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Disordered Hyperuniform Point Patterns inPhysics, Mathematics and Biology

Salvatore Torquato

Department of Chemistry,

Department of Physics,

Princeton Institute for the Science and Technology of Materials,

and Program in Applied & Computational Mathematics

Princeton University

http://cherrypit.princeton.edu

. – p. 1/50

States (Phases) of Matter

Source: www.nasa.gov

. – p. 2/50

States (Phases) of Matter

Source: www.nasa.gov

We now know there are a multitude of distinguishable states of matter, e.g.,

quasicrystals and liquid crystals , which break the continuous translational

and rotational symmetries of a liquid differently from a sol id crystal .. – p. 2/50

What Qualifies as a Distinguishable State of Matter?

Traditional Criteria

Homogeneous phase in thermodynamic equilibrium

Interacting entities are microscopic objects, e.g. atoms, molecules or spins

Often, phases are distinguished by symmetry-breaking and/ or some

qualitative change in some bulk property

. – p. 3/50







Broader Criteria

Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g.,

spin glasses and structural glasses

Interacting entities need not be microscopic, but can inclu de building blocks

across a wide range of length scales, e.g., colloids and metamaterials

Endowed with unique properties

. – p. 3/50







Broader Criteria

Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g.,

spin glasses and structural glasses

Interacting entities need not be microscopic, but can inclu de building blocks

across a wide range of length scales, e.g., colloids and metamaterials

Endowed with unique properties

New states of matter become more compelling if they

Give rise to or require new ideas and/or experimental/theor etical tools

Technologically important. – p. 3/50

Pair Statistics in Direct and Fourier SpacesFor particle systems in R

d at number density ρ , g2(r) is a nonnegative radial function that is

proportional to the probability density of pair distances r.

The nonnegative structure factor S(k) ≡ 1 + ρh(k) is obtained from the Fourier transform of

h(r) = g2(r) − 1, which we denote by h(k).

Poisson Distribution (Ideal Gas)

0 1 2 3r

0

0.5

1

1.5

2

g 2(r)

0 1 2 3k

0

0.5

1

1.5

2

S(k)

Liquid

0 1 2 3 4 5r

0

0.5

1

1.5

2

2.5

3

g 2(r)

0 5 10 15 20 25 30k

0

0.5

1

1.5

2

2.5

3

S(k)

Crystal (Angularly Averaged)

0 1 2 3r

g 2(r)

0 4π/√3 8π/√3 12π/√3k

S(k)

. – p. 4/50

CuriositiesDisordered jammed packings

0 4 8 12 16 20 24kD

0

2

4

6

8

S(k

)

40,000−particle random

φ=0.632

jammed packing

S(k) appears to vanish in the limit k → 0: very unusual behavior for a

disordered system .

Harrison-Zeldovich spectrum for density fluctuations in the early Universe:

S(k) ∼ k for sufficiently small k.

. – p. 5/50

HYPERUNIFORMITY

A hyperuniform many-particle system is one in which normalized density

fluctuations are completely suppressed at very large lengths scales .

. – p. 6/50

HYPERUNIFORMITY



Disordered hyperuniform many-particle systems can be regarded to be new

distinguishable states of disordered matter in that they

(i) behave more like crystals or quasicrystals in the manner in which they

suppress large-scale density fluctuations , and yet are also like liquids and

glasses because they are statistically isotropic structures with n o Bragg

peaks ;

(ii) can exist as both as equilibrium and quenched nonequilibrium phases ;

(iii) and, appear to be endowed with unique bulk physical properties .

Understanding such states of matter, which have technologi cal importance, require

new theoretical tools.

. – p. 6/50

HYPERUNIFORMITY








peaks ;





All perfect crystals (periodic systems) and quasicrystals are hyperuniform.

. – p. 6/50

HYPERUNIFORMITY








peaks ;





All perfect crystals (periodic systems) and quasicrystals are hyperuniform.

Thus, hyperuniformity provides a unified means of categorizing and

characterizing crystals, quasicrystals and such special disordered systems.. – p. 6/50

Local Density Fluctuations for General Point Patterns

Torquato and Stillinger, PRE (2003)

Points can represent molecules of a material, stars in a gala xy, or trees in a

forest. Let Ω represent a spherical window of radius R in d-dimensional

Euclidean space Rd.

ΩR ΩR

Denote by σ2(R) ≡ 〈N2(R)〉 − 〈N(R)〉2 the number variance .

. – p. 7/50

Local Density Fluctuations for General Point Patterns

Torquato and Stillinger, PRE (2003)

Points can represent molecules of a material, stars in a gala xy, or trees in a

forest. Let Ω represent a spherical window of radius R in d-dimensional

Euclidean space Rd.

ΩR ΩR

Denote by σ2(R) ≡ 〈N2(R)〉 − 〈N(R)〉2 the number variance .

For a Poisson point pattern and many disordered point patterns, σ2(R) ∼ Rd.

We call point patterns whose variance grows more slowly than Rd (window

volume) hyperuniform . This implies that structure factor S(k) → 0 for k → 0.

All perfect crystals and quasicrystals are hyperuniform such that

σ2(R) ∼ Rd−1: number variance grows like window surface area .

. – p. 7/50

Hyperuniformity and Crystals

-25 -20 -15 -10 -5 0 5 10 15 20 25-20

-15

-10

-5

0

5

10

15

20

Diffraction Pattern for a Square LatticeDots denote Bragg peaks at δ(k-Q) (Lattice spacing = 2π/a )

0 5 10 15 20 25

Wavenumber, k

0.00

0.05

0.10

S(k)

Angular-averaged S(k) for a Square Lattice

2π/a

All perfect crystals are trivially hyperuniform .

But the degree to which they suppress large-scale density

fluctuations varies. Which crystal in Rd minimizes such

fluctuations?

. – p. 8/50

Hidden Order on Large Length Scales

Which is the hyperuniform pattern?

. – p. 9/50

Outline

Hyperuniform Systems

“Inverted” Critical Phenomena

Minimizing Variance: Special Ground State Problem

Variance as an Order Metric at Large Length Scales

Examples of Disordered Hyperuniform Systems:

Natural and Synthetic

. – p. 10/50

ENSEMBLE-AVERAGE FORMULATIONFor an ensemble in d-dimensional Euclidean space R

d:

σ2(R) = 〈N(R)〉h

1 + ρ

Z

Rd

h(r)α(r; R)dri

α(r; R)- scaled intersection volume of 2 windows of radius R separated by r

R

r

0 0.2 0.4 0.6 0.8 1r/(2R)

0

0.2

0.4

0.6

0.8

1

α(r;

R) d=1

d=5

Spherical window of radius R

. – p. 11/50

ENSEMBLE-AVERAGE FORMULATIONFor an ensemble in d-dimensional Euclidean space R

d:

σ2(R) = 〈N(R)〉h

1 + ρ

Z

Rd

h(r)α(r; R)dri

α(r; R)- scaled intersection volume of 2 windows of radius R separated by r

R

r

0 0.2 0.4 0.6 0.8 1r/(2R)

0

0.2

0.4

0.6

0.8

1

α(r;

R) d=1

d=5


For large R, we can show

σ2(R) = 2dφh

A

„

R

D

«d

+ B

„

R

D

«d−1

+ o

„

R

D

«d−1i

,

where A and B are the “ volume ” and “ surface-area ” coefficients:

A = S(k = 0) = 1 + ρ

Z

Rd

h(r)dr, B = −c(d)

Z

Rd

h(r)rdr,

D: microscopic length scale, φ: dimensionless density

Hyperuniform : A = 0, B > 0

. – p. 11/50

INVERTED CRITICAL PHENOMENAh(r) can be divided into direct correlations , via function c(r), and indirect correlations:

c(k) =h(k)

1 + ρh(k)

. – p. 12/50


c(k) =h(k)

1 + ρh(k)

For any hyperuniform system , h(k = 0) = −1/ρ, and thus c(k = 0) = −∞. Therefore, at the

“critical” reduced density φc, h(r) is short-ranged and c(r) is long-ranged .

This is the inverse of the behavior at liquid-gas (or magnetic) critical points , where h(r) is

long-ranged (compressibility or susceptibility diverges ) and c(r) is short-ranged .

. – p. 12/50


c(k) =h(k)

1 + ρh(k)

For any hyperuniform system , h(k = 0) = −1/ρ, and thus c(k = 0) = −∞. Therefore, at the

“critical” reduced density φc, h(r) is short-ranged and c(r) is long-ranged .

This is the inverse of the behavior at liquid-gas (or magnetic) critical points , where h(r) is

long-ranged (compressibility or susceptibility diverges ) and c(r) is short-ranged .

For sufficiently large d at a disordered hyperuniform state , whether achieved via a nonequilibrium

or an equilibrium route,

c(r) ∼ −1

rd−2+η(r → ∞), c(k) ∼ −

1

k2−η(k → 0),

h(r) ∼ −1

rd+2−η(r → ∞), S(k) ∼ k2−η (k → 0),

where η is a new critical exponent .

One can think of a hyperuniform system as one resulting from an effective pair potential v(r) at

large r that is a generalized Coulombic interaction between like charges . Why? Because

v(r)

kBT∼ −c(r) ∼

1

rd−2+η(r → ∞)

However, long-range interactions are not required to drive a nonequilibrium system to a

disordered hyperuniform state.. – p. 12/50

SINGLE-CONFIGURATION FORMULATION & GROUND STATESWe showed

σ2(R) = 2dφ

„

R

D

«dh

1 − 2dφ

„

R

D

«d

+1

N

NX

i6=j

α(rij ; R)i

where α(r; R) can be viewed as a repulsive pair potential :

0 0.2 0.4 0.6 0.8 1r/(2R)

0

0.2

0.4

0.6

0.8

1

α(r;

R) d=1

d=5


. – p. 13/50


σ2(R) = 2dφ

„

R

D

«dh

1 − 2dφ

„

R

D

«d

+1

N

NX

i6=j

α(rij ; R)i


0 0.2 0.4 0.6 0.8 1r/(2R)

0

0.2

0.4

0.6

0.8

1

α(r;

R) d=1

d=5


Finding global minimum of σ2(R) equivalent to finding ground state .

. – p. 13/50


σ2(R) = 2dφ

„

R

D

«dh

1 − 2dφ

„

R

D

«d

+1

N

NX

i6=j

α(rij ; R)i


0 0.2 0.4 0.6 0.8 1r/(2R)

0

0.2

0.4

0.6

0.8

1

α(r;

R) d=1

d=5


Finding global minimum of σ2(R) equivalent to finding ground state .

For large R, in the special case of hyperuniform systems,

σ2(R) = Λ(R)

„

R

D

«d−1

+ O

„

R

D

«d−3

100 110 120 130R/D

0

0.2

0.4

0.6

0.8

1

Λ(R

)

Triangular Lattice (Average value=0.507826)

. – p. 13/50

Hyperuniformity and Number Theory

Averaging fluctuating quantity Λ(R) gives coefficient of interest:

Λ = limL→∞

1

L

∫ L

0Λ(R)dR

. – p. 14/50

Hyperuniformity and Number Theory

Averaging fluctuating quantity Λ(R) gives coefficient of interest:

Λ = limL→∞

1

L

∫ L

0Λ(R)dR

We showed that for a lattice

σ2(R) =∑

q6=0

(

2πR

q

)d

[Jd/2(qR)]2, Λ = 2dπd−1∑

q6=0

1

|q|d+1.

Epstein zeta function for a lattice is defined by

Z(s) =∑

q6=0

1

|q|2s, Re s > d/2.

Summand can be viewed as an inverse power-law potential . For

lattices , minimizer of Z(d + 1) is the lattice dual to the minimizer of Λ.

Surface-area coefficient Λ provides useful way to rank order crystals,

quasicrystals and special correlated disordered point patterns.

. – p. 14/50

Quantifying Suppression of Density Fluctuations at Large Scales: 1D

The surface-area coefficient Λ for some crystal, quasicrystaland disordered one-dimensional hyperuniform point patterns.

Pattern Λ

Integer Lattice 1/6 ≈ 0.166667

Step+Delta-Function g2 3/16 =0.1875

Fibonacci Chain ∗ 0.2011

Step-Function g2 1/4 = 0.25

Randomized Lattice 1/3 ≈ 0.333333

∗Zachary & Torquato (2009)

. – p. 15/50

Quantifying Suppression of Density Fluctuations at Large Scales: 2D

The surface-area coefficient Λ for some crystal, quasicrystaland disordered two-dimensional hyperuniform point patterns.

2D Pattern Λ/φ1/2

Triangular Lattice 0.508347

Square Lattice 0.516401

Honeycomb Lattice 0.567026

Kagom e Lattice 0.586990

Penrose Tiling ∗ 0.597798

Step+Delta-Function g2 0.600211

Step-Function g2 0.848826

∗Zachary & Torquato (2009)

. – p. 16/50

Quantifying Suppression of Density Fluctuations at Large Scales: 3DContrary to conjecture that lattices associated with the densest

sphere packings have smallest variance regardless of d, we have

shown that for d = 3, BCC has a smaller variance than FCC.

. – p. 17/50

Quantifying Suppression of Density Fluctuations at Large Scales: 3DContrary to conjecture that lattices associated with the densest

sphere packings have smallest variance regardless of d, we have

shown that for d = 3, BCC has a smaller variance than FCC.

Pattern Λ/φ2/3

BCC Lattice 1.24476

FCC Lattice 1.24552

HCP Lattice 1.24569

SC Lattice 1.28920

Diamond Lattice 1.41892

Wurtzite Lattice 1.42184

Damped-Oscillating g2 1.44837

Step+Delta-Function g2 1.52686

Step-Function g2 2.25

Carried out analogous calculations in high d (Zachary &

Torquato, 2009 ), of importance in communications. Disordered

point patterns may win in high d (Torquato & Stillinger, 2006 ).. – p. 17/50

Examples of Homogeneous and Isotropic Hyperuniform Systems

An interesting 1D hyperuniform point pattern is the distribution of the zeros of

the Riemann zeta function (eigenvalues of random Hermitian matrices and bus

arrivals in Cuernavaca) : g2(r) = 1 − sin2(πr)/(πr)2 (Dyson, 1970;

Montgomery, 1973; Krb alek & Seba, 2000).

0 1 2 3 4 5r

0

0.5

1

1.5

g 2(r)

. – p. 18/50






0 1 2 3 4 5r

0

0.5

1

1.5

g 2(r)

Constructing and/or identifying homogeneous, isotropic h yperuniform

patterns for d ≥ 2 is more challenging .

. – p. 18/50






0 1 2 3 4 5r

0

0.5

1

1.5

g 2(r)

Constructing and/or identifying homogeneous, isotropic h yperuniform

patterns for d ≥ 2 is more challenging .

In addition to the few examples discussed thus far, we now know of many

more examples.

. – p. 18/50

More Recent Examples of Disordered Hyperuniform Systems

Fermionic point processes : S(k) ∼ k as k → 0 (ground states and/or

positive temperature equilibrium states)

Maximally random jammed (MRJ) particle packings: S(k) ∼ k as k → 0

(nonequilibrium states)

Disordered classical ground states via collective coordinates

Ultracold atoms (nonequilibrium states): Lesanovsky et al. PRE (2014)

Critical absorbing states (nonequilibrium states): Hexner et al. PRL (2015)

Diffusive systems (nonequilibrium states): Jack et al. PRL (2015)

Periodically-driven emulsions (nonequilibrium states): Weijs et. al. PRL (2015)

. – p. 19/50











Natural Disordered Hyperuniform Systems

Avian Photoreceptor Cells (nonequilibrium states)

. – p. 19/50











Natural Disordered Hyperuniform Systems

Avian Photoreceptor Cells (nonequilibrium states)

Nearly Hyperuniform Disordered Systems

Amorphous Silicon (nonequilibrium states)

Supercooled Liquids and Glasses (nonequilibrium states). – p. 19/50

Hyperuniformity and Spin-Polarized Free FermionsOne can map random Hermitian matrices (GUE), fermionic gase s, and zeros of

the Riemann zeta function to a unique hyperuniform point process on R.

. – p. 20/50

Hyperuniformity and Spin-Polarized Free FermionsOne can map random Hermitian matrices (GUE), fermionic gase s, and zeros of

the Riemann zeta function to a unique hyperuniform point process on R.

We provide exact generalizations of such a point process in d-dimensional

Euclidean space Rd and the corresponding n-particle correlation functions ,

which correspond to those of spin-polarized free fermionic systems in Rd.

0 0.5 1 1.5 2r

0

0.2

0.4

0.6

0.8

1

1.2

g 2(r)

d=1d=3

0 2 4 6 8 10k

0

0.2

0.4

0.6

0.8

1

1.2

S(k

)

d=1

d=3

g2(r) = 1 −2Γ(1 + d/2) cos2 (rK − π(d + 1)/4)

K πd/2+1 rd+1(r → ∞)

S(k) =c(d)

2Kk + O(k3) (k → 0) (K : Fermi sphere radius )

Torquato, Zachary & Scardicchio, J. Stat. Mech., 2008

Scardicchio, Zachary & Torquato, PRE, 2009

. – p. 20/50

Hyperuniformity and Jammed PackingsConjecture : All strictly jammed saturated sphere packings are hyperuniform(Torquato & Stillinger, 2003 ).

. – p. 21/50


A 3D maximally random jammed (MRJ) packing is a prototypical glass in that it

is maximally disordered but perfectly rigid (infinite elastic moduli).

Such packings of identical spheres have been shown to be hyperuniform with

quasi-long-range (QLR) pair correlations in which h(r) decays as −1/r4

(Donev, Stillinger & Torquato, PRL, 2005 ).

0 0.5 1 1.5 2kD/2π

0

1

2

3

4

5

S(k)

Data

0 0.2 0.4 0.60

0.02

0.04

Linear fit

This is to be contrasted with the hard-sphere fluid with correlations that decay

exponentially fast .

. – p. 21/50


A 3D maximally random jammed (MRJ) packing is a prototypical glass in that it

is maximally disordered but perfectly rigid (infinite elastic moduli).

Such packings of identical spheres have been shown to be hyperuniform with

quasi-long-range (QLR) pair correlations in which h(r) decays as −1/r4

(Donev, Stillinger & Torquato, PRL, 2005 ).

0 0.5 1 1.5 2kD/2π

0

1

2

3

4

5

S(k)

Data

0 0.2 0.4 0.60

0.02

0.04

Linear fit

This is to be contrasted with the hard-sphere fluid with correlations that decay

exponentially fast .

Apparently, hyperuniform QLR correlations with decay −1/rd+1 are a

universal feature of general MRJ packings in Rd.

Zachary, Jiao and Torquato, PRL (2011) : ellipsoids, superballs, sphere mixtures

Berthier et al., PRL (2011); Kurita and Weeks, PRE (2011) : sphere mixtures

Jiao and Torquato, PRE (2011) : polyhedra. – p. 21/50

Slow and Rapid Cooling of a LiquidClassical ground states are those classical particleconfigurations with minimal potential energy per particle.

Vol

ume

Temperature

quenchglass

liquid

crystal

cooling

. – p. 22/50


Vol

ume

Temperature

rapidquench

glass

super−cooledliquid

liquid

freezingpoint(Tf)

glasstransition(Tg)

crystal

veryslowcooling

Typically, ground states are periodic with high crystallographicsymmetries .

. – p. 22/50


Vol

ume

Temperature

rapidquench

glass

super−cooledliquid

liquid

freezingpoint(Tf)

glasstransition(Tg)

crystal

veryslowcooling

Typically, ground states are periodic with high crystallographicsymmetries .

Can classical ground states ever be disordered ?. – p. 22/50

Creation of Disordered Hyperuniform Ground States

Uche, Stillinger & Torquato, Phys. Rev. E 2004

Batten, Stillinger & Torquato, Phys. Rev. E 2008

Collective-Coordinate Simulations

• Consider a system of N particles with configuration rN ≡ r1, r2, . . . in a fundamental region Ω

(under periodic boundary conditions) at position r. The complex collective density variable is defined by

ρ(k) =

NX

j=1

exp(ik · rj)

. – p. 23/50







ρ(k) =

NX

j=1

exp(ik · rj)

• The real nonnegative structure factor is

S(k) =|ρ(k)|2

N

. – p. 23/50







ρ(k) =

NX

j=1

exp(ik · rj)

• The real nonnegative structure factor is

S(k) =|ρ(k)|2

N

• Consider stable pair potentials v(r) that are bounded with Fourier transform v(k). The total energy is

ΦN (rN ) =X

i<j

v(rij)

=N

2|Ω|

X

k

v(k)S(k) + constant

• For v(k) positive ∀ 0 ≤ |k| ≤ K and zero otherwise, finding configurations in which S(k) is

constrained to be zero where v(k) has support is equivalent to globally minimizing Φ(rN ). These

hyperuniform ground states are called “stealthy” and generally highly degenerate .

• Stealthy patterns can be tuned by varying the parameter χ: number of independently (real) constrained

degrees of freedom to the total number of degrees of freedom d(N − 1).. – p. 23/50

Creation of Disordered Stealthy Ground StatesUnconstrained

Region

Exclusion

Zone S=0

K

. – p. 24/50

Creation of Disordered Stealthy Ground StatesUnconstrained

Region

Exclusion

Zone S=0

K

One class of stealthy potentials involves the following power-law form:

v(k) = v0(1 − k/K)m Θ(K − k),

where n is any whole number. The special case n = 0 is just the simple step function.

10 20r

0

0.01v(r)

m=0m=2m=4

d=3

0 0.5 1 1.5k

0

0.5

1

1.5

v(k)

m=0m=2m=4

~

In the large-system ( thermodynamic ) limit with m = 0 and m = 4, we have the following large-r

asymptotic behavior , respectively:v(r) ∼

cos(r)

r2(m = 0)

v(r) ∼1

r4(m = 4)

While the specific forms of these stealthy potentials lead to the same convergent ground-state

energies, this will not be the case for the pressure and other thermodynamic quantities.. – p. 24/50

Creation of Disordered Stealthy Ground States

In our previous simulations, we began with an initial random distribution of N

points and then found the energy minimizing configurations (with extremely

high precision) using optimization techniques.

. – p. 25/50





For 0 ≤ χ < 0.5, the stealthy ground states are degenerate, disordered and

isotropic .

0 1 2 3k

0

0.5

1

1.5

S(k)

Note the “hard-core” nature of S(k) for stealthy ground states.

The success rate to achieve any disordered ground state is 100%, independent

of the dimension. It turns out that χ = 1/2 is the limit at which there are no

longer degrees of freedom that can be constrained independe ntly.

. – p. 25/50





For 0 ≤ χ < 0.5, the stealthy ground states are degenerate, disordered and

isotropic .

0 1 2 3k

0

0.5

1

1.5

S(k)

Note the “hard-core” nature of S(k) for stealthy ground states.

The success rate to achieve any disordered ground state is 100%, independent

of the dimension. It turns out that χ = 1/2 is the limit at which there are no

longer degrees of freedom that can be constrained independe ntly.

For χ > 1/2, the system undergoes a transition to a crystal phase and the

energy landscape becomes considerably more complex.. – p. 25/50

Creation of Disordered Stealthy Ground StatesTwo Dimensions

(a) χ= 0.04167 (b) χ = 0.41071

Three Dimensions

0 0.1 0.2 0.3 0.4r / L

x

0

1

2

g 2(r)

. – p. 26/50

Creation of Disordered Stealthy Ground StatesTwo Dimensions

(a) χ= 0.04167 (b) χ = 0.41071

Three Dimensions

0 0.1 0.2 0.3 0.4r / L

x

0

1

2

g 2(r)

Increasing χ induces strong short-range ordering , and eventually the system runs out of degrees

of freedom that can be constrained and crystallizes .

0 1 2 3k/K

S(k)

Maximum χ

. – p. 26/50

Ensemble Theory of Disordered Ground StatesTorquato, Zhang & Stillinger, Phys. Rev. X, 2015

Nontrivial : Dimensionality of the configuration space depends on the nu mber density ρ (or χ) and

there is a multitude of ways of sampling the ground-state man ifold, each with its own probability

measure. Which ensemble? How are entropically favored states determined?

Derived general exact relations for thermodynamic propert ies that apply to any ground-state

ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise.

. – p. 27/50







From previous considerations, we that an important contrib ution to S(k) is a simple hard-core

step function Θ(k − K), which can be viewed as a disordered hard-sphere system in Fourier

space in the limit that χ ∼ 1/ρ tends to zero, i.e., as the number density ρ tends to infinity.

0 1 2 3k

0

0.5

1

1.5

S(k

)

0 1 2 3r

0

0.5

1

1.5

g 2(r)

That the structure factor must have the behavior

S(k) → Θ(k − K), χ → 0

is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = 1 for all k.

. – p. 27/50







From previous considerations, we that an important contrib ution to S(k) is a simple hard-core

step function Θ(k − K), which can be viewed as a disordered hard-sphere system in Fourier

space in the limit that χ ∼ 1/ρ tends to zero, i.e., as the number density ρ tends to infinity.

0 1 2 3k

0

0.5

1

1.5

S(k

)

0 1 2 3r

0

0.5

1

1.5

g 2(r)

That the structure factor must have the behavior

S(k) → Θ(k − K), χ → 0

is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = 1 for all k.

We make the ansatz that for sufficiently small χ, S(k) in the canonical ensemble for a stealthy

potential can be mapped to g2(r) for an effective disordered hard-sphere system for sufficiently

small density .. – p. 27/50

Pseudo-Hard Spheres in Fourier SpaceLet us define

H(k) ≡ ρh(k) = hHS(r = k)

There is an Ornstein-Zernike integral eq. that defines FT of appropriate direct correlation function , C(k):

H(k) = C(k) + η H(k) ⊗ C(k),

where η is an effective packing fraction . Therefore,

H(r) =C(r)

1 − (2π)d η C(r).

This mapping enables us to exploit the well-developed accur ate theories of standard Gibbsian

disordered hard spheres in direct space .

0 1 2 3 4k

0

0.5

1

1.5

S(k

)

TheorySimulation

d=3, χ=0.05

0 1 2 3 4k

0

0.5

1

1.5S

(k)

TheorySimulation

d=3, χ=0.1

0 1 2 3 4k

0

0.5

1

1.5

S(k

)

TheorySimulation

d=3, χ=0.143

0 5 10r

0

0.5

1

1.5

g 2(r)

d=1, Simulationd=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory

χ=0.05

0 5 10r

0

0.5

1

1.5

g 2(r)


χ=0.1

0 5 10r

0

0.5

1

1.5

g 2(r)


χ=0.143

. – p. 28/50

Stealthy Disordered Ground States and Novel Materials

Until recently, it was believed that Bragg scattering was required to achieve

metamaterials with complete photonic band gaps .

. – p. 29/50




Have used disordered, isotropic “stealthy” ground-state configurat ions to

design photonic materials with large complete (both polarizations and all

directions) band gaps .

Florescu, Torquato and Steinhardt, PNAS (2009)

. – p. 29/50








These metamaterial designs have been fabricated for microwave regime.

Man et. al., PNAS (2013)

Because band gaps are isotropic , such photonic materials offer advantages

over photonic crystals (e.g., free-form waveguides ).

. – p. 29/50








These metamaterial designs have been fabricated for microwave regime.

Man et. al., PNAS (2013)

Because band gaps are isotropic , such photonic materials offer advantages

over photonic crystals (e.g., free-form waveguides ).

Other applications include new phononic devices.. – p. 29/50

Which is the Cucumber Image?

. – p. 30/50

Avian Cone PhotoreceptorsOptimal spatial sampling of light requires that photoreceptors be arranged in

the triangular lattice (e.g., insects and some fish).

Birds are highly visual animals, yet their cone photoreceptor patterns are

irregular .

. – p. 31/50

Avian Cone PhotoreceptorsOptimal spatial sampling of light requires that photoreceptors be arranged in

the triangular lattice (e.g., insects and some fish).

Birds are highly visual animals, yet their cone photoreceptor patterns are

irregular .

5 Cone Types

Jiao, Corbo & Torquato, PRE (2014).. – p. 31/50

Avian Cone PhotoreceptorsDisordered mosaics of both total population and individual cone types are

effectively hyperuniform , which has been never observed in any system before

(biological or not). We term this multi-hyperuniformity .

Jiao, Corbo & Torquato, PRE (2014). – p. 32/50

Amorphous Silicon is Nearly HyperuniformHighly sensitive transmission X-ray scattering measureme nts performed at

Argonne on amorphous-silicon (a-Si) samples reveals that t hey are nearly

hyperuniform with S(0) = 0.0075.

Long, Roorda, Hejna, Torquato, and Steinhardt (2013)

This is significantly below the putative lower bound recentl y suggested by de

Graff and Thorpe (2009) but consistent with the recently pro posed nearly

hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).

0 5 10 15 200.0

0.5

1.0

1.5

2.0

k(A )-1

S(k)

a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX]

. – p. 33/50

Amorphous Silicon is Nearly HyperuniformHighly sensitive transmission X-ray scattering measureme nts performed at

Argonne on amorphous-silicon (a-Si) samples reveals that t hey are nearly

hyperuniform with S(0) = 0.0075.

Long, Roorda, Hejna, Torquato, and Steinhardt (2013)

This is significantly below the putative lower bound recentl y suggested by de

Graff and Thorpe (2009) but consistent with the recently pro posed nearly

hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).

0 5 10 15 200.0

0.5

1.0

1.5

2.0

k(A )-1

S(k)

a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX]

Increasing the degree of hyperuniformity of a-Si appears to be correlated with

a larger electronic band gap (Hejna, Steinhardt and Torquato, 2013).. – p. 33/50

Structural Glasses and Growing Length ScalesImportant question in glass physics: Do growing relaxation times under

supercooling have accompanying growing structural length scales? Lubchenko

& Wolynes (2006); Berthier et al. (2007); Karmakar, Dasgupt a & Sastry (2009); Chandler &

Garrahan (2010); Hocky, Markland & Reichman (2012)

. – p. 34/50





We studied glass-forming liquid models that support an alte rnative view:

existence of growing static length scales (due to increase of the degree of

hyperuniformity ) as the temperature T of the supercooled liquid is decreased

to and below Tg that is intrinsically nonequilibrium in nature.

0 1 2 3 4 5Dimensionless temperature, T / T

g

3

4

5

6

7

8L

engt

h sc

ale,

ξc

. – p. 34/50





We studied glass-forming liquid models that support an alte rnative view:

existence of growing static length scales (due to increase of the degree of

hyperuniformity ) as the temperature T of the supercooled liquid is decreased

to and below Tg that is intrinsically nonequilibrium in nature.

0 1 2 3 4 5Dimensionless temperature, T / T

g

3

4

5

6

7

8L

engt

h sc

ale,

ξc

The degree of deviation from thermal equilibrium is determi ned from a

nonequilibrium indexX =

S(k = 0)

ρkBTκT− 1,

which increases upon supercooling .Marcotte, Stillinger & Torquato (2013)

. – p. 34/50

CONCLUSIONSDisordered hyperuniform many-particle systems can be regarded to be a new

distinguishable state of disordered matter .

Hyperuniformity provides a unified means of categorizing and characterizing

crystals, quasicrystals and special correlated disordere d systems.

The degree of hyperuniformity provides an order metric for the extent to which

large-scale density fluctuations are suppressed in such systems.

Disordered hyperuniform systems appear to be endowed with unusual

physical properties that we are only beginning to discover.

Hyperuniformity has connections to physics (e.g., ground states, quantum

systems, random matrices, novel materials, etc.), mathematics (e.g., geometry

and number theory), and biology .

. – p. 35/50

CONCLUSIONSDisordered hyperuniform many-particle systems can be regarded to be a new

distinguishable state of disordered matter .

Hyperuniformity provides a unified means of categorizing and characterizing

crystals, quasicrystals and special correlated disordere d systems.

The degree of hyperuniformity provides an order metric for the extent to which

large-scale density fluctuations are suppressed in such systems.

Disordered hyperuniform systems appear to be endowed with unusual

physical properties that we are only beginning to discover.

Hyperuniformity has connections to physics (e.g., ground states, quantum

systems, random matrices, novel materials, etc.), mathematics (e.g., geometry

and number theory), and biology .

CollaboratorsRobert Batten (Princeton) Etienne Marcotte (Princeton)

Paul Chaikin (NYU) Weining Man (San Francisco State)

Joseph Corbo (Washington Univ.) Sjoerd Roorda (Montreal)

Marian Florescu (Surrey) Antonello Scardicchio (ICTP)

Miroslav Hejna (Princeton) Paul Steinhardt (Princeton)

Yang Jiao (Princeton/ASU) Frank Stillinger (Princeton)

Gabrielle Long (NIST) Chase Zachary (Princeton)

Ge Zhang (Princeton). – p. 35/50

SCATTERING AND DENSITY FLUCTUATIONS

. – p. 36/50

DENSITY FLUCTUATIONS

Density fluctuations of many-particle systems are of great fundamental

interest, containing important thermodynamic and structu ral information.

For systems in equilibrium at number density ρ,

ρkBTκT =〈N2〉 − 〈N〉2

〈N〉= S(k = 0) = 1 + ρ

∫

Rd

h(r)dr

where h(r) ≡ g2(r) − 1 is the total correlation function, g2(r) is the pair

correlation function, and S(k) = 1 + ρh(k) is the structure factor .

Note that generally ρkTκT 6= S(k = 0).

X =S(k = 0)

ρkBTκT− 1 : Nonequilibrium index

. – p. 37/50

DENSITY FLUCTUATIONS

Density fluctuations of many-particle systems are of great fundamental

interest, containing important thermodynamic and structu ral information.

For systems in equilibrium at number density ρ,

ρkBTκT =〈N2〉 − 〈N〉2

〈N〉= S(k = 0) = 1 + ρ

∫

Rd

h(r)dr

where h(r) ≡ g2(r) − 1 is the total correlation function, g2(r) is the pair

correlation function, and S(k) = 1 + ρh(k) is the structure factor .

Note that generally ρkTκT 6= S(k = 0).

X =S(k = 0)

ρkBTκT− 1 : Nonequilibrium index

Large-scale structure of the universe ( Peebles 1993 )

Probe structure and collective motions in vibrated granula r media ( Warr

and Hansen 1996 )

Coulombic systems ( Martin & Yalcin 1980; Lebowitz 1983 )

Integrable quantum systems ( Bleher, Dyson and Lebowitz 1993 )

. – p. 37/50

Hyperuniform Diffraction Patterns

. – p. 38/50

Hypothesized Disordered Hyperuniform Statesg2-invariant process ≡ one in which a given nonnegative pair correlation g2(r) function remains

invariant as the density varies, for all r, over the range of densities

0 ≤ φ ≤ φ∗

At the terminal density φ∗, the hypothesized system is hyperuniform .

. – p. 39/50



0 ≤ φ ≤ φ∗


Step-function g2

0 1 2 3 4 5r

0

0.5

1

1.5

2

g 2(r)

Asymptotic large- r behavior of c(r) is given by the infinite-space Green’s function for the

d-dimensional Laplace equation:

c(r) =

8

>

>

<

>

>

:

−6r, d = 1,

4 ln(r), d = 2,

−2(d+2)d(d−2)

1rd−2

, d ≥ 3.

. – p. 39/50



0 ≤ φ ≤ φ∗


Step-function g2

0 1 2 3 4 5r

0

0.5

1

1.5

2

g 2(r)

Asymptotic large- r behavior of c(r) is given by the infinite-space Green’s function for the

d-dimensional Laplace equation:

c(r) =

8

>

>

<

>

>

:

−6r, d = 1,

4 ln(r), d = 2,

−2(d+2)d(d−2)

1rd−2

, d ≥ 3.

Step + delta function g2

0 1 2 3 4 5r

0

0.5

1

1.5

2

g 2(r)

. – p. 39/50

Pseudo-Hard Spheres in Fourier SpaceLet us define

H(k) ≡ ρh(k) = hHS(r = k)

There is an Ornstein-Zernike integral equation that defines the appropriatedirect correlation function C(k):

H(k) = C(k) + η H(k) ⊗ C(k),

where η = bχ is an effective packing fraction , where b is a constant to be

determined. Therefore,

H(r) =C(r)

1 − (2π)d η C(r).

This mapping enables us to exploit the well-developed accur ate theories of

standard Gibbsian disordered hard spheres in direct space .

To get an idea of the large- r asymptotics , consider case χ → 0 for any d:

ρh(r) ∼ −1

rd/2Jd/2(r) (χ → 0)

ρh(r) ∼ −1

r(d+1)/2cos(r − (d + 1)π/4) (r → +∞)

. – p. 40/50

Stealthy Disordered Ground State Configurations

Compression Dilation

Direct Space Reciprocal Space

S(k)

. – p. 41/50

General Scaling Behaviors

Hyperuniform particle distributions possess structure factorswith a small-wavenumber scaling

S(k) ∼ kα, α > 0,

including the special case α = +∞ for periodic crystals.

Hence, number variance σ2(R) increases for large R

asymptotically as ( Zachary and Torquato, 2011 )

σ2(R) ∼

Rd−1 ln R, α = 1

Rd−α, α < 1

Rd−1, α > 1

(R → +∞).

Until recently, all known hyperuniform configurations pertainedto α ≥ 1.

. – p. 42/50

Targeted SpectraS ∼ kα

0

k0

S(k)

k4

k2

k

k1/2

K

Configurations are ground states of an interactingmany-particle system with up to four-body interactions .

. – p. 43/50

Targeted SpectraS ∼ kα with α ≥ 1

Uche, Stillinger & Torquato (2006)

Figure 1: One of them is for S(k) ∼ k6 and other for S(k) ∼ k.

. – p. 44/50

Targeted SpectraS ∼ kα with α < 1

Zachary & Torquato (2011)

(a) (b)

Figure 2: Both configurations exhibit strong local clustering of poin ts and possess a highly irreg-

ular local structure; however, only one of them is hyperuniform (with S ∼ k1/2).

. – p. 45/50

DEFINITIONS OF THE CRITICAL EXPONENTS

In the vicinity of or at a hyperuniform state , we have

Exponent Asymptotic behavior

γ S−1(0) ∼ (1 − φφc

)−γ (φ → φ−c )

γ ′ S−1(0) ∼ ( φφc

− 1)−γ′

(φ → φ+c )

ν ξ ∼ (1 − φφc

)−ν (φ → φ−c )

ν ′ ξ ∼ ( φφc

− 1)−ν′

(φ → φ+c )

η c(r) ∼ r2−d−η (φ = φc)

where γ = (2 − η)ν

. – p. 46/50

Definitions

A lattice in d-dimensional Euclidean space Rd is the set of points that are

integer linear combinations of d basis (linearly independent) vectors, i.e., for

basis vectors a1, . . . ,ad,

n1a1 + n2a2 + · · · + ndad | n1, . . . , nd ∈ Z

The space Rd can be geometrically divided into identical regions F called

fundamental cells , each of which contains just one point.

In R2:

A periodic point distribution in Rd is a fixed configuration of N points (where

N ≥ 1) in each fundamental cell of a lattice.

. – p. 47/50

Stealthy Hyperuniform Ground and Excited StatesThese soft, long-ranged potentials in two and three dimensions exhibit

unusual low- T behavior.

Batten, Stillinger and Torquato, Phys. Rev. Lett. (2009)

Have used disordered “stealthy” materials to design photon ic materials with

large complete band gaps .


. – p. 48/50







Disordered Hyperuniform Biological Systems

. – p. 48/50







Disordered Hyperuniform Biological Systems

Which is the Cucumber Image?

. – p. 48/50

Pair Correlation (Radial Distribution) Function g2(r)

g2(r) ≡ Probability density function associated with finding

a point at a radial distance r from a given point.

Expected cumulative coordination number at number density ρ

Z(r) = ρs1(1)∫ r

0xd−1g2(x)dx: Expected number

of points contained in a spherical region of radius r.

. – p. 49/50

Pair Correlation (Radial Distribution) Function g2(r)

g2(r) ≡ Probability density function associated with finding

a point at a radial distance r from a given point.

Expected cumulative coordination number at number density ρ

Z(r) = ρs1(1)∫ r

0xd−1g2(x)dx: Expected number

of points contained in a spherical region of radius r.

Poisson Point Distribution in R3 Maximally Random Jammed State in R

3

0 1 2 3

r

0

1

2

3

4

5

g 2(r)

0 1 2 3r

0

1

2

3

4

5

g 2(r)

. – p. 49/50

Diffraction Pattern for Poisson Point Distribution (Ideal Gas)Ground states are highly degenerate.

0 2 4 6 8 10

k

0.6

0.8

1

1.2

1.4

1.6

1.8

2

S(k)

Thermodynamic limit

. – p. 50/50

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