Disordered Hyperuniform Point Patterns in Physics, 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 solid 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 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

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

Endowed with unique properties

. – p. 3/50 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

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 include 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/theoretical tools

Technologically important . – p. 3/50 Pair Statistics in Direct and Fourier Spaces d For particle systems in R 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) 2 2

1.5 1.5

(r) 1 1 2 S(k) g

0.5 0.5

0 0 0 1 2 3 0 1r 2 3 k Liquid 3 3 2.5 2.5 2 2 1.5 (r) 1.5 S(k) 2

g 1 1 0.5 0.5

0 0 0 1 2 3 4 0 5 10 15 20 25 30 r 5 k Crystal (Angularly Averaged) (r) 2 S(k) g

0 0 1 2 3 4π/√3 8π/√3 12π/√3 r k . – p. 4/50 Curiosities Disordered jammed packings

8 40,000−particle random jammed packing 6 φ=0.632

4 S(k)

2

0 0 4 8 12 16 20 24 kD

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

A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales.

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 no 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 technological importance, require new theoretical tools.

. – p. 6/50 HYPERUNIFORMITY

A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales.

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 no 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 technological importance, require new theoretical tools.

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

. – p. 6/50 HYPERUNIFORMITY

A hyperuniform many-particle system is one in which normalized density fluctuations are completely suppressed at very large lengths scales.

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 no 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 technological importance, require new theoretical tools.

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 galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d-dimensional Euclidean space Rd.

Ω R R Ω

Denote by σ2(R) ≡N 2(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 galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d-dimensional Euclidean space Rd.

Ω R R Ω

Denote by σ2(R) ≡N 2(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

Diffraction Pattern for a Square Lattice Angular-averaged S(k) for a Square Lattice Dots denote Bragg peaks at δ(k-Q) (Lattice spacing = 2π/a ) 20

15 0.10

10

5 S(k) 0 0.05 2π/a -5

-10

-15 0.00 0 5 10 15 20 25 -20 -25 -20 -15 -10 -5 0 5 10 15 20 25 Wavenumber, k

All perfect crystals are trivially hyperuniform.

But the degree to which they suppress large-scale density d fluctuations varies. Which crystal in R 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 FORMULATION

For an ensemble in d-dimensional Euclidean space Rd: σ2(R) = N(R) 1+ ρ h(r)α(r; R)dr h ZRd i α(r; R)- scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8

R 0.6 d=1 (r;R) r α 0.4 d=5 0.2

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

. – p. 11/50 ENSEMBLE-AVERAGE FORMULATION

For an ensemble in d-dimensional Euclidean space Rd: σ2(R) = N(R) 1+ ρ h(r)α(r; R)dr h ZRd i α(r; R)- scaled intersection volume of 2 windows of radius R separated by r 1 Spherical window of radius R 0.8

R 0.6 d=1 (r;R) r α 0.4 d=5 0.2

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

For large R, we can show R d R d−1 R d−1 σ2(R)=2dφ A + B + o , h „ D « „ D « „ D « i where A and B are the “volume” and “surface-area” coefficients:

A = S(k = 0)=1+ ρ h(r)dr, B = −c(d) h(r)rdr, ZRd ZRd D: microscopic length scale, φ: dimensionless density

Hyperuniform: A =0, B > 0

. – p. 11/50 INVERTED CRITICAL PHENOMENA h(r) can be divided into direct correlations, via function c(r), and indirect correlations: h˜(k) c˜(k)= 1+ ρh˜(k)

. – p. 12/50 INVERTED CRITICAL PHENOMENA h(r) can be divided into direct correlations, via function c(r), and indirect correlations: h˜(k) c˜(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 INVERTED CRITICAL PHENOMENA h(r) can be divided into direct correlations, via function c(r), and indirect correlations: h˜(k) c˜(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, 1 1 c(r) ∼ − (r → ∞), c(k) ∼ − (k → 0), rd−2+η k2−η 1 h(r) ∼ − (r → ∞), S(k) ∼ k2−η (k → 0), rd+2−η 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) 1 ∼ −c(r) ∼ (r → ∞) d−2+η kB T 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 STATES We showed d d N 2 d R d R 1 σ (R)=2 φ 1 − 2 φ + α(rij ; R) „ D « „ D « N h Xi= j i where α(r; R) can be viewed as a repulsive pair potential:

1 Spherical window of radius R 0.8

0.6 d=1 (r;R) α 0.4

d=5 0.2

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

. – p. 13/50 SINGLE-CONFIGURATION FORMULATION & GROUND STATES We showed d d N 2 d R d R 1 σ (R)=2 φ 1 − 2 φ + α(rij ; R) „ D « „ D « N h Xi= j i where α(r; R) can be viewed as a repulsive pair potential:

1 Spherical window of radius R 0.8

0.6 d=1 (r;R) α 0.4

d=5 0.2

0 0 0.2 0.4 0.6 0.8 1 r/(2R) Finding global minimum of σ2(R) equivalent to finding ground state.

. – p. 13/50 SINGLE-CONFIGURATION FORMULATION & GROUND STATES We showed d d N 2 d R d R 1 σ (R)=2 φ 1 − 2 φ + α(rij ; R) „ D « „ D « N h Xi= j i where α(r; R) can be viewed as a repulsive pair potential:

1 Spherical window of radius R 0.8

0.6 d=1 (r;R) α 0.4

d=5 0.2

0 0 0.2 0.4 0.6 0.8 1 r/(2R) Finding global minimum of σ2(R) equivalent to finding ground state. For large R, in the special case of hyperuniform systems, R d−1 R d−3 σ2(R) = Λ(R) + O „ D « „ D «

Triangular Lattice (Average value=0.507826)

1

0.8

0.6 (R)

Λ 0.4

0.2

0 100 110 120 130 R/D . – p. 13/50 Hyperuniformity and Number Theory

Averaging fluctuating quantity Λ(R) gives coefficient of interest: 1 L Λ = lim Λ(R)dR L→∞ L 0

. – p. 14/50 Hyperuniformity and Number Theory

Averaging fluctuating quantity Λ(R) gives coefficient of interest: 1 L Λ = lim Λ(R)dR L→∞ L 0 We showed that for a lattice 2πR d 1 σ2(R) = [J (qR)]2, Λ = 2dπd−1 . q d/2 |q|d+1 q=0 q=0

Epstein zeta function for a lattice is defined by 1 Z(s) = , Re s > d/2. |q|2s q=0 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, quasicrystal and 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, quasicrystal and disordered two-dimensional hyperuniform point patterns.

2D Pattern Λ/φ1/2

Triangular Lattice 0.508347

Square Lattice 0.516401

Honeycomb Lattice 0.567026

Kagome´ 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: 3D Contrary 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: 3D Contrary 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 2 2 arrivals in Cuernavaca): g2(r)=1 − sin (πr)/(πr) (Dyson, 1970; Montgomery, 1973; Krbalek` & Seba,˘ 2000).

1.5

1 (r) 2 g

0.5

0 0 1 2 3 4 5 r

. – p. 18/50 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 2 2 arrivals in Cuernavaca): g2(r)=1 − sin (πr)/(πr) (Dyson, 1970; Montgomery, 1973; Krbalek` & Seba,˘ 2000).

1.5

1 (r) 2 g

0.5

0 0 1 2 3 4 5 r Constructing and/or identifying homogeneous, isotropic hyperuniform patterns for d ≥ 2 is more challenging.

. – p. 18/50 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 2 2 arrivals in Cuernavaca): g2(r)=1 − sin (πr)/(πr) (Dyson, 1970; Montgomery, 1973; Krbalek` & Seba,˘ 2000).

1.5

1 (r) 2 g

0.5

0 0 1 2 3 4 5 r Constructing and/or identifying homogeneous, isotropic hyperuniform 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 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)

Natural Disordered Hyperuniform Systems

Avian Photoreceptor Cells (nonequilibrium states)

. – p. 19/50 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)

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 Fermions One can map random Hermitian matrices (GUE), fermionic gases, and zeros of the Riemann zeta function to a unique hyperuniform point process on R.

. – p. 20/50 Hyperuniformity and Spin-Polarized Free Fermions One can map random Hermitian matrices (GUE), fermionic gases, 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.

1.2 1.2

1 1 d=1 d=3 0.8 0.8 d=3

(r) 0.6 0.6 2 S(k) g

0.4 0.4 d=1

0.2 0.2

0 0 0 0.5 1 1.5 2 0 2 4 6 8 10 r k

2Γ(1 + d/2) cos2 (rK − π(d + 1)/4) g2(r)=1 − (r → ∞) K πd/2+1 rd+1 c(d) S(k)= k + O(k3) (k → 0) (K : Fermi sphere radius) 2K

Torquato, Zachary & Scardicchio, J. Stat. Mech., 2008 Scardicchio, Zachary & Torquato, PRE, 2009

. – p. 20/50 Hyperuniformity and Jammed Packings Conjecture: All strictly jammed saturated sphere packings are hyperuniform (Torquato & Stillinger, 2003).

. – p. 21/50 Hyperuniformity and Jammed Packings Conjecture: All strictly jammed saturated sphere packings are hyperuniform (Torquato & Stillinger, 2003).

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). 5 0.04

4 Linear fit

0.02 3

S(k) 0 0 0.2 0.4 0.6 2

1 Data

0 0 0.5 1 1.5 2 kD/2π This is to be contrasted with the hard-sphere fluid with correlations that decay exponentially fast.

. – p. 21/50 Hyperuniformity and Jammed Packings Conjecture: All strictly jammed saturated sphere packings are hyperuniform (Torquato & Stillinger, 2003).

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). 5 0.04

4 Linear fit

0.02 3

S(k) 0 0 0.2 0.4 0.6 2

1 Data

0 0 0.5 1 1.5 2 kD/2π 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 liquid quench glass cooling

Volume crystal

Temperature

Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle.

. – p. 22/50 Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle.

super− cooled liquid liquid rapid quench glass very slow cooling

Volume crystal

glass freezing transition point (Tg) (Tf) Temperature

Typically, ground states are periodic with high crystallographic symmetries.

. – p. 22/50 Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle.

super− cooled liquid liquid rapid quench glass very slow cooling

Volume crystal

glass freezing transition point (Tg) (Tf) Temperature

Typically, ground states are periodic with high crystallographic symmetries.

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 N • Consider a system of N particles with configuration r ≡ r1, r2,... in a fundamental region Ω (under periodic boundary conditions) at position r. The complex collective density variable is defined by N ρ(k)= exp(ik rj ) jX=1

. – p. 23/50 Creation of Disordered Hyperuniform Ground States

Uche, Stillinger & Torquato, Phys. Rev. E 2004 Batten, Stillinger & Torquato, Phys. Rev. E 2008

Collective-Coordinate Simulations N • Consider a system of N particles with configuration r ≡ r1, r2,... in a fundamental region Ω (under periodic boundary conditions) at position r. The complex collective density variable is defined by N ρ(k)= exp(ik rj ) jX=1

• The real nonnegative structure factor is |ρ(k)|2 S(k)= N

. – p. 23/50 Creation of Disordered Hyperuniform Ground States

Uche, Stillinger & Torquato, Phys. Rev. E 2004 Batten, Stillinger & Torquato, Phys. Rev. E 2008

Collective-Coordinate Simulations N • Consider a system of N particles with configuration r ≡ r1, r2,... in a fundamental region Ω (under periodic boundary conditions) at position r. The complex collective density variable is defined by N ρ(k)= exp(ik rj ) jX=1

• The real nonnegative structure factor is |ρ(k)|2 S(k)= N • Consider stable pair potentials v(r) that are bounded with Fourier transform v˜(k). The total energy is

N ΦN (r ) = v(rij ) Xi

• Stealthy patterns can be tuned by varying the parameter χ: number of independently (real) constrained . – p. 23/50 degrees of freedom to the total number of degrees of freedom d(N − 1). Creation of Disordered Stealthy Ground States

Unconstrained Region

K

Exclusion Zone S=0

. – p. 24/50 Creation of Disordered Stealthy Ground States

Unconstrained Region

K

Exclusion Zone S=0

One class of stealthy potentials involves the following power-law form: m v˜(k)= v0(1 − k/K) Θ(K − k), where n is any whole number. The special case n =0 is just the simple step function. 1.5 d=3 m=0 m=0 m=2 m=2 m=4 0.01 m=4 1 v(r) v(k)~ 0.5

0 0 10 20 0 0.5 1 1.5 r k In the large-system (thermodynamic) limit with m =0 and m =4, we have the following large-r asymptotic behavior, respectively: cos(r) v(r) ∼ (m = 0) r2 1 v(r) ∼ (m = 4) r4 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 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.

For 0 ≤ χ< 0.5, the stealthy ground states are degenerate, disordered and isotropic. 1.5

1 S(k) 0.5

0 0 1 2 3 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 independently.

. – p. 25/50 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.

For 0 ≤ χ< 0.5, the stealthy ground states are degenerate, disordered and isotropic. 1.5

1 S(k) 0.5

0 0 1 2 3 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 independently.

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 States Two Dimensions

(a) χ= 0.04167 (b) χ = 0.41071

Three Dimensions

2

(r) 1 2 g

0 0 0.1 0.2 0.3 0.4 r / Lx

. – p. 26/50 Creation of Disordered Stealthy Ground States Two Dimensions

(a) χ= 0.04167 (b) χ = 0.41071

Three Dimensions

2

(r) 1 2 g

0 0 0.1 0.2 0.3 0.4 r / Lx Increasing χ induces strong short-range ordering, and eventually the system runs out of degrees of freedom that can be constrained and crystallizes.

Maximum χ S(k)

0 1 2 3 k/K . – p. 26/50 Ensemble Theory of Disordered Ground States Torquato, Zhang & Stillinger, Phys. Rev. X, 2015 Nontrivial: Dimensionality of the configuration space depends on the number density ρ (or χ) and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure. Which ensemble? How are entropically favored states determined?

Derived general exact relations for thermodynamic properties 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 Ensemble Theory of Disordered Ground States Torquato, Zhang & Stillinger, Phys. Rev. X, 2015 Nontrivial: Dimensionality of the configuration space depends on the number density ρ (or χ) and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure. Which ensemble? How are entropically favored states determined?

Derived general exact relations for thermodynamic properties that apply to any ground-state ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise. From previous considerations, we that an important contribution 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.

1.5 1.5

1 1 (r) 2 S(k) g 0.5 0.5

0 0 0 1 2 3 0 1 2 3 k 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 Ensemble Theory of Disordered Ground States Torquato, Zhang & Stillinger, Phys. Rev. X, 2015 Nontrivial: Dimensionality of the configuration space depends on the number density ρ (or χ) and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure. Which ensemble? How are entropically favored states determined?

Derived general exact relations for thermodynamic properties that apply to any ground-state ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise. From previous considerations, we that an important contribution 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.

1.5 1.5

1 1 (r) 2 S(k) g 0.5 0.5

0 0 0 1 2 3 0 1 2 3 k 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 Space Let 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, C(r) H(r)= . 1 − (2π)d ηC(r) This mapping enables us to exploit the well-developed accurate theories of standard Gibbsian disordered hard spheres in direct space.

1.5 1.5 1.5 d=3, χ=0.05 d=3, χ=0.1 d=3, χ=0.143 1 1 1 S(k) S(k) Theory S(k) Theory Theory 0.5 Simulation 0.5 Simulation 0.5 Simulation

00 1 2 3 4 00 1 2 3 4 00 1 2 3 4 k k k

1.5 1.5 1.5 χ=0.05 χ=0.1 χ=0.143 1 1 1 (r) d=1, Simulation (r) d=1, Simulation (r) d=1, Simulation 2 d=2, Simulation 2 d=2, Simulation 2 d=2, Simulation g d=3, Simulation g d=3, Simulation g d=3, Simulation 0.5 d=1, Theory 0.5 d=1, Theory 0.5 d=1, Theory d=2, Theory d=2, Theory d=2, Theory d=3, Theory d=3, Theory d=3, Theory 00 5 10 00 5 10 00 5 10 r r r . – 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 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.

Have used disordered, isotropic “stealthy” ground-state configurations to design photonic materials with large complete (both polarizations and all directions) band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

. – p. 29/50 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.

Have used disordered, isotropic “stealthy” ground-state configurations to design photonic materials with large complete (both polarizations and all directions) band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

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 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.

Have used disordered, isotropic “stealthy” ground-state configurations to design photonic materials with large complete (both polarizations and all directions) band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

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 Photoreceptors Optimal 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 Photoreceptors Optimal 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 Photoreceptors Disordered 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 Hyperuniform Highly sensitive transmission X-ray scattering measurements performed at Argonne on amorphous-silicon (a-Si) samples reveals that they are nearly hyperuniform with S(0) = 0.0075. Long, Roorda, Hejna, Torquato, and Steinhardt (2013)

This is significantly below the putative lower bound recently suggested by de Graff and Thorpe (2009) but consistent with the recently proposed nearly hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).

2.0 a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX] 1.5

S(k) 1.0

0.5

0.0 0 5 10 15 20 -1 k(A )

. – p. 33/50 Amorphous Silicon is Nearly Hyperuniform Highly sensitive transmission X-ray scattering measurements performed at Argonne on amorphous-silicon (a-Si) samples reveals that they are nearly hyperuniform with S(0) = 0.0075. Long, Roorda, Hejna, Torquato, and Steinhardt (2013)

This is significantly below the putative lower bound recently suggested by de Graff and Thorpe (2009) but consistent with the recently proposed nearly hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).

2.0 a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX] 1.5

S(k) 1.0

0.5

0.0 0 5 10 15 20 -1 k(A ) 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 Scales Important question in glass physics: Do growing relaxation times under supercooling have accompanying growing structural length scales? Lubchenko

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

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

. – p. 34/50 Structural Glasses and Growing Length Scales Important question in glass physics: Do growing relaxation times under supercooling have accompanying growing structural length scales? Lubchenko

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

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

We studied glass-forming liquid models that support an alternative 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. 8

7 c ξ 6

5 Length scale, 4

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

. – p. 34/50 Structural Glasses and Growing Length Scales Important question in glass physics: Do growing relaxation times under supercooling have accompanying growing structural length scales? Lubchenko

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

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

We studied glass-forming liquid models that support an alternative 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. 8

7 c ξ 6

5 Length scale, 4

3 0 1 2 3 4 5 Dimensionless temperature, T / Tg The degree of deviation from thermal equilibrium is determined from a nonequilibrium index S(k = 0) X = − 1, ρkBT κT which increases upon supercooling. Marcotte, Stillinger & Torquato (2013) . – p. 34/50 CONCLUSIONS Disordered 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 disordered 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 CONCLUSIONS Disordered 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 disordered 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. Collaborators Robert 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 structural information.

For systems in equilibrium at number density ρ,

N 2−N2 ρkBT κT = = S(k = 0)=1+ ρ h(r)dr N Rd

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 ρkT κT = S(k = 0). S(k = 0) X = − 1: Nonequilibrium index ρkBT κT

. – p. 37/50 DENSITY FLUCTUATIONS

Density fluctuations of many-particle systems are of great fundamental interest, containing important thermodynamic and structural information.

For systems in equilibrium at number density ρ,

N 2−N2 ρkBT κT = = S(k = 0)=1+ ρ h(r)dr N Rd

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 ρkT κT = S(k = 0). S(k = 0) X = − 1: Nonequilibrium index ρkBT κT Large-scale structure of the universe (Peebles 1993) Probe structure and collective motions in vibrated granular 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 States g2-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 Hypothesized Disordered Hyperuniform States g2-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.

Step-function g2

2

1.5 (r) 2

g 1

0.5

0 0 1 2 3 4 5 r Asymptotic large-r behavior of c(r) is given by the infinite-space Green’s function for the d-dimensional Laplace equation: −6r, d =1, 8 c(r)= 4ln(r), d =2, > < 2(d+2) 1 − d(d−2) d−2 , d ≥ 3. > r :>

. – p. 39/50 Hypothesized Disordered Hyperuniform States g2-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.

Step-function g2

2

1.5 (r) 2

g 1

0.5

0 0 1 2 3 4 5 r Asymptotic large-r behavior of c(r) is given by the infinite-space Green’s function for the d-dimensional Laplace equation: −6r, d =1, 8 c(r)= 4ln(r), d =2, > < 2(d+2) 1 − d(d−2) d−2 , d ≥ 3. > r Step + delta function g2 :>

2

1.5 (r) 2

g 1

0.5

0 0 1 2 3 4 5 r . – p. 39/50 Pseudo-Hard Spheres in Fourier Space Let us define H˜ (k) ≡ ρh˜(k)= hHS(r = k)

There is an Ornstein-Zernike integral equation that defines the appropriate direct 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, C(r) H(r)= . 1 − (2π)d ηC(r) This mapping enables us to exploit the well-developed accurate 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: 1 ρh(r) ∼ − J (r) (χ → 0) rd/2 d/2 1 ρh(r) ∼ − cos(r − (d + 1)π/4) (r → +∞) r(d+1)/2

. – p. 40/50 Stealthy Disordered Ground State Configurations

Direct Space Reciprocal Space S(k)

Compression Dilation

. – p. 41/50 General Scaling Behaviors

Hyperuniform particle distributions possess structure factors with a small-wavenumber scaling

α S(k) ∼ k , α > 0, including the special case α = +∞ for periodic crystals. 2 Hence, number variance σ (R) increases for large R asymptotically as (Zachary and Torquato, 2011)

Rd−1 ln R, α =1 2 σ (R) ∼ Rd−α, α < 1 (R → +∞).  Rd−1, α > 1

Until recently, all known hyperuniform configurations pertained to α ≥ 1.

. – p. 42/50 Targeted Spectra S ∼ kα

1/2 k

k S(k) 2 k

4 k

0 0 k K

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

. – p. 43/50 Targeted Spectra S ∼ 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 Spectra S ∼ kα with α < 1 Zachary & Torquato (2011)

(a) (b)

Figure 2: Both configurations exhibit strong local clustering of points 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 Asymptoticbehavior γ S−1(0) ∼ (1 − φ )−γ (φ → φ−) φc c γ ′ S−1(0) ∼ ( φ − 1)−γ′ (φ → φ+) φc c ν ξ ∼ (1 − φ )−ν (φ → φ−) φc c ν ′ ξ ∼ ( φ − 1)−ν′ (φ → φ+) φc c 2−d−η η c(r) ∼ r (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 States These 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 photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

. – p. 48/50 Stealthy Hyperuniform Ground and Excited States These 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 photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

Disordered Hyperuniform Biological Systems

. – p. 48/50 Stealthy Hyperuniform Ground and Excited States These 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 photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (2009)

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 r − d 1 Expected number Z(r)= ρs1(1) 0 x g2(x)dx: 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 r − d 1 Expected number Z(r)= ρs1(1) 0 x g2(x)dx: of points contained in a spherical region of radius r.

Poisson Point Distribution in R3 Maximally Random Jammed State in R3

5 5

4 4

3 3 (r) 2 (r) 2 g g 2 2

1 1

0 0 0 1 2 3 0 1 2 3 r r . – p. 49/50 Diffraction Pattern for Poisson Point Distribution (Ideal Gas)

Ground states are highly degenerate.

2

1.8 Thermodynamic limit

1.6

1.4

1.2 S(k)

1

0.8

0.6

0 2 4 6 8 10 k . – p. 50/50