigidity result while presenting a theorem to extend isospectrality to all pair distances in select 2D cases. The study finds $n_{\min}(1)=4$, $n_{\min}(2)=3$, and $n_{\min}(3)=2$, with 2D and 3D examples demonstrating isospectral crystals and distinct higher-order correlations, and it demonstrates ground-state degeneracy under tailored isotropic pair potentials via inverse design. These results imply that pair statistics alone cannot uniquely determine crystal structures and have implications for diffraction interpretation and the design of degenerate crystalline ground states; they also align with the decorrelation principle, suggesting richer degeneracy in higher dimensions.

Abstract

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question ``Can one hear the shape of a drum?'' concerns whether different shaped drums can have the same vibrational modes. The isospectrality of a lattice in $d$-dimensional Euclidean space $\mathbb{R}^d$ is a tantamount to whether it is uniquely determined by its theta series, i.e., the radial distribution function $g_2(r)$. While much is known about the isospectrality of Bravais lattices across dimensions, little is known about this question of more general crystal (periodic) structures with an $n$-particle basis ($n \ge 2$). Here, we ask, What is $n_{\text{min}}(d)$, the minimum value of $n$ for inequivalent (i.e., unrelated by isometric symmetries) crystals with the same theta function in space dimension $d$? To answer these questions, we use rigorous methods as well as a precise numerical algorithm that enables us to determine the minimum multi-particle basis of inequivalent isospectral crystals. Our algorithm identifies isospectral 4-, 3- and 2-particle bases in one, two and three spatial dimensions, respectively. For many of these isospectral crystals, we rigorously show that they indeed possess identical $g_2(r)$ up to infinite $r$. Based on our analyses, we conjecture that $n_{\text{min}}(d) = 4, 3, 2$ for $d = 1, 2, 3$, respectively. The identification of isospectral crystals enables one to study the degeneracy of the ground-state under the action of isotropic pair potentials.

Can one hear the shape of a crystal?

TL;DR

The paper investigates whether inequivalent crystals can share the same theta series by focusing on the minimum multi-particle basis size in dimensions . It develops a precise multi-objective algorithm that combines and geometric separations to locate inequivalent isospectral crystals, and proves a 1D rigidity result while presenting a theorem to extend isospectrality to all pair distances in select 2D cases. The study finds , , and , with 2D and 3D examples demonstrating isospectral crystals and distinct higher-order correlations, and it demonstrates ground-state degeneracy under tailored isotropic pair potentials via inverse design. These results imply that pair statistics alone cannot uniquely determine crystal structures and have implications for diffraction interpretation and the design of degenerate crystalline ground states; they also align with the decorrelation principle, suggesting richer degeneracy in higher dimensions.

Abstract

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question ``Can one hear the shape of a drum?'' concerns whether different shaped drums can have the same vibrational modes. The isospectrality of a lattice in -dimensional Euclidean space is a tantamount to whether it is uniquely determined by its theta series, i.e., the radial distribution function . While much is known about the isospectrality of Bravais lattices across dimensions, little is known about this question of more general crystal (periodic) structures with an -particle basis (). Here, we ask, What is , the minimum value of for inequivalent (i.e., unrelated by isometric symmetries) crystals with the same theta function in space dimension ? To answer these questions, we use rigorous methods as well as a precise numerical algorithm that enables us to determine the minimum multi-particle basis of inequivalent isospectral crystals. Our algorithm identifies isospectral 4-, 3- and 2-particle bases in one, two and three spatial dimensions, respectively. For many of these isospectral crystals, we rigorously show that they indeed possess identical up to infinite . Based on our analyses, we conjecture that for , respectively. The identification of isospectral crystals enables one to study the degeneracy of the ground-state under the action of isotropic pair potentials.

Paper Structure

This paper contains 15 sections, 1 theorem, 29 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Lattices, Crystals and Theta Series
  4. Distance Metrics
  5. Algorithm to Search for Inequivalent Isospectral Crystals
  6. Results for Isospectral Crystals for $d = 1, 2, 3$
  7. 1D Cases
  8. 2D Cases
  9. 3D Cases
  10. Rigorous Results on Isospectrality for All Pair Distances
  11. Many-Body System with Degenerate Crystalline Ground States
  12. Conclusions and Discussion
  13. Theta series for some 2D crystals
  14. Parametrization of Crystals
  15. Parameters of the Isotropic Pair Potential with Degenerate Crystalline Ground States

Key Result

Theorem 1

Let $C_1$ and $C_2$ be two $d$-dimensional crystals of $n$-particle basis with common underlying basis vectors $\mathbf{a}_1, \dots, \mathbf{a}_d$. One has $\Theta_{C_1} \subseteq \Theta_{C_2}$, if for any pair displacement vector $\mathbf{r}_{ij} = \mathbf{p}_i - \mathbf{p}_{j} = \lambda_1 \mathbf{ We remark that by definition [Eq. (thetaC)], $C_1$ and $C_2$ are isospectral if $\Theta_{C_1} \subs

Figures (7)

  • Figure 1: Illustration showing the objective of the algorithm to search for pairs of inequivalent isospectral crystals in the case $d = 2, n = 4$. The orange particles are particles in the fundamental cell $\mathbf{p}_1, \dots \mathbf{p}_4$ [see Eq. (\ref{['pj']})], and the gray particles are images of $\mathbf{p}_1 = \mathbf{0}$. (a) A case with high $D_{g_2}$ and high $\xi$. (b) A case with low $D_{g_2}$ and low $\xi$, which yields crystal structures that are approximately equivalent. (c) A case with low $D_{g_2}$ and high $\xi$, which yields crystals that are close to being inequivalent isospectral. The optimization algorithm aims to search for cases (c) by minimizing the objective function $\Psi$ that involves both $D_{g_2}$ and $\xi$ and attains its deep local minima at cases of inequivalent isospectral crystals. (d) Schematic of the "energy landscape" of the optimization objective $\Psi$ as a function of the crystal parameters of $C_1$ and $C_2$. The cases (a)--(c) are indicated as local minima in the energy landscape.
  • Figure 2: (a) The case of 1D inequivalent isospectral crystals with 4-particle bases identified via our algorithm. (b) Illustration showing that two 1D isospectral crystals with 3-particle bases are equivalent, as they are related by a translation and an inversion; see the configurations in the fundamental cells indicated by the yellow boxes.
  • Figure 3: (a) Configurations of the pair of 2D inequivalent isospectral crystals with 3-particle bases that attains the lowest value of $\Psi(C_1, C_2)$ (\ref{['Psi']}) among all isospectral 2D 3-particle bases identified via our algorithm using deformable fundamental cells. The fundamental cells are oblique and are are indicated with parallelograms with red borders. (b) Radial distribution function $g_2(r)$ of both crystals in (a).
  • Figure 4: Configurations of 2D inequivalent isospectral crystals with 3-particle bases identified via our algorithm using fixed fundamental cells that have more symmetry elements than simple oblique cells. The fundamental cells are indicated with parallelograms with red borders. (a) A pair of isospectral crystals with hexagonal fundamental cells, with cmm and p3m1 symmetries, respectively. (b) Radial distribution function $g_2(r)$ of both crystals in (a). (c) A pair of isospectral crystals with square fundamental cells, both with pm symmetry. (d) $g_2(r)$ of both crystals in (c). (e) A pair of isospectral crystals with rhombic fundamental cells, with pgg and pmm symmetries, respectively. (f) $g_2(r)$ of both crystals in (e). All radial distribution functions are plotted with bin size $0.05\rho^{-1/2}$.
  • Figure 5: (a) A pair of 3D inequivalent isospectral crystals with 2-particle bases, identified via our algorithm using deformable fundamental cells. The fundamental cells are triclinic. (b) Radial distribution function of both crystals shown in (a), plotted with bin size $0.05\rho^{-1/3}$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof