• Open Access

Ensemble Theory for Stealthy Hyperuniform Disordered Ground States

S. Torquato, G. Zhang, and F. H. Stillinger
Phys. Rev. X 5, 021020 – Published 29 May 2015

Abstract

It has been shown numerically that systems of particles interacting with isotropic “stealthy” bounded long-ranged pair potentials (similar to Friedel oscillations) have classical ground states that are (counterintuitively) disordered, hyperuniform, and highly degenerate. Disordered hyperuniform systems have received attention recently because they are distinguishable exotic states of matter poised between a crystal and liquid that are endowed with novel thermodynamic and physical properties. The task of formulating an ensemble theory that yields analytical predictions for the structural characteristics and other properties of stealthy degenerate ground states in d-dimensional Euclidean space Rd is highly nontrivial because the dimensionality of the configuration space depends on the number density ρ and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure for finding a particular ground-state configuration. The purpose of this paper is to take some initial steps in this direction. Specifically, we derive general exact relations for thermodynamic properties (energy, pressure, and isothermal compressibility) that apply to any ground-state ensemble as a function of ρ in any d, and we show how disordered degenerate ground states arise as part of the ground-state manifold. We also derive exact integral conditions that both the pair correlation function g2(r) and structure factor S(k) must obey for any d. We then specialize our results to the canonical ensemble (in the zero-temperature limit) by exploiting an ansatz that stealthy states behave remarkably like “pseudo”-equilibrium hard-sphere systems in Fourier space. Our theoretical predictions for g2(r) and S(k) are in excellent agreement with computer simulations across the first three space dimensions. These results are used to obtain order metrics, local number variance, and nearest-neighbor functions across dimensions. We also derive accurate analytical formulas for the structure factor and thermal expansion coefficient for the excited states at sufficiently small temperatures for any d. The development of this theory provides new insights regarding our fundamental understanding of the nature and formation of low-temperature states of amorphous matter. Our work also offers challenges to experimentalists to synthesize stealthy ground states at the molecular level.

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  • Received 9 January 2015

DOI:https://doi.org/10.1103/PhysRevX.5.021020

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

S. Torquato*

  • Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

G. Zhang and F. H. Stillinger

  • Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

  • *torquato@electron.princeton.edu

Popular Summary

When slowly cooling a typical liquid we expect that it will undergo a freezing transition to a solid phase at some temperature and then attain a unique perfectly ordered crystal ground-state configuration. Particles interacting with “stealthy” long-ranged pair potentials have classical ground states that are—counterintuitively—disordered, hyperuniform, and highly degenerate. These exotic amorphous states of matter are endowed with novel thermodynamic and physical properties. Previous investigations of these unusual systems relied heavily on computer-simulation techniques. The task of formulating a theory that yields analytical predictions for the structural characteristics and physical properties of stealthy degenerate ground states in multiple dimensions presents many theoretical challenges. A new type of statistical-mechanical theory, as we show here, must be invented to characterize these exotic states of matter.

We derive general exact relations for the thermodynamic (energy, pressure, and compressibility) and structural properties that apply to any ground-state ensemble as a function of its density and dimension in the zero-temperature limit. We show how disordered, degenerate ground states arise as part of the ground-state manifold. We then specialize our results to the canonical ensemble by exploiting an ansatz that stealthy states behave remarkably like “pseudo”-equilibrium hard-sphere systems in Fourier space. This mapping enables us to obtain theoretical predictions for the pair correlation function, structure factor, and other structural characteristics that are in excellent agreement with computer simulations across one, two, and three dimensions. We also derive accurate analytical formulas for the properties of the excited states.

Our results provide new insights into the nature and formation of low-temperature states of amorphous matter. Our work also offers challenges to experimentalists to synthesize stealthy ground states at the molecular level.

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Vol. 5, Iss. 2 — April - June 2015

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