A common theme of Torquato’s research work is the search for unifying and rigorous principles to elucidate a broad range of physical phenomena.

[23][24] [25][26] A study in 2019 has uncovered that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which are called effectively limit-periodic.

Random media abound in nature and synthetic situations, and include composites, thin films, colloids, packed beds, foams, microemulsions, blood, bone, animal and plant tissue, sintered materials, and sandstones.

Torquato broke this impasse by providing a unified rigorous means of characterizing the microstructures and macroscopic properties of widely diverse random heterogeneous media.

In an article published in Physical Review X in 2021, Torquato and Jaeuk Kim formulated the first “nonlocal” exact formula for the effective dynamic dielectric constant tensor for general composite microstructures that accounts for multiple scattering of electromagnetic waves to all orders.

[35][36][37] An outcome is the quantitative and definitive demonstration that pair information of a disordered many-particle system is insufficient to uniquely determine a representative configuration and identified more sensitive structural descriptors beyond the standard three-, four-body distribution functions, which is of enormous significance in the study of liquid and glassy states of matter.

He has made seminal contributions[peacock prose] to the understanding of the venerable hard-sphere model, which has been invoked to study local molecular order, transport phenomena, glass formation, and freezing behavior in liquids.

Toward the quantification of randomness Torquato and colleagues pioneered the powerful notion[peacock prose] of "order metrics and maps" to characterize the degree of order/disorder in many-particle systems.

Torquato along with his co-workers have used order metrics to provide novel insights into the structural, thermodynamical, and dynamical nature of molecular systems, such as Lennard-Jones liquids and glasses,[43] water[44] and disordered ground states of matter,[45] among other examples.

[50] Growing length dcales upon supercooling a liquid In 2013, Marcotte, Stillinger and Torquato demonstrated that a sensitive signature of the glass transition of atomic liquid models is apparent well before the transition temperature Tc is reached upon supercooling as measured by a length scale determined from the volume integral of the direct correlation function c(r), as defined by the Ornstein-Zernike equation.

Subsequently, it was shown computationally that perfect glasses possess unique disordered classical ground states up to trivial symmetries and hence have vanishing entropy: a highly counterintuitive situation.

[54] This was made possible by pioneering the idea of scalar metrics of order (or disorder), which opened new avenues of research in condensed-matter physics, and by introducing mathematically precise jamming categories.

Torquato’s work on polyhedra spurred a flurry of activity in the physics and mathematics communities to determine the densest possible packings of such solids, including dramatic improvements on the density of regular tetrahedra.

[60][61][62] Disordered sphere packings may win in high dimensions Torquato and Stillinger derived a conjectural lower bound on the maximal density of sphere packings in arbitrary Euclidean space dimension d whose large-d asymptotic behavior is controlled by 2-(0:77865...)d. This work may remarkably provide the putative exponential improvement on Minkowski’s 100-year-old bound for Bravais lattices, the dominant asymptotic term of which is 1/2d.

[66] In a seminal article[peacock prose] published in 2003, Torquato and Stillinger introduced the "hyperuniformity" concept to characterize the large-scale density fluctuations of ordered and disordered point configurations.

Torquato and co-workers have contributed to these developments[68] by showing that these exotic states of matter can be obtained via both equilibrium and nonequilibrium routes and come in both quantum mechanical and classical varieties.

The study of hyperuniform states of matter is an emerging multidisciplinary field, influencing and linking developments across the physical sciences, mathematics and biology.

In particular, the hybrid crystal-liquid attribute of disordered hyperuniform materials endows them with unique or nearly optimal, direction-independent physical properties and robustness against defects, which makes them an intense subject of research.

[72] Recently, Duyu Chen and Torquato formulated a Fourier space-based optimization approach to construct, at will, two-phase hyperuniform media with prescribed spectral densities.

[73] To more completely characterize density fluctuations of point configurations, Torquato, Kim and Klatt carried out an extensive theoretical and computational study of the higher-order moments or cumulants, including the skewness, excess kurtosis, and the corresponding probability distribution function of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order, and determined when a central limit theorem was achieved.

Elsewhere, Lomba, Torquato and co-workers presented the first statistical-mechanical model that rigorously achieves disordered multihyperuniformity in ternary mixtures to sample the three primary colors: red, blue and green.

[79] Stealthy and hyperuniform disordered ground states Torquato, Stillinger and colleagues pioneered the collective-coordinate numerical optimization approach to generate systems of particles interacting with isotropic "stealthy" bounded long-ranged pair potentials (similar to Friedel oscillations) whose classical ground states are counterintuitively disordered, hyperuniform, and highly degenerate across space dimensions.

A singular feature of such systems is that dimensionality of the configuration space depends on the fraction of such constrained wave vectors compared to the number of degrees of freedom.

[87] Torquato and Chen discovered that the effective thermal (or electrical) conductivities and elastic moduli of 2D disordered hyperuniform low-density cellular networks are optimal under the constraint of statistical isotropy.

Kim and Torquato formulated a new tessellation-based computational procedure to design extremely large perfectly hyperuniform disordered dispersions (more than 108 particles) for materials discovery via 3D printing techniques.

More recently, Ma, Lomba and Torquato a feasible experimental protocol to create very large hyperuniform systems was proposed using binary paramagnetic colloidal particles.