Minimization of lattice energies From old to new results in - - PowerPoint PPT Presentation

Minimization of lattice energies

From old to new results in dimensions 2 and 3

Laurent B´ etermin

Institute for Applied Mathematics, University of Heidelberg Joint works with P. Zhang and M. Petrache

OPCOP 2017 19th April 2017, CIEM, Castro Urdiales

Laurent B´ etermin Review Lattice Energies 04/19/2017 1 / 20

Introduction: minimization among Bravais lattices

Definitions Potential: f : (0, +∞) → R such that f (r) = O(r −η), η > d/2 Energy per point of a Bravais lattice L =

d

• i=1

Zui is given by Ef [L] :=

• p∈L\{0}

f

• |p|2

< +∞ Let V (or A) be the volume (area) of L, i.e. the volume of its unit cell. Problems Minimizing L → Ef [L] among all the Bravais lattices L with (or without) a fixed volume V . Motivation: Crystallization problems; Vortices in superconductors

Laurent B´ etermin Review Lattice Energies 04/19/2017 2 / 20

Main examples

Epstein zeta function If f (r) = r −s/2, s > d, then ζL(s) :=

• p∈L\{0}

1 |p|s . Theta function If f (r) = e−απr, α > 0, then θL(α) :=

• p∈L

e−πα|p|2. Lennard-Jones energy If f (r) = a1 r x1 − a2 r x2 , ai > 0, x2 > x1 > d/2, then Ef [L] = a2ζL(2x2) − a1ζL(2x1).

Laurent B´ etermin Review Lattice Energies 04/19/2017 3 / 20

Triangular lattice

Laurent B´ etermin Review Lattice Energies 04/19/2017 4 / 20

FCC and BCC lattices

Face-Centred-Cubic lattice Body-Centred-Cubic lattice

Laurent B´ etermin Review Lattice Energies 04/19/2017 5 / 20

1

Epstein zeta function and theta function: old results

2

New results for Lennard-Jones energy

3

Theta function in 3d: Local minimality of FCC/BCC lattices

Laurent B´ etermin Review Lattice Energies 04/19/2017 6 / 20

Epstein zeta function and theta function: old results

The Epstein zeta function in dimensions 2 and 3

The volume of lattices should be fixed. Rankin ’53, Cassels ’59, Diananda ’64, Ennola ’64: in 2d, for any s > 0, the minimizer of the Epstein zeta function L → ζL(s) :=

• p∈L\{0}

1 |p|s is a triangular lattice, for any fixed area; Ennola ’64: in 3d, for any s > 0, the FCC and the BCC lattices are local minimizers of L → ζL(s), for any fixed volume. [Sarnak-Str¨

• mbergsson ’06] - Conjecture for L → ζL(s), V = 1

For any s > 3/2, the FCC lattice is the unique minimizer; For any 0 < s < 3/2, the BCC lattice is the unique minimizer.

Laurent B´ etermin Review Lattice Energies 04/19/2017 7 / 20

Epstein zeta function and theta function: old results

The theta function in dimensions 2 and 3

The volume of lattices should be fixed. Montgomery ’88: in 2d, for any α > 0, the minimizer of the theta function L → θL(α) :=

• p∈L

e−πα|p|2 is a triangular lattice, for any fixed area. [Sarnak-Str¨

• mbergsson ’06] - Conjecture for L → θL(α), V = 1

For any α > 1, the FCC lattice is the unique minimizer; For any 0 < α < 1, the BCC lattice is the unique minimizer. Applications: Heat equation, Ginzburg-Landau Vortices, Bose-Einstein Condensates, Cryptography...

Laurent B´ etermin Review Lattice Energies 04/19/2017 8 / 20

Epstein zeta function and theta function: old results

Completely monotone functions and the triangular lattice

By Bernstein Theorem, we get: Proposition (Minimality at any fixed volume in 2d) Let f be a completely monotone function such that f (r) = O(r −p), p > 1, then for any A > 0, the triangular lattice ΛA is the unique minimizer of L → Ef [L] among Bravais lattices of fixed area A. Also true for long-range potentials (Ewald summation method). In particular, the triangular lattice is the minimizer of

• p∈L

e−|p|2α, 0 < α ≤ 1;

• p∈L\{0}

K0(|p|);

• p∈L\{0}

e−a|p| |p| , a > 0. [Cohn-Kumar ’06] - Conjecture If f is completely monotone, then ΛA is the unique minimizer of L → Ef [L] among periodic lattices of fixed area A.

Laurent B´ etermin Review Lattice Energies 04/19/2017 9 / 20

Epstein zeta function and theta function: old results

Convexity

Proposition [LB ’15] (Example of non optimality of ΛA) Let f be defined by f (r) = 14 r 2 − 40 r 3 + 35 r 4 , then f is strictly convex, strictly decreasing and strictly positive; there exist A1, A2 such that ΛA is not a minimizer of Ef among all Bravais lattices of fixed area A ∈ (A1, A2). Remark: A1 ≈ 2.3152307 and A2 ≈ 3.759353.

Laurent B´ etermin Review Lattice Energies 04/19/2017 10 / 20

New results for Lennard-Jones energy

Lennard-Jones in 2d: results about the global minimizer

For (a1, a2) ∈ (R∗

+)2 and 1 < x1 < x2, let

f LJ

a,x(r) := a2

r x2 − a1 r x1 and Ef LJ

a,x [L] = a2ζL(2x2) − a1ζL(2x1).

Proposition [LB-Zhang ’14, LB ’15] (High/low densities)

• If A ≤ π

a2Γ(x1) a1Γ(x2)

• 1

x2−x1 , then ΛA is the unique minimizer of Ef LJ a,x

among Bravais lattices of fixed area A.

• Triangular lattice ΛA is a minimizer of Ef LJ

a,x among Bravais lattices of

fixed area A if and only if A ≤ inf

|L|=1,L=Λ1

a2(ζL(2x2) − ζΛ1(2x2)) a1(ζL(2x1) − ζΛ1(2x1))

• 1

x2−x1 ,

i.e. if A is sufficiently large, then ΛA is NOT a minimizer.

Laurent B´ etermin Review Lattice Energies 04/19/2017 11 / 20

New results for Lennard-Jones energy

Lennard-Jones in 2d: results about the global minimizer

Theorem [LB ’15] (Global minimizer for LJ type potentials) Let h(t) := π−tΓ(t)t. If h(x2) ≤ h(x1), then the minimizer La,x of Ef LJ

a,x

among all Bravais lattices is unique and triangular. Furthermore, its area is |La,x| = a2x2ζΛ1(2x2) a1x1ζΛ1(2x1)

• 1

x2−x1 .

Remark: True for (x1, x2) ∈ {(1.5, 2); (1.5, 2.5); (1.5, 3); (2, 2.5); (2, 3)}. Cannot be used for the classical Lennard-Jones potential (x1, x2) = (3, 6).

Laurent B´ etermin Review Lattice Energies 04/19/2017 12 / 20

New results for Lennard-Jones energy

Lennard-Jones in 2d: local study

By lattice reduction, u1 = √ A y , 0

• and u2 =
• x

√ A y , √ A√y,

• , where

(x, y) ∈ D :=

• (x, y) ∈ R2; 0 ≤ x ≤ 1/2, y > 0, x2 + y 2 ≥ 1
• .

Each (x, y, A) is associated with one Bravais lattice L = Zu1 ⊕ Zu2. Theorem [LB ’16] - Local study of Ef LJ

a,x

Given (a, x), there exist A0 < A1 < A2 (explicit in terms of infinite sums) such that: if 0 < A < A0, then ΛA is a local minimizer; if A > A0, then ΛA is a local maximizer; if A1 < A < A2, then √ AZ2 is a local minimizer; if A ∈ [A1, A2], then √ AZ2 is a saddle point.

Laurent B´ etermin Review Lattice Energies 04/19/2017 13 / 20

New results for Lennard-Jones energy

Degeneracy as A → +∞

Theorem [LB ’16] - Minimizer for large A Let f LJ

a,x(r 2) = 1

r 12 − 2 r 6 , then there exists A3 such that for any A > A3, the minimizer of Ef LJ

a,x is a rectangular lattice, i.e.

(x, y) = (0, yA) ∈ D. Furthermore, lim

A→+∞ yA = +∞.

Thanks Doug!

Laurent B´ etermin Review Lattice Energies 04/19/2017 14 / 20

New results for Lennard-Jones energy

Lennard-Jones in 2d: numerical investigation

We consider the classic case f LJ

a,x(r 2) = 1

r 12 − 2 r 6 . ABZ = inf

|L|=1 L=Λ1

ζL(12) − ζΛ1(12) 2(ζL(6) − ζΛ1(6)) 1/3 ≈ 1.138, A1 ≈ 1.143, A2 ≈ 1.268.

Laurent B´ etermin Review Lattice Energies 04/19/2017 15 / 20

New results for Lennard-Jones energy

Lennard-Jones in 3d: local study

By lattice reduction, L = Zu1 ⊕ Zu2 ⊕ Zu3 of volume V is such that u1 = √ C 1 √u , 0, 0

• , u2 =

√ C x √u , v √u , 0

• , u3 =

√ C y √u , vz √u , u v √ 2

• where C = V 2/321/3. We have 5 parameters (u, v, x, y, z).

Theorem [LB ’16] - Local optimality of BCC and FCC For any f , BCC and FCC lattices are critical points of L → Ef [L]. Given (a, x), there exist V0 < V1 (explicit) such that if 0 < V < V0, then BCC and FCC are local minimizers of Ef LJ

a,x ;

if V0 < V < V1, then BCC and FCC are saddle points of Ef LJ

a,x ;

if V > V1, then BCC and FCC are local maximizers of Ef LJ

a,x .

Proof: following Ennola’s proof for L → ζL(s).

Laurent B´ etermin Review Lattice Energies 04/19/2017 16 / 20

Theta function in 3d: Local minimality of FCC/BCC lattices

L → θL(α) in 3d: minimality of BCC among BCO lattices

Body-Centred-Orthorhombic (BCO) lattice Ly, t = 1, y ≥ 1. Theorem [LB-Petrache ’16] - Optimality of BCC/FCC There exists α0 such that, for any α > α0, y = 1 is not a minimizer

• f y → θLy (α).

If α ∈ {0.001k; k ∈ N, 1 ≤ k ≤ 1000}, then the BCC lattice is the unique minimizer of y → θLy (α). Proof: Asymptotics of the energy + Computer assistant.

Laurent B´ etermin Review Lattice Energies 04/19/2017 17 / 20

Theta function in 3d: Local minimality of FCC/BCC lattices

Consequence for the local minimality of BCC/FCC

Using the previous result and [Baernstein ’97] - Optimality of the barycenter in the triangular lattice case For any α > 0, the minimizer of z → θΛ1+z(α) is, up to symmetry, the barycenter of a primitive triangle. we get Theorem [LB-Petrache ’16] - Local minimality for some α There exists α0 > 0 such that for any α > α0, the BCC lattice is not a local minimizer; for any 0 < α < 1/α0, the FCC lattice is not a local minimizer. If α ∈ {0.001k; k ∈ N, 1 ≤ k ≤ 1000}, then the BCC lattice is a local minimizer. The same holds for the FCC lattice if α ∈ {k; k ∈ N, 1 ≤ k ≤ 1000}.

Laurent B´ etermin Review Lattice Energies 04/19/2017 18 / 20

Theta function in 3d: Local minimality of FCC/BCC lattices

L → θL(α) in 3d: local minimality of FCC/BCC, V = 1

Using Ennola’s method, we get: Theorem [LB ’16] - Local minimality of BCC/FCC for α small/large There exists α1 < α2 such that for any 0 < α < α1 (resp. α > 1/α1), the FCC (resp. BCC) lattice is a saddle point; for any α > α2 (resp. 0 < α < 1/α2), the FCC (resp. BCC) lattice is a local minimizer.

Laurent B´ etermin Review Lattice Energies 04/19/2017 19 / 20

Theta function in 3d: Local minimality of FCC/BCC lattices

Thank you for your attention! Simon Beck, Brean Beach (UK) , 2014 Constructed with a rake and a compass...

Laurent B´ etermin Review Lattice Energies 04/19/2017 20 / 20