Photonic network solids exhibit surprising universal behavior

Together with Paul Steinhardt and Salvatore Torquato, I performed an intensive simulation study of the photonic band gaps of crystal and disordered networks. Our surprising discovering has now been published in Physical Review Letters.

M. A. Klatt, P. J. Steinhardt, S. Torquato. Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks. Phys. Rev. Lett. 127:037401-1–6 (2021).

Photonic solids possessing large complete band gaps that block light waves in all directions and polarizations for a range of frequencies are key characteristics in photonic devices because they act as semiconductors for light. One of the computational challenges is finding the largest band gap possible for a given symmetry and structure type, e.g., a three-dimensional heterostructure in the form of a network of high dielectric material. The problem is inherently nonlinear, time-consuming to compute numerically, and requires many repetitions in order to optimize over the numerous network parameters (such as the network rod radius and the dielectric constant of the material). Hence, it is both surprising and of great practical use to discover universal behaviors shared by a wide class of photonics solids, as is identified in this Letter for the case of network solids, since it greatly reduces the computations needed to find optimal structures. In particular, we have identified a quantity called the gap-sensitivity (a measure of how much the optimal band gap changes with dielectric constant) that is to good approximation the same for a wide range of crystal networks and, for high dielectric constants, the same for a wide range of disordered networks. Even more specifically, the universal behavior of the gap sensitivity for these three-dimensional networks is well described by the analytic formula for an optimal solid composed of a periodic stack of alternating planar slabs of different dielectrics. As a result, given a new network, one only has to compute the band gap at a single value of the dielectric constant to predict the optimal band gap for all values. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.

Photonic crystal networks: the crystal diamond network (left) and a nearly-hyperuniform network (right).

The Supplemental Videos also show how the band structure plots change with the dielectric contrast.


Predicting flow through porous media via pore-space descriptors

Together with Robert Ziff and Salvatore Torquato, I determined the void percolation thresholds around hard and overlapping sphere models, including MRJ sphere packings and our amorphous inherent structures of the quantizer energy. The latter have a remarkably low critical porosity.

M. A. Klatt, R. M. Ziff, S. Torquato. Critical pore radius and transport properties of disordered hard- and overlapping-sphere models. Phys. Rev. E 104:014127-1–10 (2021).

Fluid flow through porous media plays a crucial role in many applications, from groundwater hydrology to industrial filtration. Our aim is to find convenient yet reliable estimates of the fluid permeability based on the structural and topological characteristics of the complex tortuous pore space. Such predictions can facilitate the design of porous media with desirable transport properties. For porous media with a well-connected pore space, a recent study suggested the second moment of the pore-size distribution as a convenient alternative to the often-used critical pore radius, which requires a sophisticated percolation analysis. We determine both descriptors for disordered and ordered model microstructures, including maximally random jammed (MRJ) spheres, overlapping spheres, equilibrium hard spheres, quantizer configurations, and lattice packings. Interestingly. we find that the second moment of the pore-size distribution is — to a good approximation — proportional to the critical pore radius. In fact, in contrast to the latter, the former predicts the correct ranking of the permeability for our models. Moreover, we find that the hyperuniform structures, which are characterized by an anomalous suppression of volume-fraction fluctuations, tend to have lower values of the permeability, including MRJ sphere packings, quantizer configurations, and BCC sphere packings.


Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems

Our systematic study of local number fluctuations in both hyperuniform and nonhyperuniform systems has revealed some intriguing behaviors and provides useful tools for further studies:

Popular Summary by Physical Review X:
“The emerging field of ‘hyperuniformity’ provides mathematical tools for classifying large ensembles of particles based on the degree to which density fluctuations are suppressed according to their first two moments (the expected value and the variance). Originally formulated to describe how atoms are positioned in crystals, quasicrystals, and amorphous solids, the framework is useful in many studies, from the distribution of prime numbers to photoreceptor cells in avian retina. Here, we stress the importance of characterizing the higher-order moments of any system, hyperuniform or not.

To more completely characterize density fluctuations, we carry out an extensive theoretical and computational study of higher-order moments and the corresponding probability distribution function of a large class of models across the first three space dimensions. These models describe both hyperuniform and nonhyperuniform systems, that is, those in which density fluctuations are greatly suppressed and those in which they are not. Remarkably, we discover that diverse systems—encompassing the majority of our models—can be described by ‘universal’ distribution functions.

Our work elucidates the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus it has broad implications for condensed-matter physics, engineering, mathematics, and biology.”


A few more details:
We derived rigorous bounds on higher-moments of the number distributions, beyond the number variance, and we developed a robust Gaussian “distance” metric for number distributions. We inferred the presence of three- and higher-body correlations in some of our models by the behaviors of the higher-order moments. Moreover, we showed for models that decorrelate or correlate, as the space dimension increases, that fluctuations become increasingly Gaussian or non-Gaussian, respectively. Finally, we found a universal behavior at intermediate sizes of the observation windows in that the number distributions are well approximated by gamma distributions for all of our distinctly different models that obey a CLT. For our models, we thus found that the convergence to a CLT is slower for standard nonhyperuniform models and slowest for the ‘antihyperuniform’ model studied here.

Supplementary dataset:
We have published all results of our analysis on Zenodo:


Cloaking the underlying long-range order of randomly perturbed lattices

How can we cloak the Bragg peaks of a lattice with random perturbations and will traces of the underlying long-range order persist nevertheless? Our affirmative answer to both questions has been published as an Editors’ Suggestion:

M. A. Klatt, J. Kim, S. Torquato. Cloaking the underlying long-range order of randomly perturbed lattices. Phys. Rev. E 101, 032118-1–9 (2020)

The Editors of Physical Review E have suggested our paper as a Highlight:
“When a regular lattice is perturbed, traces of the original lattice usually remain visible as Bragg peaks in the diffraction pattern. The authors of this paper consider uniformly randomized lattices and show that fine-tuned distributions of perturbations can hide the Bragg peaks. Interestingly, as the strength of the perturbations increases, long-range order oscillates and Bragg peaks appear and disappear.”


Keynote on a hidden order among disorder

It has been my great pleasure to open the 15th International Congress for Stereology and Image Analysis in Aarhus, Denmark, with my keynote on:

How to find a hidden order among disorder

where I gave an introduction and overview on hyperuniformity. The study of this anomalous suppression of density fluctuations on large length scales has shed light on a variety of seemingly unrelated fields, from the eyes of chicken to exotic many-particle ensembles and random matrices. The unique properties of hyperuniform amorphous materials (with a hidden order such that the system remains macroscopically uniform, despite not being crystalline) have recently led to intense research in physics, mathematics, material science, and biology. Aiming for an intuitive understanding of the rigorous mathematical definitions, I presented both basic concepts and recent examples.

I thank the organizers for a great conference in the last week of May 2019 with many stimulating discussions across disciplines.


Focus Session at the APS March Meeting 2019

It has been my great pleasure to organize together with Lisa Manning, Gregory Grason, and Gerd Schröder-Turk a GSOFT Focus Session at the APS March Meeting in Boston this year:

"Hyperuniformity and Optimal Tessellations: Structure, Formation and Properties"

After recent breakthroughs in the search for ordered optimal tessellations (for example, including Frank-Kasper phases in copolymer melts), now findings of the optimal properties of amorphous tessellations are emerging, e.g., in biological tissues.

At the same time, there have been intensive studies of amorphous systems with an anomalous suppression of density fluctuations on large length scales, known as hyperuniformity. This geometric concept qualitatively and quantitatively characterizes a hidden-order in amorphous states that allows for unique physical properties – combining those of crystalline and disordered phases. Thus it offers candidates for optimal amorphous tessellations of space.

The two invited speakers of our session on Thursday, March 7, were Jasna Brujic and Salvatore Torquato.


Universal hidden order in amorphous cellular geometries

In this paper, we analyze the evolution of random point patterns and their Voronoi diagrams as “quantizer” of space. Applying an iterative local optimization of their so-called Quantizer energy, we show that the patterns converge to the apparently same effectively hyperuniform state regardless of their initial conditions.

M.A. Klatt, J. Lovrić, D. Chen, S. C. Kapfer, F. M. Schaller, P. W. A. Schönhöfer, B. S. Gardiner, A.-S. Smith, G. E. Schröder-Turk, S. Torquato. Universal hidden order in amorphous cellular geometries. Nature Communications 10, 811 (2019)

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.