Derivation of Pekar s polaron from a microscopic model of quantum crystal

Derivation of Pekar s polaron from a microscopic model of quantum crystal Mathieu LEWIN Mathieu.Lewin@math.cnrs.fr (CNRS & University of Cergy-Pontoise) joint work with Nicolas Rougerie (Grenoble, France) EMS Week End Bilbao Oct. 9, 2011 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 1 / 15

What is an (N )polaron? N electrons in a polar crystal can overcome Coulomb repulsion + dispersion of kinetic energy & form bound states using lattice distortions induced by own charge Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 2 / 15

Pekar s polaron Pekar [Pek54]: replace crystal by linear continuous polarizable medium, with dielectric constant ε M > 1 [ D ext = ε M E ] possible to calculate effective field felt by the electrons Case of 1 electron: E ε M (1) = inf {ˆ ψ(x) 2 dx 1 ε M R 3 2 1 ˆ ˆR3 ψ(x) 2 ψ(y) 2 dx dy, R 3 x y ˆ } ψ H 1 (R 3 ), ψ 2 = 1 R 3 Lieb [Lie77]: existence + uniqueness (up to translations) ε M > 1 ( (1 ε M 1 ) ψ 2 x 1) ψ = λ ψ [= Choquard-Pekar, Schrödinger-Newton] [Pek54] Pekar, Untersuchungen fiber die Elektronen Theorie der Kristalle (1954). [Lie77] Lieb, Studies in Applied Mathematics 57 (1977). Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 3 / 15

Pekar s multi-polarons Case of N electrons: {ˆ ˆ ( N E ε M (N) = inf dx 1 dx N R 3 R 3 xj Ψ(x 1,...,x N ) 2 + 1 k<l N 1 x k x l Ψ(x 1,...,x N ) 2 j=1 ) 1 ε M 1 2 ˆ ˆ with ρ Ψ (x) := N dx 2 dx N Ψ(x,x 2,...,x N ) 2 R 3 R 3 ˆ R 3 ˆ ρ Ψ (x)ρ Ψ (y) R x y 3 dxdy, } Ψ H 1 ((R 3 ) N ), Ψ antisymmetric, R 3N Ψ 2 = 1 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 4 / 15

Pekar s multi-polarons Case of N electrons: {ˆ ˆ ( N E ε M (N) = inf dx 1 dx N R 3 R 3 xj Ψ(x 1,...,x N ) 2 + 1 k<l N 1 x k x l Ψ(x 1,...,x N ) 2 j=1 ) 1 ε M 1 2 ˆ ˆ with ρ Ψ (x) := N dx 2 dx N Ψ(x,x 2,...,x N ) 2 R 3 R 3 ˆ R 3 ˆ ρ Ψ (x)ρ Ψ (y) R x y 3 dxdy, } Ψ H 1 ((R 3 ) N ), Ψ antisymmetric, R 3N Ψ 2 = 1 Theorem (Existence of N-polarons [FraLieSeiTho-11,Lew11]) α > 1 s.t. E ε M(N) has no minimizer for ε M α and N 2 N 2, β(n) s.t. E ε M(N) has one minimizer for ε M > β(n) [FraLieSeiTho-11] Frank, Lieb, Seiringer, Thomas. Publ. Math. IHES 113 (2011). [Lew11] M.L. J. Func. Anal. 260 (2011). Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 4 / 15

Macroscopic limit Intuitive physical picture: Pekar valid if the polaron lives at much larger scale than typical size of crystal Goal: show that Pekar s model emerges from microscopic model of quantum crystal, with some ε M depending on this microscopic structure introduce a parameter m = micro macro 1 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 5 / 15

Crystal fixed Z 3 -periodic arrangement of nuclei (isotropic) optimized quantum electrons Total Coulomb potential of periodic crystal: V per = 4π ( ρ per µ nuc ) per, with [0,1) ρ 3 per = [0,1) µ nuc 3 per Reaction of electrons in presence of external density ν ext (x): ρ(x) = ρ per (x)+δρ(x) where δρ solves {ˆ ˆ ν ext (x)δρ(y) [ ] } F crys [ν ext ] = inf dx dy +F crys δρ δρ ρ 0 e R 3 R 3 x y 0 F crys [ δρ ] = energy cost to displace the electrons ρper ρ per +δρ We use Hartree model of [CanDelLew08] [CanDelLew08] Cancès, Deleurence & M.L. Commun. Math. Phys. 281 (2008) Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 6 / 15

Coupled system In macroscopic variables (for one electron): E m (1) = inf {ˆ R 3 ( ψ 2 + 1 ( ) m V per ψ 2) + 1 m m F [ crys m 3 ψ(m ) 2], ˆ } ψ H 1 (R 3 ), ψ 2 = 1 R 3 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 7 / 15

Coupled system In macroscopic variables (for one electron): E m (1) = inf {ˆ R 3 ( ψ 2 + 1 ( ) m V per ψ 2) + 1 m m F [ crys m 3 ψ(m ) 2], ˆ } ψ H 1 (R 3 ), ψ 2 = 1 R 3 Rmk 1. ( m 1 V per ( /m) ) = m 3 4π ( ρ per µ nuc per) ( /m) Rmk 2. In microscopic variables, with ψ = m 3/2 ψ(m ), (ˆ ( 1 E m (1) = m 1 inf ψ m ψ 2 +V per ψ )+F 2 crys [ ψ 2]) Rmk 3. Similar definition for E m (N) (N-polaron). R 3 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 7 / 15

The fast microscopic oscillations Definition Let u m per be the unique periodic > 0 solution to ( +m 1 V per ( /m) ) u m per( /m) = E m per u m per( /m) ( +mv per )u m per = m 2 E m per u m per Perturbation theory: u m per 1+mf per and lim m 0 E m per = E per where ˆ f per = 2V per, E per = V per f per = 1 ˆ f per 2 [0,1) 3 2 [0,1) 3 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 8 / 15

Main result Theorem (Derivation of Pekar s polarons [LewRou11]) Assume F crys is given by the Hartree model of [CanDelLew08]. There exists a constant ε M > 1 depending only on µ nuc per, such that: lim m 0 E m (N) = NE per +E ε M (N) Let (Ψ m ) be any sequence of approximate minimizers for E m (N), and define (Φ m ) by N ( Ψ m (x 1,...,x N ) = uper m xj ) Φ m (x 1,...,x N ). m j=1 Then (Φ m ) is a minimizing sequence for the Pekar problem E ε M(N). If N = 1, or N > 1 and ε M > β(n), then (Φ m ) converges strongly in H 1 to a minimizer of E ε M(N), up to a subsequence and up to translations. M.L. & Rougerie, preprint arxiv:1108.5931 Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 9 / 15

A multiscale structure Polaron (1 electron): ψ m ( ( x )) 1+mf per ψ ε M m Pek (x) m m Rmk. ψ m (x) ψ ε M Pek (x) but ψ m (x) ψ ε M Pek (x)+ψ ε ( M Pek (x) f x ) per m Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 10 / 15

Strategy of proof 1 Easy decoupling of fast oscillations Insert Ψ m = Φ m uper m ( /m) in energy Use equation/properties of uper m energy decouples 2 Get estimates on (Φ m ) Prove exact upper bound Some lower bound (Φ m ) is bounded in H 1 (R 3 ) 3 Approximate nonlinear crystal by its linear response m 3/2 Φ m ( /m) L 12/5 0 enough to justify 1st order perturbation theory F crys replaced by simpler (explicit) quadratic form 4 Do macroscopic limit on linear response Generalize results of [CanLew10], where ε M was already defined for sequences of the type m 3/2 Φ( /m) Use bubble decomposition to treat m-dependent sequence Φ m [CanLew10] Cancès & M.L. Arch. Rat. Mech. Anal. 197 (2010) Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 11 / 15

Hartree Theory for crystals I State of the electrons: an orthogonal projection (=density matrix) γ = γ = γ 2 acting on L 2 (R 3 ), with tr(γ) =nb of electrons Perfect crystal: γ per solution of nonlinear eq. ε F ε F ( ) γ per = 1 (,εf )( +V per ) V per = 4π ( ρ γper µ nuc ) per ˆ ( ργper µ nuc ) per =ˆ V per = 0 [0,1) 3 [0,1) 3 Insulator / semi-conductor Conductor N = 2 N = 3 We assume the crystal is an insulator [CatBriLio01] Catto, Le Bris, Lions, Ann. I. Henri Poincaré 18 (2001). [CanDelLew08] Cancès, Deleurence, M.L., Commun. Math. Phys. 281 (2008). Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 12 / 15

Hartree Theory for crystals II Perturbed crystal in external density ν ext : ( ) { γ = 1(,εF )( +V γ ) V γ = 4π ( ρ γ µ nuc per +ν ) ext Theorem (Existence [CanDelLew08]) Assume the periodic crystal is an insulator and fix ε F band gap. For every ν ext L 6/5 (R 3 ) with ν ext L 6/5 small enough, Eq. ( ) admits a unique solution γ, such that Q := γ γ per is Hilbert-Schmidt, and ρ Q L 2 (R 3 ), R 3 R 3 ρ Q (x)ρ Q (y) x y 1 dx dy <. ( [ ] F crys νext := inf tr( +V per ε F )Q + 1 ˆ ˆ ρ Q (x)ρ Q (y) Q 2 R 3 R x y 3 ˆ ˆ + R 3 dxdy ν ext(x)ρ Q (y) R x y 3 ) dxdy Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 13 / 15

Linear response of crystals in Hartree Theory I Assuming ν ext is small (in L 6/5 (R 3 )), we can expand 1 (,εf )( +Vper +(ν ext +δρ) x 1) 1 (,εf )( +V per ) = 1 1 2iπ +V per z (ν ext +δρ) x 1 1 dz + +V per z C C σ( +V per ) Hence δρ = ρ γ ρ per = L ( ν ext +δρ ) ( +o L 6/5 νext +δρ ) ε M L can be explicitly computed, it holds L 0 δρ L(1+L) 1 ν ext Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 14 / 15

Linear response of crystals in Hartree Theory II Theorem (Macroscopic dielectric tensor [CanLew10,LewRou11]) ε M > 1 s.t., for any (ψ m ) bd in H 1 (R 3 ), we have with ν m = m 3 ψ m (m ) 2 ( ) 1 νm(x)((1+l) 1 ν m (y) ) m x y dx dy ε 1 ψ m(x) 2 ψ m(y) 2 M x y dx dy = 0 lim m 0 Rmk. For anisotropic crystals, ε M is a 3 3 symmetric real matrix s.t. ε M > 1. Everything works similarly. Conclusion Couple N quantum particles to a quantum crystal living at a smaller scale, described by Hartree s theory For the ground state, Pekar s model is obtained in macroscopic limit There are also explicit fast oscillations, which contribute to the energy at the same order Extension to time-dependent systems Mathieu LEWIN (CNRS / Cergy) Derivation of Pekar s polarons Bilbao, 09/10/11 15 / 15