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 d � Energy per point of a Bravais lattice L = Z u i is given by i =1 � | p | 2 � � E f [ L ] := f < + ∞ p ∈ L \{ 0 } Let V (or A ) be the volume (area) of L , i.e. the volume of its unit cell. Problems Minimizing L �→ E f [ 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 1 � If f ( r ) = r − s / 2 , s > d , then ζ L ( s ) := | p | s . p ∈ L \{ 0 } Theta function e − πα | p | 2 . � If f ( r ) = e − απ r , α > 0, then θ L ( α ) := p ∈ L Lennard-Jones energy If f ( r ) = a 1 r x 1 − a 2 r x 2 , a i > 0, x 2 > x 1 > d / 2, then E f [ L ] = a 2 ζ L (2 x 2 ) − a 1 ζ L (2 x 1 ) . 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
Epstein zeta function and theta function: old results 1 New results for Lennard-Jones energy 2 Theta function in 3d: Local minimality of FCC/BCC lattices 3 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 1 � L �→ ζ L ( s ) := | p | s p ∈ L \{ 0 } 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 �→ ζ L ( s ), for any fixed volume. [Sarnak-Str¨ ombergsson ’06] - Conjecture for L �→ ζ L ( 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 e − πα | p | 2 � L �→ θ L ( α ) := p ∈ L is a triangular lattice , for any fixed area. [Sarnak-Str¨ ombergsson ’06] - Conjecture for L �→ θ L ( 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 Λ A is the unique minimizer of L �→ E f [ 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 e − a | p | e −| p | 2 α , 0 < α ≤ 1; � � � K 0 ( | p | ); | p | , a > 0 . p ∈ L p ∈ L \{ 0 } p ∈ L \{ 0 } [Cohn-Kumar ’06] - Conjecture If f is completely monotone, then Λ A is the unique minimizer of L �→ E f [ 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 Λ A ) 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 A 1 , A 2 such that Λ A is not a minimizer of E f among all Bravais lattices of fixed area A ∈ ( A 1 , A 2 ). Remark: A 1 ≈ 2 . 3152307 and A 2 ≈ 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 + ) 2 and 1 < x 1 < x 2 , let For ( a 1 , a 2 ) ∈ ( R ∗ a , x ( r ) := a 2 r x 2 − a 1 f LJ and E f LJ a , x [ L ] = a 2 ζ L (2 x 2 ) − a 1 ζ L (2 x 1 ) . r x 1 Proposition [LB-Zhang ’14, LB ’15] ( High/low densities ) 1 � a 2 Γ( x 1 ) � x 2 − x 1 , then Λ A is the unique minimizer of E f LJ • If A ≤ π a 1 Γ( x 2 ) a , x among Bravais lattices of fixed area A . • Triangular lattice Λ A is a minimizer of E f LJ a , x among Bravais lattices of fixed area A if and only if 1 � a 2 ( ζ L (2 x 2 ) − ζ Λ 1 (2 x 2 )) � x 2 − x 1 , A ≤ inf a 1 ( ζ L (2 x 1 ) − ζ Λ 1 (2 x 1 )) | L | =1 , L � =Λ 1 i.e. if A is sufficiently large , then Λ A 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 − t Γ( t ) t . If h ( x 2 ) ≤ h ( x 1 ), then the minimizer L a , x of E f LJ a , x among all Bravais lattices is unique and triangular . Furthermore, its area is 1 � a 2 x 2 ζ Λ 1 (2 x 2 ) � x 2 − x 1 . | L a , x | = a 1 x 1 ζ Λ 1 (2 x 1 ) Remark: True for ( x 1 , x 2 ) ∈ { (1 . 5 , 2); (1 . 5 , 2 . 5); (1 . 5 , 3); (2 , 2 . 5); (2 , 3) } . Cannot be used for the classical Lennard-Jones potential ( x 1 , x 2 ) = (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 � √ √ � � � √ A x A A √ y , By lattice reduction, u 1 = y , 0 and u 2 = , , where y ( x , y ) ∈ R 2 ; 0 ≤ x ≤ 1 / 2 , y > 0 , x 2 + y 2 ≥ 1 � � ( x , y ) ∈ D := . Each ( x , y , A ) is associated with one Bravais lattice L = Z u 1 ⊕ Z u 2 . Theorem [LB ’16] - Local study of E f LJ a , x Given ( a , x ), there exist A 0 < A 1 < A 2 (explicit in terms of infinite sums) such that: if 0 < A < A 0 , then Λ A is a local minimizer ; if A > A 0 , then Λ A is a local maximizer ; √ A Z 2 is a local minimizer ; if A 1 < A < A 2 , then √ A Z 2 is a saddle point . if A �∈ [ A 1 , A 2 ], then 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 a , x ( r 2 ) = 1 r 12 − 2 Let f LJ r 6 , then there exists A 3 such that for any A > A 3 , the minimizer of E f LJ a , x is a rectangular lattice , i.e. ( x , y ) = (0 , y A ) ∈ D . Furthermore, A → + ∞ y A = + ∞ . lim 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 a , x ( r 2 ) = 1 r 12 − 2 We consider the classic case f LJ r 6 . � ζ L (12) − ζ Λ 1 (12) � 1 / 3 A BZ = inf ≈ 1 . 138, A 1 ≈ 1 . 143, A 2 ≈ 1 . 268. 2( ζ L (6) − ζ Λ 1 (6)) | L | =1 L � =Λ1 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 = Z u 1 ⊕ Z u 2 ⊕ Z u 3 of volume V is such that � 1 � x � y √ √ √ � √ u , v � √ u , vz u � u 1 = C √ u , 0 , 0 , u 2 = C √ u , 0 , u 3 = C √ u , √ v 2 where C = V 2 / 3 2 1 / 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 �→ E f [ L ]. Given ( a , x ), there exist V 0 < V 1 (explicit) such that if 0 < V < V 0 , then BCC and FCC are local minimizers of E f LJ a , x ; if V 0 < V < V 1 , then BCC and FCC are saddle points of E f LJ a , x ; if V > V 1 , then BCC and FCC are local maximizers of E f LJ a , x . Proof: following Ennola’s proof for L �→ ζ L ( s ). Laurent B´ etermin Review Lattice Energies 04/19/2017 16 / 20
Theta function in 3d: Local minimality of FCC/BCC lattices L �→ θ L ( α ) in 3d: minimality of BCC among BCO lattices Body-Centred-Orthorhombic (BCO) lattice L y , t = 1, y ≥ 1. Theorem [LB-Petrache ’16] - Optimality of BCC/FCC There exists α 0 such that, for any α > α 0 , y = 1 is not a minimizer of y �→ θ L y ( α ). If α ∈ { 0 . 001 k ; k ∈ N , 1 ≤ k ≤ 1000 } , then the BCC lattice is the unique minimizer of y �→ θ L y ( α ). Proof: Asymptotics of the energy + Computer assistant. Laurent B´ etermin Review Lattice Energies 04/19/2017 17 / 20
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