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:


Low-temperature statistical mechanics of the Quantizer problem

Following up on our finding of a universal hidden order in tessellations with the Quantizer energy functionals (where the cells minimize the second moment of the volume distribution) using Lloyd’s algorithm, we now compared these states with quenches from equilibrium states at finite temperature.

Upon sufficiently fast non-equilibrium cooling, the system adopts similar states as the effectively hyperuniform inherent structures found by Lloyd’s algorithm and prevents the ordering transition into a body-centered cubic ordered structure.


Hyperuniform and rigid stable matchings

With respect to the degree of order and disorder, a snapshot of the ideal gas with completely independent particle positions (in mathematics known as the realization of a Poisson point process) is the exact opposite of a lattice with a perfect short- and long-range order.

We have defined a stable matching between these two extremes and proved that the resulting pattern can locally be virtually indistinguishable from the ideal gas but at large scales it inherits two remarkable global properties of the lattice: hyperuniformity and number rigidity, as demonstrated in our supplementary video.

M. A. Klatt, G. Last, D. Yogeshwaran. Hyperuniform and rigid stable matchings. Random Struct Alg. 57:439–473 (2020)

In the stable matching, points of the lattice and the ideal gas prefer to be close to each other. Assuming that there are on average more points per unit area in the ideal gas than in the lattice, the matched points in the ideal gas form a new point process that inherits properties from both the lattice and the ideal gas.

In fact, if the mean number of points per unit area converge, the new point pattern becomes almost indistinguishable from the ideal gas in any finite observation window, while its large-scale density fluctuations remain anomalously suppressed similar to the lattice.

We prove that the resulting point process is hyperuniform and number rigidity also for a stable matching between a lattice and not only the ideal gas but a class of correlated point processes, known as determinantal point processes. For a one-sided matching in 1D, our proof applies to quite general point processes.

Our paper solves an open question from the seminal paper by Salvatore Torquato and Frank Stillinger that introduced the notion of hyperuniformity. Our matched point processes generalizes a 1D hyperuniform point process by Goldstein, Lebowitz, Speer.

You can easily match two arbitrary point patterns using my R-package: matchingpp


Geometric Functionals of Fractal Percolation

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly
from a dust-like set to a system-spanning cluster. Where this transition occurs is unknown
and notoriously difficult to estimate. In continuum percolation, the percolation thresholds have been approximated well using Minkowski functionals. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process.

M. A. Klatt and S. Winter. Geometric functionals of fractal percolation. Adv. Appl. Probab. 52:1085–1126 (2020)

We establish the existence of the limit functionals and obtain explicit formulas for them as well as for their finite approximations (in excellent agreement with simulation result). They provide new geometric insights into fractal percolation. For example, we provide a mathematically rigorous notion of fractal subdimensions that have been previously defined in physics as an approach to refined information about fractal sets beyond fractal dimension.

My code to simulate fractal percolation and to analyze our geometric functionals is freely available at GitHub.


Detecting structured sources in noisy images via Minkowski maps

Distinguishing a feature from background noise is often the key to discovering physical phenomena in spatial data for a wide range of applications. We show how the sensitivity of a null hypothesis test can be dramatically increased by additional geometric information without the need for any prior knowledge about potential sources.

M. A. Klatt and K. Mecke. Detecting structured sources in noisy images via Minkowski maps. EPL (Europhysics Letters) 128:60001-p1–8 (2019)

We present a morphometric analysis technique, based on Minkowski functionals, to quantify the shape of structural deviations in greyscale images. It identifies important features in noisy spatial data, especially for short observation times and low statistics.
Without assuming any prior knowledge about potential sources, the additional shape information can increase the sensitivity by 14 orders of magnitude compared to previous methods. Rejection rates can increase by an order of magnitude.

As a key ingredient to such a dramatic increase, we accurately describe the distribution of the homogeneous background noise in terms of the density of states for the area, perimeter, and Euler characteristic of random black-and-white images. Remarkably, the density of states can vary by 64 orders of magnitude. We apply our technique to data of the H.E.S.S. experiment.

The density of states and my code for Minkowski maps are freely available at GitHub.

All simulated data and parameters underlying our study are available at 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.”


Phoamtonics: Photonic band gaps of 3D foams

In our paper, published in PNAS, we show how to construct a foam-based photonic crystal with a substantial band gap using the famous Weaire-Phelan foam.

M. A. Klatt, P. J. Steinhardt, S. Torquato. Phoamtonic designs yield sizeable 3D photonic band gaps. Proc. Natl. Acad. Sci. U.S.A. 116:23480–23486 (2019)

A report by Steven Schultz about our study has been featured on the main webpage of Princeton University:


See also, the original post of the engineering department:

The Weaire-Phelan foam has the smallest surface area among all known tessellations with equal-volume cells. While it has only a slightly smaller surface area than the Kelvin foam. Its corresponding photonic network provides a distinctly larger band gap and a much smaller critical refractive index. To determine the latter, we have created a detailed map of gap sizes.

Among the promising prospects that phoamtonics offers for applications are multifunctional characteristics, the self-organization of large photonic networks, and a high degree of isotropy.

We published all data generated or analyzed for our phoamtonics study, including configurations, parameter files, raw output, and postprocessed data, in a Zenodo
( DOI: 10.5281/zenodo.3401635


Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry

In this paper, we combine physics, microbiology and mathematics to study how bacteria attach to nanostructured surfaces, and how topography of nanostructured substrates can directly quantify the adhesion.

C. Spengler, F. Nolle, J. Mischo, T. Faidt, S. Grandthyll, N. Thewes, P. Jung, M.
Koch, F. Müller, M. Bischoff, M. A. Klatt, K. Jacobs. Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry. Nanoscale 11:19713–19722 (2019)

Aiming for 3D-structured materials for or against bio-adhesion, we studied mechanisms by which the bacteria adhere to different types of materials. Our insights help, for example, to prevent adhesion of pathogens (like Staphylococcus aureus) before the bacteria can begin to grow a biofilm, which protects individual bacteria from attack by antibiotics.

We found that the measured forces with which the individual bacterial cells adhered to nanostructured surfaces decrease with an increasing size of these nanostructures. In fact, with our morphometric analysis based on Minkowski functionals, we found that the strength of the adhesion force can be predicted from measurements of the surface area that is accessible to the tethering proteins with which the bacteria adhere to the surface. More precisely, we found that the ratio of the experimentally measured adhesion forces at the rough surface compared to a smooth surface agrees with the fraction of surface area that is accessible to the tethering proteins, which are about 50 nm long.

There are press releases in English and German.