{{Short description|State similar to a liquid and a crystal}} {{use dmy dates|date=March 2021}} [[File:Number-variance.png|thumb|590x590px|Hyperuniformity is defined by the scaling of the variance of the number of points that are within a disk of radius ''R''. For the ideal gas (left), this variance scales like the area of the disk. For a hyperuniform system (center), it scales slower than the area of the disk;<ref name="PRE-68"/> for example, for a crystal (right), it scales like the boundary length of the disk. Adapted after Figure 1 of Ref.<ref name=":3"/>]]
'''Hyperuniform''' materials are characterized by an anomalous suppression of density fluctuations at large scales. More precisely, the vanishing of density fluctuations in the long-wavelength limit (like for crystals) distinguishes ''hyperuniform'' systems from typical gases, liquids, or amorphous solids.<ref name="PRE-68">{{cite journal |last1=Torquato |first1=Salvatore |last2=Stillinger |first2=Frank H. |date=Oct 29, 2003 |title=Local density fluctuations, hyperuniformity, and order metrics|journal=Physical Review E |volume=68 |issue=4 |article-number=041113 |arxiv=cond-mat/0311532 |bibcode=2003PhRvE..68d1113T |doi=10.1103/PhysRevE.68.041113 |pmid=14682929 |s2cid=9162488}}</ref><ref name=":3">{{cite journal |last=Torquato |first=Salvatore |year=2018 |title=Hyperuniform states of matter |journal=Physics Reports |language=en |volume=745 |pages=1–95 |doi=10.1016/j.physrep.2018.03.001 |arxiv=1801.06924 |bibcode=2018PhR...745....1T |s2cid=119378373}}</ref> Examples of ''hyperuniformity'' include all perfect crystals,<ref name="PRE-68"/> perfect quasicrystals,<ref name=":15"/><ref name=":16">{{cite journal |last1=Oğuz |first1=Erdal C. |last2=Socolar |first2=Joshua E.S. |last3=Steinhardt |first3=Paul J. |last4=Torquato |first4=Salvatore |date=2017-02-23 |title=Hyperuniformity of quasicrystals |journal=Physical Review B |language=en |volume=95 |issue=5 |article-number=054119 |doi=10.1103/PhysRevB.95.054119 |arxiv=1612.01975 |bibcode=2017PhRvB..95e4119O |s2cid=85522310 |issn=2469-9950}}</ref> and exotic amorphous states of matter.<ref name=":3">{{cite journal |last=Torquato |first=Salvatore |year=2018 |title=Hyperuniform states of matter |journal=Physics Reports |language=en |volume=745 |pages=1–95 |doi=10.1016/j.physrep.2018.03.001 |arxiv=1801.06924 |bibcode=2018PhR...745....1T |s2cid=119378373}}</ref>
Quantitatively, a many-particle system is said to be ''hyperuniform'' if the variance of the number of points within a spherical observation window grows more slowly than the volume of the observation window. This definition is equivalent to a vanishing of the structure factor in the long-wavelength limit,<ref name="PRE-68"/> and it has been extended to include heterogeneous materials as well as scalar, vector, and tensor fields.<ref name=":17">{{Cite journal |last=Torquato |first=Salvatore |date=2016-08-15 |title=Hyperuniformity and its generalizations |url=https://link.aps.org/doi/10.1103/PhysRevE.94.022122 |journal=Physical Review E |language=en |volume=94 |issue=2 |article-number=022122 |doi=10.1103/PhysRevE.94.022122 |pmid=27627261 |arxiv=1607.08814 |bibcode=2016PhRvE..94b2122T |s2cid=30459937 |issn=2470-0045}}</ref> Disordered hyperuniform systems have been shown to be poised at an "inverted" critical point.<ref name="PRE-68"/> They can be obtained via equilibrium or nonequilibrium routes, and are found in both classical physical and quantum-mechanical systems.<ref name="PRE-68"/><ref name=":3"/> Hence the concept of ''hyperuniformity'' now connects a broad range of topics in physics,<ref name=":3" /><ref name=":5" /><ref name=":4" /><ref name="auto10" /><ref name="auto4" /> mathematics,<ref name=":9">{{cite journal |last1=Ghosh |first1=Subhroshekhar |last2=Lebowitz |first2=Joel L. |year=2017 |title=Fluctuations, large deviations and rigidity in hyperuniform systems: A brief survey |journal=Indian Journal of Pure and Applied Mathematics |language=en |volume=48 |issue=4 |pages=609–631 |arxiv=1608.07496 |doi=10.1007/s13226-017-0248-1 |issn=0019-5588 |s2cid=8709357}}</ref><ref name=":10">{{cite journal |last1=Ghosh |first1=Subhroshekhar |last2=Lebowitz |first2=Joel L. |year=2018 |title=Generalized stealthy hyperuniform processes: Maximal rigidity and the bounded holes conjecture |journal=Communications in Mathematical Physics |language=en |volume=363 |issue=1 |pages=97–110 |arxiv=1707.04328 |bibcode=2018CMaPh.363...97G |doi=10.1007/s00220-018-3226-5 |issn=0010-3616 |s2cid=6243545}}</ref><ref name=":11">{{cite journal |last1=Torquato |first1=Salvatore |last2=Zhang |first2=Ge |last3=De Courcy-Ireland |first3=Matthew |date=2019-03-29 |title=Hidden multiscale order in the primes |journal=Journal of Physics A: Mathematical and Theoretical |volume=52 |issue=13 |page=135002 |arxiv=1804.06279 |bibcode=2019JPhA...52m5002T |doi=10.1088/1751-8121/ab0588 |issn=1751-8113 |s2cid=85508362}}</ref><ref name=":12">{{Cite journal |last1=Brauchart |first1=Johann S. |last2=Grabner |first2=Peter J. |last3=Kusner |first3=Wöden |last4=Ziefle |first4=Jonas |year=2020 |title=Hyperuniform point sets on the sphere: probabilistic aspects |journal=Monatshefte für Mathematik |language=en |volume=192 |issue=4 |pages=763–781 |arxiv=1809.02645 |doi=10.1007/s00605-020-01439-y |issn=0026-9255 |s2cid=119179807}}</ref><ref name=":13">{{Cite journal |last1=Baake |first1=Michael |last2=Grimm |first2=Uwe|author2-link=Uwe Grimm |date=2020-09-01 |title=Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction |journal=Acta Crystallographica Section A |volume=76 |issue=5 |pages=559–570 |arxiv=2004.03256 |doi=10.1107/S2053273320007421 |issn=2053-2733 |pmc=7459767 |pmid=32869753 |s2cid=220404667}}</ref><ref name=":14">{{Cite journal |last1=Klatt |first1=Michael Andreas |last2=Last |first2=Günter |last3=Yogeshwaran |first3=D. |year=2020 |title=Hyperuniform and rigid stable matchings |journal=Random Structures & Algorithms |language=en |volume=57 |issue=2 |pages=439–473 |arxiv=1810.00265 |doi=10.1002/rsa.20923 |issn=1098-2418 |s2cid=119678948}}</ref> biology,<ref name=":0" /><ref>{{Cite journal |last1=Mayer |first1=Andreas |last2=Balasubramanian |first2=Vijay |last3=Mora |first3=Thierry |last4=Walczak |first4=Aleksandra M. |date=2015-05-12 |title=How a well-adapted immune system is organized |journal=Proceedings of the National Academy of Sciences |language=en |volume=112 |issue=19 |pages=5950–5955 |arxiv=1407.6888 |bibcode=2015PNAS..112.5950M |doi=10.1073/pnas.1421827112 |issn=0027-8424 |pmc=4434741 |pmid=25918407 |doi-access=free}}</ref><ref>{{Cite journal |last1=Huang |first1=Mingji |last2=Hu |first2=Wensi |last3=Yang |first3=Siyuan |last4=Liu |first4=Quan-Xing |last5=Zhang |first5=H. 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The concept of ''hyperuniformity'' generalizes the traditional notion of long-range order and thus defines an exotic state of matter. A disordered ''hyperuniform'' many-particle system can be statistically isotropic like a liquid, with no Bragg peaks and no conventional type of long-range order. Nevertheless, at large scales, ''hyperuniform'' systems resemble crystals in their suppression of large-scale density fluctuations. This unique combination is known to endow disordered ''hyperuniform'' materials with novel physical properties that are, e.g., both nearly optimal and direction independent (in contrast to those of crystals that are anisotropic).<ref name=":3">{{cite journal |last=Torquato |first=Salvatore |year=2018 |title=Hyperuniform states of matter |journal=Physics Reports |language=en |volume=745 |pages=1–95 |doi=10.1016/j.physrep.2018.03.001 |arxiv=1801.06924 |bibcode=2018PhR...745....1T |s2cid=119378373}}</ref>
==History== The term ''hyperuniformity'' (also independently called ''super-homogeneity'' in the context of cosmology<ref name="PRD-65">{{cite journal |last1=Gabrielli |first1=Andrea |last2=Joyce |first2=Michael |last3=Sylos Labini |first3=Francesco |date=Apr 11, 2002 |title=Glass-like universe: Real-space correlation properties of standard cosmological models |journal=Physical Review D |volume=65 |issue=4 |article-number=083523 |doi=10.1103/PhysRevD.65.083523 |s2cid=9162488 |arxiv=astro-ph/0110451|bibcode=2002PhRvD..65h3523G }}</ref>) was coined and studied by Salvatore Torquato and Frank Stillinger in a 2003 paper,<ref name="PRE-68"/> in which they showed that, among other things, hyperuniformity provides a unified framework to classify and structurally characterize crystals, quasicrystals, and exotic disordered varieties. In that sense, hyperuniformity is a long-range property that can be viewed as generalizing the traditional notion of long-range order (e.g., translational / orientational order of crystals or orientational order of quasicrystals) to also encompass exotic disordered systems.<ref name=":3"/>
Hyperuniformity was first introduced for point processes<ref name="PRE-68" /> and later generalized to two-phase materials (or porous media)<ref name=":15">{{Cite journal |last1=Zachary |first1=Chase E. |last2=Torquato |first2=Salvatore |date=2009-12-21 |title=Hyperuniformity in point patterns and two-phase random heterogeneous media |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2009 |issue=12 |article-number=12015 |doi=10.1088/1742-5468/2009/12/P12015 |arxiv=0910.2172 |bibcode=2009JSMTE..12..015Z |s2cid=18838058 |issn=1742-5468}}</ref> and random scalar or vectors fields.<ref name=":17" /> It has been observed in theoretical models, simulations, and experiments, see list of examples below.<ref name=":3">{{cite journal |last=Torquato |first=Salvatore |year=2018 |title=Hyperuniform states of matter |journal=Physics Reports |language=en |volume=745 |pages=1–95 |doi=10.1016/j.physrep.2018.03.001 |arxiv=1801.06924 |bibcode=2018PhR...745....1T |s2cid=119378373}}</ref>
==Definition==
A many-particle system in <math>d</math>-dimensional Euclidean space <math>\mathbb{R}^d</math> is said to be ''hyperuniform'' if the number of points in a spherical observation window with radius <math>R</math> has a variance <math>\sigma_N^2(R)</math> that scales slower than the volume of the observation window:<ref name="PRE-68" /> <math display="block">\lim_{R\to \infty} \frac{\sigma_N^2(R)}{R^d} = 0.</math> This definition is (essentially) equivalent to the vanishing of the structure factor at the origin:<ref name="PRE-68" /> <math display="block">\lim_{\mathbf{k}\to 0} S(\mathbf{k}) = 0</math> for wave vectors <math>\mathbf{k} \in \mathbb{R}^d</math>.
Similarly, a two-phase medium consisting of a solid and a void phase is said to be ''hyperuniform'' if the volume of the solid phase inside the spherical observation window has a variance that scales slower than the volume of the observation window. This definition is, in turn, equivalent to a vanishing of the spectral density at the origin.<ref name=":15" />
An essential feature of hyperuniform systems is their scaling of the number variance <math>\sigma_N^2(R)</math> for large radii or, equivalently, of the structure factor <math>S(k)</math> for small wave numbers. If we consider hyperuniform systems that are characterized by a power-law behavior of the structure factor close to the origin:<ref name=":3" /> <math display="block">S(\mathbf{k}) \sim |\mathbf{k}|^{\alpha} \text{ for } \mathbf{k}\to 0</math> with a constant <math>0<\alpha<\infty</math>, then there are three distinct scaling behaviors that define ''three classes of hyperuniformity'': <math display="block">\sigma_N^2(R) \sim \begin{cases} R^{d-1}, &\alpha>1 & \text{(CLASS I)}\\ R^{d-1} \ln R, &\alpha=1 & \text{(CLASS II)}\\ R^{d-\alpha}, &0<\alpha<1 & \text{(CLASS III)}\\ \end{cases}</math> Examples are known for all three classes of hyperuniformity.<ref name=":3" />
==Examples== Examples of disordered hyperuniform systems in physics are disordered ground states,<ref name=":4">{{Cite journal |last1=Torquato |first1=S. |last2=Zhang |first2=G. |last3=Stillinger |first3=F.H. |date=2015-05-29 |title=Ensemble theory for stealthy hyperuniform disordered ground states |journal=Physical Review X |language=en |volume=5 |issue=2 |article-number=021020 |doi=10.1103/PhysRevX.5.021020 |arxiv=1503.06436 |bibcode=2015PhRvX...5b1020T |s2cid=17275490 |issn=2160-3308}}</ref> jammed disordered sphere packings,<ref name=":5">{{Cite journal |last1=Donev |first1=Aleksandar |last2=Stillinger |first2=Frank H. |last3=Torquato |first3=Salvatore |date=2005-08-26 |title=Unexpected density fluctuations in jammed disordered sphere packings |journal=Physical Review Letters |language=en |volume=95 |issue=9 |article-number=090604 |doi=10.1103/PhysRevLett.95.090604 |pmid=16197201 |arxiv=cond-mat/0506406 |bibcode=2005PhRvL..95i0604D |s2cid=7887194 |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Zachary |first1=Chase E. |last2=Jiao |first2=Yang |last3=Torquato |first3=Salvatore |date=2011-04-29 |title=Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings |journal=Physical Review Letters |language=en |volume=106 |issue=17 |article-number=178001 |doi=10.1103/PhysRevLett.106.178001 |pmid=21635063 |arxiv=1008.2548 |bibcode=2011PhRvL.106q8001Z |s2cid=15587068 |issn=0031-9007}}</ref><ref name="auto8">{{Cite journal |last1=Weijs |first1=Joost H. |last2=Jeanneret |first2=Raphaël |last3=Dreyfus |first3=Rémi |last4=Bartolo |first4=Denis |date=2015-09-03 |title=Emergent Hyperuniformity in Periodically Driven Emulsions |journal=Physical Review Letters |language=en |volume=115 |issue=10 |article-number=108301 |doi=10.1103/PhysRevLett.115.108301 |pmid=26382706 |arxiv=1504.04638 |bibcode=2015PhRvL.115j8301W |s2cid=10340709 |issn=0031-9007}}</ref><ref name="auto2">{{Cite journal |last1=Jack |first1=Robert L. |last2=Thompson |first2=Ian R. |last3=Sollich |first3=Peter |date=2015-02-09 |title=Hyperuniformity and Phase Separation in Biased Ensembles of Trajectories for Diffusive Systems |journal=Physical Review Letters |language=en |volume=114 |issue=6 |article-number=060601 |doi=10.1103/PhysRevLett.114.060601 |pmid=25723197 |arxiv=1409.3986 |bibcode=2015PhRvL.114f0601J |s2cid=3132460 |issn=0031-9007}}</ref><ref name="auto11">{{Cite journal |last1=Weijs |first1=Joost H. |last2=Bartolo |first2=Denis |date=2017-07-27 |title=Mixing by Unstirring: Hyperuniform Dispersion of Interacting Particles upon Chaotic Advection |journal=Physical Review Letters |language=en |volume=119 |issue=4 |article-number=048002 |doi=10.1103/PhysRevLett.119.048002 |pmid=29341775 |arxiv=1702.02395 |bibcode=2017PhRvL.119d8002W |s2cid=12229553 |issn=0031-9007}}</ref><ref name="auto5">{{Cite journal |last1=Ricouvier |first1=Joshua |last2=Pierrat |first2=Romain |last3=Carminati |first3=Rémi |last4=Tabeling |first4=Patrick |last5=Yazhgur |first5=Pavel |date=2017-11-15 |title=Optimizing Hyperuniformity in Self-Assembled Bidisperse Emulsions |journal=Physical Review Letters |language=en |volume=119 |issue=20 |article-number=208001 |doi=10.1103/PhysRevLett.119.208001 |pmid=29219379 |arxiv=1711.00719 |bibcode=2017PhRvL.119t8001R |s2cid=28177098 |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Chieco |first1=A. 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|issn=1286-4854}}</ref><ref name="auto12">{{Cite journal |last1=Ness |first1=Christopher |last2=Cates |first2=Michael E. |date=2020-02-27 |title=Absorbing-State Transitions in Granular Materials Close to Jamming |journal=Physical Review Letters |language=en |volume=124 |issue=8 |article-number=088004 |doi=10.1103/PhysRevLett.124.088004 |pmid=32167320 |arxiv=2001.10228 |bibcode=2020PhRvL.124h8004N |s2cid=210932396 |issn=0031-9007}}</ref><ref name="auto4">{{cite journal |last1=Wilken |first1=Sam |last2=Guerra |first2=Rodrigo E. |last3=Pine |first3=David J. |last4=Chaikin |first4=Paul M. |date=2020-02-11 |title=Hyperuniform Structures Formed by Shearing Colloidal Suspensions |journal=Physical Review Letters |volume=125 |issue=14 |article-number=148001 |doi=10.1103/PhysRevLett.125.148001 |pmid=33064537 |arxiv=2002.04499|bibcode=2020PhRvL.125n8001W |s2cid=211075881 }}</ref> perturbed lattices,<ref name="auto1">{{Cite journal |last1=Gabrielli |first1=Andrea |last2=Joyce |first2=Michael |last3=Sylos Labini |first3=Francesco |date=2002-04-11 |title=Glass-like universe: Real-space correlation properties of standard cosmological models |journal=Physical Review D |language=en |volume=65 |issue=8 |article-number=083523 |doi=10.1103/PhysRevD.65.083523 |arxiv=astro-ph/0110451 |bibcode=2002PhRvD..65h3523G |s2cid=119442907 |issn=0556-2821}}</ref><ref name="auto7">{{Cite journal |last=Gabrielli |first=Andrea |year=2004 |title=Point processes and stochastic displacement fields |journal=Physical Review E |language=en |volume=70 |issue=6 |article-number=066131 |doi=10.1103/PhysRevE.70.066131 |pmid=15697458 |arxiv=cond-mat/0409594 |bibcode=2004PhRvE..70f6131G |s2cid=33621420 |issn=1539-3755}}</ref><ref>{{Cite journal |last1=Le Thien |first1=Q. |last2=McDermott |first2=D. |last3=Reichhardt |first3=C.J.O. |last4=Reichhardt |first4=C. |date=2017-09-15 |title=Enhanced pinning for vortices in hyperuniform pinning arrays and emergent hyperuniform vortex configurations with quenched disorder |journal=Physical Review B |language=en |volume=96 |issue=9 |article-number=094516 |doi=10.1103/PhysRevB.96.094516 |bibcode=2017PhRvB..96i4516L |s2cid=18031713 |issn=2469-9950 |doi-access=free|arxiv=1611.01532 }}</ref><ref name="auto3">{{Cite journal |last1=Klatt |first1=Michael A. |last2=Kim |first2=Jaeuk |last3=Torquato |first3=Salvatore |date=2020-03-13 |title=Cloaking the underlying long-range order of randomly perturbed lattices |journal=Physical Review E |language=en |volume=101 |issue=3 |article-number=032118 |doi=10.1103/PhysRevE.101.032118 |pmid=32289999 |arxiv=2001.08161 |bibcode=2020PhRvE.101c2118K |s2cid=210859161 |issn=2470-0045}}</ref> and avian photoreceptor cells.<ref name=":0"/>
In mathematics, disordered hyperuniformity has been studied in the context of probability theory,<ref name=":9" /><ref>{{Cite journal |last1=Ghosh |first1=Subhro |last2=Lebowitz |first2=Joel |year=2017 |title=Number Rigidity in Superhomogeneous Random Point Fields |journal=Journal of Statistical Physics |language=en |volume=166 |issue=3–4 |pages=1016–1027 |doi=10.1007/s10955-016-1633-6 |arxiv=1601.04216 |bibcode=2017JSP...166.1016G |s2cid=19675015 |issn=0022-4715}}</ref><ref name=":10" /> geometry,<ref name=":12" /><ref name=":13"/> and number theory,<ref>{{Cite journal |last1=Zhang |first1=G |last2=Martelli |first2=F |last3=Torquato |first3=S |date=2018-03-16 |title=The structure factor of primes |url=https://iopscience.iop.org/article/10.1088/1751-8121/aaa52a |journal=Journal of Physics A: Mathematical and Theoretical |volume=51 |issue=11 |page=115001 |doi=10.1088/1751-8121/aaa52a |arxiv=1801.01541 |bibcode=2018JPhA...51k5001Z |s2cid=67819480 |issn=1751-8113}}</ref><ref name=":11" /><ref>{{Cite journal |last1=Baake |first1=Michael |last2=Coons |first2=Michael |year=2021 |title=Scaling of the Diffraction Measure of $k$ -Free Integers Near the Origin |journal=Michigan Mathematical Journal |volume=70 |pages=213–221 |language=en |doi=10.1307/mmj/1592877613 |issn=0026-2285 |arxiv=1904.00279|s2cid=90260746 }}</ref> where the prime numbers have been found to be effectively limit periodic and hyperuniform in a certain scaling limit.<ref name=":11" /> Further examples include certain random walks<ref>{{Cite journal |last1=Casini |first1=Emanuele |last2=Le Caër |first2=Gérard |last3=Martinelli |first3=Andrea |year=2015 |title=Short Hyperuniform Random Walks |journal=Journal of Statistical Physics |language=en |volume=160 |issue=1 |pages=254–273 |doi=10.1007/s10955-015-1244-7 |bibcode=2015JSP...160..254C |s2cid=45170541 |issn=0022-4715 |url=https://hal.archives-ouvertes.fr/hal-01139661/file/JSP-2015-%28GLC%29.pdf}}</ref> and stable matchings of point processes.<ref name=":14"/><ref name="auto8"/><ref name="auto2"/><ref name="auto11"/><ref name="auto5"/><ref>{{Cite journal |last1=Chieco |first1=A.T. |last2=Zu |first2=M. |last3=Liu |first3=A.J. |last4=Xu |first4=N. |last5=Durian |first5=D.J. |date=2018-10-17 |title=Spectrum of structure for jammed and unjammed soft disks |journal=Physical Review E |language=en |volume=98 |issue=4 |article-number=042606 |doi=10.1103/PhysRevE.98.042606 |arxiv=1806.10118 |bibcode=2018PhRvE..98d2606C |s2cid=119448635 |issn=2470-0045 |author5-link=Douglas Durian}}</ref>
===Ordered hyperuniformity=== Examples of ordered, hyperuniform systems include all crystals,<ref name="PRE-68" /> all quasicrystals,<ref name=":15" /><ref name=":16" /><ref>{{cite journal |last1=Lin |first1=C. |last2=Steinhardt |first2=P.J. |last3=Torquato |first3=S. |date=2017-04-13 |title=Hyperuniformity variation with quasicrystal local isomorphism class |journal=Journal of Physics: Condensed Matter |volume=29 |issue=20 |page=204003 |bibcode=2017JPCM...29t4003L |doi=10.1088/1361-648x/aa6944 |issn=0953-8984 |pmid=28345537 |s2cid=46764513}}</ref> and limit-periodic sets.<ref>{{cite journal |last1=Baake |first1=Michael |last2=Grimm |first2=Uwe|author2-link=Uwe Grimm |date=2019-05-23 |title=Scaling of diffraction intensities near the origin: Some rigorous results |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2019 |issue=5 |page=054003 |arxiv=1905.04177 |bibcode=2019JSMTE..05.4003B |doi=10.1088/1742-5468/ab02f2 |issn=1742-5468 |doi-access=free}}</ref> While weakly correlated noise typically preserves hyperuniformity, correlated excitations at finite temperature tend to destroy hyperuniformity.<ref>{{cite journal |last1=Kim |first1=Jaeuk |last2=Torquato |first2=Salvatore |date=2018-02-12 |title=Effect of imperfections on the hyperuniformity of many-body systems |journal=Physical Review B |language=en |volume=97 |issue=5 |article-number=054105 |doi=10.1103/PhysRevB.97.054105 |bibcode=2018PhRvB..97e4105K |issn=2469-9950 |doi-access=free}}</ref>
Hyperuniformity was also reported for fermionic quantum matter in correlated electron systems as a result of cramming.<ref>{{cite journal |last1=Gerasimenko |display-authors=etal |year=2019 |title=Quantum jamming transition to a correlated electron glass in 1T-TaS2 |journal=Nature Materials |volume=317 |issue=10 |pages=1078–1083 |doi=10.1038/s41563-019-0423-3 |pmid=31308513 |bibcode=2019NatMa..18.1078G |s2cid=196810837 |arxiv=1803.00255}}</ref>
===Disordered hyperuniformity=== Torquato (2014)<ref name=":6"/> gives an illustrative example of the hidden order found in a "shaken box of marbles",<ref name=":6"/> which fall into an arrangement called ''maximally random jammed packing''.<ref name=":5"/><ref>{{Cite journal |last1=Atkinson |first1=Steven |last2=Stillinger |first2=Frank H. |last3=Torquato |first3=Salvatore |date=2014-12-30 |title=Existence of isostatic, maximally random jammed monodisperse hard-disk packings |journal=Proceedings of the National Academy of Sciences |language=en |volume=111 |issue=52 |pages=18436–18441 |doi=10.1073/pnas.1408371112 |issn=0027-8424 |pmc=4284597 |pmid=25512529 |bibcode=2014PNAS..11118436A|doi-access=free }}</ref> Such hidden order may eventually be used for self-organizing colloids or optics with the ability to transmit light with an efficiency like a crystal but with a highly flexible design.<ref name=":6">{{cite press release |first=Morgan |last=Kelly |date=2014-02-24 |title=In the eye of a chicken, a new state of matter comes into view |publisher=Princeton University |place=Princeton, NJ |url=https://www.princeton.edu/main/news/archive/S39/32/02E70/index.xml |access-date=2021-03-08}}</ref>
It has been found that disordered hyperuniform systems possess unique optical properties. For example, disordered hyperuniform photonic networks have been found to exhibit complete photonic band gaps that are comparable in size to those of photonic crystals but with the added advantage of isotropy, which enables free-form waveguides not possible with crystal structures.<ref name=":1"/><ref name=":2"/><ref>{{cite journal |last1=Froufe-Pérez |first1=Luis S. |last2=Engel |first2=Michael |last3=Sáenz |first3=Juan José |last4=Scheffold |first4=Frank |date=2017-09-05 |title=Band gap formation and Anderson localization in disordered photonic materials with structural correlations |journal=Proceedings of the National Academy of Sciences |language=en |volume=114 |issue=36 |pages=9570–9574 |doi=10.1073/pnas.1705130114 |issn=0027-8424 |pmc=5594660 |pmid=28831009 |arxiv=1702.03883 |bibcode=2017PNAS..114.9570F|doi-access=free }}</ref><ref>{{cite journal |last1=Milošević |first1=Milan M. |last2=Man |first2=Weining |last3=Nahal |first3=Geev |last4=Steinhardt |first4=Paul J. |last5=Torquato |first5=Salvatore |last6=Chaikin |first6=Paul M. |last7=Amoah |first7=Timothy |last8=Yu |first8=Bowen |last9=Mullen |first9=Ruth Ann |last10=Florescu |first10=Marian |year=2019 |title=Hyperuniform disordered waveguides and devices for near infrared silicon photonics |journal=Scientific Reports |language=en |volume=9 |issue=1 |page=20338 |issn=2045-2322 |pmc=6937303 |doi=10.1038/s41598-019-56692-5 |bibcode=2019NatSR...920338M |pmid=31889165}}</ref> Moreover, in stealthy hyperuniform systems,<ref name=":4"/> light of any wavelength longer than a value specific to the material is able to propagate forward without loss (due to the correlated disorder) even for high particle density.<ref name=":7"/>
By contrast, in conditions where light is propagated through an uncorrelated, disordered material of the same density, the material would appear opaque due to multiple scattering. "Stealthy" hyperuniform materials can be theoretically designed for light of any wavelength, and the applications of the concept cover a wide variety of fields of wave physics and materials engineering.<ref name=":7">{{cite journal |doi=10.1364/OPTICA.3.000763 |title=High-density hyperuniform materials can be transparent |journal=Optica |volume=3 |issue=7 |page=763 |year=2016 |last1=Leseur |first1=O. |last2=Pierrat |first2=R. |last3=Carminati |first3=R. |arxiv=1510.05807 |bibcode=2016Optic...3..763L |s2cid=118443561}}</ref><ref>{{cite journal |last1=Gorsky |first1=S. |last2=Britton |first2=W. A. |last3=Chen |first3=Y. |last4=Montaner |first4=J. |last5=Lenef |first5=A. |last6=Raukas |first6=M. |last7=Dal Negro |first7=L. |date=2019-11-01 |title=Engineered hyperuniformity for directional light extraction |journal=APL Photonics |language=en |volume=4 |issue=11 |page=110801 |doi=10.1063/1.5124302 |bibcode=2019APLP....4k0801G |issn=2378-0967 |doi-access=free}}</ref>
Disordered hyperuniformity was recently discovered in amorphous 2‑D materials, including amorphous silica<ref name="2D_DHU_Si">{{cite journal |last1=Yu |display-authors=etal |year=2020 |title=Disordered hyperuniformity in two-dimensional amorphous silica |journal=Science Advances |volume=6 |issue=16 |article-number=eaba0826 |doi=10.1126/sciadv.aba0826 |pmid=32494625 |pmc=7164937 |bibcode=2020SciA....6..826Z |s2cid=218844271 |doi-access=free}}</ref> as well as amorphous graphene,<ref name="2D_DHU_C">{{cite journal |last1=Chen |display-authors=etal |year=2021 |title=Stone–Wales defects preserve hyperuniformity in amorphous two-dimensional networks |journal=Proceedings of the National Academy of Sciences |volume=118 |issue=3 |page=e2016862118 |doi=10.1073/pnas.201686211|doi-broken-date=12 July 2025 |doi-access=free }}</ref> which was shown to enhance electronic transport in the material.<ref name="2D_DHU_Si"/> It was shown that the Stone-Wales topological defects, which transform two-pair of neighboring hexagons to a pair of pentagons and a pair of heptagons by flipping a bond, preserves the hyperuniformity of the parent honeycomb lattice.<ref name="2D_DHU_C"/>
==Disordered hyperuniformity in biology==
Disordered hyperuniformity was found in the photoreceptor cell patterns in the eyes of chickens.<ref name=":0">{{cite journal |last1=Jiao |display-authors=etal |title=Avian Photoreceptor Patterns Represent a Disordered Hyperuniform Solution to a Multiscale Packing Problem |journal=Physical Review E |year=2014 |volume=89 |issue=2 |article-number=022721 |doi=10.1103/PhysRevE.89.022721 |pmid=25353522 |pmc=5836809 |arxiv=1402.6058 |bibcode=2014PhRvE..89b2721J}}</ref> This is thought to be the case because the light-sensitive cells in chicken or other bird eyes cannot easily attain an optimal crystalline arrangement but instead form a disordered configuration that is as uniform as possible.<ref name=":0"/><ref>{{cite web |url=https://gizmodo.com/disordered-hyperuniformity-a-weird-new-state-of-matter-1548659862 |title=Disordered hyperuniformity: A weird new state of matter in chicken eyes |author=Melissa |website=TodayIFoundOut.com |date=March 21, 2014 |publisher=Gawker Media |via=Gizmodo}}</ref><ref>{{cite news |url=https://www.huffingtonpost.com/2014/02/26/chicken-eye-weird-state-of-matter_n_4854897.html |title=Scientists Look In Chicken's Eye And Discover Weird New State Of Matter |author=David Freeman |date=26 February 2014 |work=The Huffington Post |access-date=20 December 2015}}</ref> Indeed, it is the remarkable property of "mulithyperuniformity" of the avian cone patterns that enables birds to achieve acute color sensing.<ref name=":0"/>
It may also emerge in the mysterious biological patterns known as fairy circles—circles and patterns of circles that emerge in arid places.<ref>{{cite news |newspaper=The Washington Post |date=2017-01-18 |url=https://www.washingtonpost.com/news/speaking-of-science/wp/2017/01/18/the-astonishing-science-behind-namibias-mysterious-fairy-circles/ |title=Dragons, aliens, bugs? Scientists may have solved the mystery of the desert's 'fairy circles' |quote=The thing that immediately caught my eye about what they had was it seemed to fall into an exotic type of patterning I call ''<nowiki />'hyperuniformity'<nowiki />''. — Salvatore Torquato}}</ref><ref>{{cite journal |last1=Getzin |first1=Stephan |display-authors=etal |year=2016 |title=Discovery of fairy circles in Australia supports self-organization theory1 |journal=Proceedings of the National Academy of Sciences |volume=113 |issue=13 |pages=3551–3556 |bibcode=2016PNAS..113.3551G |doi=10.1073/pnas.1522130113 |pmc=4822591 |pmid=26976567 |doi-access=free }}</ref> It is believed such vegetation patterns can optimize the efficiency of water utility, which is crucial for the survival of the plants.
A universal hyperuniform organization was observed in the looped vein network of tree leaves, including ficus religiosa, ficus caulocarpa, ficus microcarpa, smilax indica, populus rotundifolia, yulania denudate, and others.<ref name="DHU_leaf">{{cite journal |last1=Liu |display-authors=etal |year=2024 |title=Universal Hyperuniform Organization in Looped Leaf Vein Networks |journal=Physical Review Letters|volume=133 |issue=2 |article-number=028401 |doi=10.1103/PhysRevLett.133.028401|pmid=39073952 |arxiv=2311.09551 |bibcode=2024PhRvL.133b8401L }}</ref> It was shown the hyperuniform network optimizes the diffusive transport of water and nutrients from the vein to the leaf cells.<ref name="DHU_leaf"/> The hyperuniform vein network organization was believed to result from a regulation of growth factor uptake during vein network development.<ref name="DHU_leaf"/>
==Making disordered, but highly uniform, materials== The challenge of creating disordered hyperuniform materials is partly attributed to the inevitable presence of imperfections, such as defects and thermal fluctuations. For example, the fluctuation-compressibility relation dictates that any compressible one-component fluid in thermal equilibrium cannot be strictly hyperuniform at finite temperature.<ref name=":3"/>
Recently Chremos & Douglas (2018) proposed a design rule for the practical creation of hyperuniform materials at the molecular level.<ref name="PRL-18">{{cite journal |last1=Chremos |first1=Alexandros |last2=Douglas |first2=Douglas F. |title=Hidden hyperuniformity in soft polymeric materials |journal=Physical Review Letters |date=Dec 21, 2018 |volume=121 |issue=25 |article-number=258002 |doi=10.1103/PhysRevLett.121.258002 |pmid=30608782 |bibcode=2018PhRvL.121y8002C |doi-access=free}}</ref><ref name=":8">{{cite journal |last=Chremos |first=Alexandros |date=2020-08-07 |title=Design of nearly perfect hyperuniform polymeric materials |journal=The Journal of Chemical Physics |language=en |volume=153 |issue=5 |page=054902 |doi=10.1063/5.0017861 |pmid=32770903 |bibcode= |issn=0021-9606|pmc=7530914 }}</ref> Specifically, effective hyperuniformity as measured by the hyperuniformity index is achieved by specific parts of the molecules (e.g., the core of the star polymers or the backbone chains in the case of bottlebrush polymers).<ref>{{cite journal |last1=Atkinson |first1=Steven |last2=Zhang |first2=Ge |last3=Hopkins |first3=Adam B. |last4=Torquato |first4=Salvatore |date=2016-07-08 |title=Critical slowing down and hyperuniformity on approach to jamming |journal=Physical Review E |language=en |volume=94 |issue=1 |article-number=012902 |doi=10.1103/PhysRevE.94.012902 |pmid=27575201 |arxiv=1606.05227 |bibcode=2016PhRvE..94a2902A |s2cid=12103288 |issn=2470-0045}}</ref><ref name=":3" /> The combination of these features leads to molecular packings that are highly uniform at both small and large length scales.<ref name="PRL-18" /><ref name=":8"/> ==Non-equilibrium hyperuniform fluids and length scales== Disordered hyperuniformity implies a long-ranged direct correlation function (the Ornstein–Zernike equation).<ref name="PRE-68"/> In an equilibrium many-particle system, this requires delicately designed effectively long-ranged interactions, which are not necessary for the dynamic self-assembly of non-equilibrium hyperuniform states. In 2019, Ni and co-workers theoretically predicted a non-equilibrium strongly hyperuniform fluid phase that exists in systems of circularly swimming active hard spheres,<ref name="SA-19"/> which was confirmed experimentally in 2022.<ref name="PRL-22">{{cite journal |last1=Zhang |first1=Bo |last2= Snezhko |first2= Alexey |title=Hyperuniform Active Chiral Fluids with Tunable Internal Structure |journal=Physical Review Letters |date=May 27, 2022 |volume=128 |issue=21 |article-number= 218002 |doi=10.1103/PhysRevLett.128.218002 |pmid=35687470 |arxiv=2205.12384 |bibcode=2022PhRvL.128u8002Z |s2cid=249063085 }}</ref>
This new hyperuniform fluid features a special length scale, i.e., the diameter of the circular trajectory of active particles, below which large density fluctuations are observed. Moreover, based on a generalized random organising model, Lei and Ni (2019)<ref name="HUF-19"/> formulated a hydrodynamic theory for non-equilibrium hyperuniform fluids, and the length scale above which the system is hyperuniform is controlled by the inertia of the particles. The theory generalizes the mechanism of fluidic hyperuniformity as the damping of the stochastic harmonic oscillator, which indicates that the suppressed long-wavelength density fluctuation can exhibit as either acoustic (resonance) mode or diffusive (overdamped) mode.<ref name="HUF-19"/> In the Lei-Ni reactive hard-sphere model,<ref name="HUF-19"/> it was found that the discontinuous absorbing transition of metastable hyperuniform fluid into an immobile absorbing state does not have the kinetic pathway of nucleation and growth, and the transition rate decreases with increasing the system size. This challenges the common understanding of metastability in discontinuous phase transitions and suggests that non-equilibrium hyperuniform fluid is fundamentally different from conventional equilibrium fluids.<ref name="pnas-23">{{cite journal |last1=Lei |first1=Yusheng |last2= Ni |first2= Ran |title=How does a hyperuniform fluid freeze? |journal=Proceedings of the National Academy of Sciences of the United States of America |date=November 21, 2023 |volume=120 |issue=48 |article-number= e2312866120 |doi=10.1073/pnas.2312866120 |doi-access=free |pmid=37988461 |pmc=10691242 |arxiv=2306.02753 |bibcode=2023PNAS..12012866L }}</ref>
== See also == * Crystal * Quasicrystal * Amorphous solid * State of matter
==References== {{reflist|25em}}
==External links== * {{cite magazine |url=https://www.quantamagazine.org/20160712-hyperuniformity-found-in-birds-math-and-physics/ |title=A bird's-eye view of nature's hidden order |first=Natalie |last=Wolchover |magazine=Quanta Magazine}} * {{cite magazine |url=https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/ |title=A chemist shines light on a surprising prime number pattern |first=Natalie |last=Wolchover |magazine=Quanta Magazine}}
Category:Liquids Category:Concepts in physics Category:Materials science category:Statistical mechanics