Mean field evolution of fermionic systems Marcello Porta University of Z¨ urich, Institute for Mathematics QMath13: Mathematical Results in Quantum Physics Georgia Tech, 8-11 October, 2016
Outline Outline • Introduction. • Results: 1 Derivation of the time-dependent Hartree-Fock equation, for pure and mixed states, with bounded interaction potentials. 2 Extension to Coulomb interactions. • Conclusions. Marcello Porta Mean field evolution of fermions October 8, 2016 1 / 12
Introduction Introduction Marcello Porta Mean field evolution of fermions October 8, 2016 1 / 12
Introduction Fermionic mean field regime • N interacting fermionic particles in R 3 , ψ N ∈ L 2 a ( R 3 N ). V ( x i − x j ) = pair interaction potential, V ext ( x i ) = confining potential. System confined in Λ ⊂ R 3 , | Λ | = O (1). Density = O ( N ). Marcello Porta Mean field evolution of fermions October 8, 2016 2 / 12
Introduction Fermionic mean field regime • N interacting fermionic particles in R 3 , ψ N ∈ L 2 a ( R 3 N ). V ( x i − x j ) = pair interaction potential, V ext ( x i ) = confining potential. System confined in Λ ⊂ R 3 , | Λ | = O (1). Density = O ( N ). • Mean field regime. V varies on scale O (1), and V → N − 1 / 3 V . In fact: E int = � ψ N , � N i<j V ( x i − x j ) ψ N � = O ( N 2 ) E kin = � ψ N , � N i =1 − ∆ x i ψ N � � N 5 / 3 (by Lieb-Thirring inequality) Marcello Porta Mean field evolution of fermions October 8, 2016 2 / 12
Introduction Fermionic mean field regime • N interacting fermionic particles in R 3 , ψ N ∈ L 2 a ( R 3 N ). V ( x i − x j ) = pair interaction potential, V ext ( x i ) = confining potential. System confined in Λ ⊂ R 3 , | Λ | = O (1). Density = O ( N ). • Mean field regime. V varies on scale O (1), and V → N − 1 / 3 V . In fact: E int = � ψ N , � N i<j V ( x i − x j ) ψ N � = O ( N 2 ) E kin = � ψ N , � N i =1 − ∆ x i ψ N � � N 5 / 3 (by Lieb-Thirring inequality) • Mean field Hamiltonian: N N � � � � H trap + N − 1 / 3 := − ∆ j + V ext ( x j ) V ( x i − x j ) N j =1 i<j Marcello Porta Mean field evolution of fermions October 8, 2016 2 / 12
Introduction Hartree-Fock theory • Hartree-Fock ground state energy: E N HF := inf ψ Slater � ψ Slater , H trap ψ Slater � N 1 √ ψ Slater ( x 1 , . . . , x N ) = det f i ( x j ) , � f i , f j � = δ ij . N ! • Setting ω N := N tr 2 ,...,N | ψ Slater �� ψ Slater | = � N i =1 | f i �� f i | , � ψ Slater , H trap ψ Slater � = tr( − ∆ + V ext ) ω N N � 1 V ( x − y )( ω N ( x ; x ) ω N ( y ; y ) − | ω N ( x ; y ) | 2 ) + 2 N 1 / 3 One expects that, as N → ∞ : � ψ, H trap ψ � E N N = E N GS := inf HF + smaller order terms , � ψ, ψ � ψ ∈ L 2 a ( R 3 N ) Proven for large atoms: Bach ’92, Graf-Solovej ’94 . Marcello Porta Mean field evolution of fermions October 8, 2016 3 / 12
Introduction Hartree-Fock theory • Hartree-Fock ground state energy: E N HF := inf ψ Slater � ψ Slater , H trap ψ Slater � N 1 √ ψ Slater ( x 1 , . . . , x N ) = det f i ( x j ) , � f i , f j � = δ ij . N ! • Setting ω N := N tr 2 ,...,N | ψ Slater �� ψ Slater | = � N i =1 | f i �� f i | , � ψ Slater , H trap ψ Slater � = tr( − ∆ + V ext ) ω N N � 1 V ( x − y )( ω N ( x ; x ) ω N ( y ; y ) − | ω N ( x ; y ) | 2 ) + 2 N 1 / 3 One expects that, as N → ∞ : � ψ, H trap ψ � E N N = E N GS := inf HF + smaller order terms , � ψ, ψ � ψ ∈ L 2 a ( R 3 N ) Proven for large atoms: Bach ’92, Graf-Solovej ’94 . • Next: Thomas-Fermi theory ( Lieb-Simon ’73, Fournais-Lewin-Solovej ’15 ). Marcello Porta Mean field evolution of fermions October 8, 2016 3 / 12
Introduction Fermionic mean-field dynamics • Suppose that V ext = 0 at t = 0. Dynamics: � N N � � � − ∆ x j + N − 1 / 3 V ( x i − x j ) i∂ t ψ N,τ = ψ N,τ j =1 i<j • E kin ∼ N 5 / 3 ⇒ velocity ∼ N 1 / 3 . Time scale: τ ∼ N − 1 / 3 . Marcello Porta Mean field evolution of fermions October 8, 2016 4 / 12
Introduction Fermionic mean-field dynamics • Suppose that V ext = 0 at t = 0. Dynamics: � N N � � � − ∆ x j + N − 1 / 3 V ( x i − x j ) i∂ t ψ N,τ = ψ N,τ j =1 i<j • E kin ∼ N 5 / 3 ⇒ velocity ∼ N 1 / 3 . Time scale: τ ∼ N − 1 / 3 . • Introducing the rescaled time t = N 1 / 3 τ : � N � N � � iN 1 / 3 ∂ t ψ N,t = − ∆ j + N − 1 / 3 V ( x i − x j ) ψ N,t j =1 i<j Marcello Porta Mean field evolution of fermions October 8, 2016 4 / 12
Introduction Fermionic mean-field dynamics • Suppose that V ext = 0 at t = 0. Dynamics: � N N � � � − ∆ x j + N − 1 / 3 V ( x i − x j ) i∂ t ψ N,τ = ψ N,τ j =1 i<j • E kin ∼ N 5 / 3 ⇒ velocity ∼ N 1 / 3 . Time scale: τ ∼ N − 1 / 3 . • Introducing the rescaled time t = N 1 / 3 τ : � N � N � � iN 1 / 3 ∂ t ψ N,t = − ∆ j + N − 1 / 3 V ( x i − x j ) ψ N,t j =1 i<j • Let ε = N − 1 / 3 . Multiplying LHS and RHS by ε 2 : � N � � N � − ε 2 ∆ j + N − 1 V ( x i − x j ) iε∂ t ψ N,t = ψ N,t j =1 i<j Mean-field limit coupled with a semiclassical scaling. Marcello Porta Mean field evolution of fermions October 8, 2016 4 / 12
Introduction Hartree-Fock and Vlasov dynamics • Let γ (1) N = N tr 2 ,...,N | ψ N �� ψ N | ≃ ω N , with ω N = ω 2 N (Slater det.). • Expect: for N ≫ 1, γ (1) N,t ≃ ω N,t = solution of time dep. HF equation: iε∂ t ω N,t = [ − ε 2 ∆ + V ∗ ρ t − X t , ω N,t ] where ρ t ( x ) = N − 1 ω N,t ( x ; x ) and X t ( x ; y ) = N − 1 V ( x − y ) ω N,t ( x ; y ). Marcello Porta Mean field evolution of fermions October 8, 2016 5 / 12
Introduction Hartree-Fock and Vlasov dynamics • Let γ (1) N = N tr 2 ,...,N | ψ N �� ψ N | ≃ ω N , with ω N = ω 2 N (Slater det.). • Expect: for N ≫ 1, γ (1) N,t ≃ ω N,t = solution of time dep. HF equation: iε∂ t ω N,t = [ − ε 2 ∆ + V ∗ ρ t − X t , ω N,t ] where ρ t ( x ) = N − 1 ω N,t ( x ; x ) and X t ( x ; y ) = N − 1 V ( x − y ) ω N,t ( x ; y ). • Wigner transform of ω N,t : � � � ε 3 x + εy 2 , x − εy e − ip · y W N,t ( x, p ) := dy ω N,t (2 π ) 3 2 As N → ∞ , Vlasov equation: � � ∂ t W ∞ ,t ( x, p ) + p · ∇ x W ∞ ,t ( x, p ) = ∇ x V ∗ ρ t ( x ) · ∇ p W ∞ ,t ( x, p ) Marcello Porta Mean field evolution of fermions October 8, 2016 5 / 12
Introduction Hartree-Fock and Vlasov dynamics • Let γ (1) N = N tr 2 ,...,N | ψ N �� ψ N | ≃ ω N , with ω N = ω 2 N (Slater det.). • Expect: for N ≫ 1, γ (1) N,t ≃ ω N,t = solution of time dep. HF equation: iε∂ t ω N,t = [ − ε 2 ∆ + V ∗ ρ t − X t , ω N,t ] where ρ t ( x ) = N − 1 ω N,t ( x ; x ) and X t ( x ; y ) = N − 1 V ( x − y ) ω N,t ( x ; y ). • Wigner transform of ω N,t : � � � ε 3 x + εy 2 , x − εy e − ip · y W N,t ( x, p ) := dy ω N,t (2 π ) 3 2 As N → ∞ , Vlasov equation: � � ∂ t W ∞ ,t ( x, p ) + p · ∇ x W ∞ ,t ( x, p ) = ∇ x V ∗ ρ t ( x ) · ∇ p W ∞ ,t ( x, p ) • Narnhofer-Sewell ’81, Spohn ’81; Elgart-Erd˝ os-Schlein-Yau ’04; Bardos-Golse-Gottlieb-Mauser ’03, Fr¨ ohlich-Knowles ’11 Marcello Porta Mean field evolution of fermions October 8, 2016 5 / 12
Results Results Marcello Porta Mean field evolution of fermions October 8, 2016 5 / 12
Pure states Hartree-Fock dynamics of pure states � V ( p ) | (1 + | p | ) 2 < ∞ . dp | � 1 (interaction) V ∈ L 1 ( R 3 ), such that a ( R 3 N ) s.t. tr | γ (1) 2 (initial data) ψ N ∈ L 2 N − ω N | ≤ C , with ω N = ω 2 N and tr | [ e iq · x , ω N ] | ≤ CNε (1 + | q | ) , tr | [ ε ∇ , ω N ] | ≤ CNε Theorem (Benedikter-P-Schlein, Comm. Math. Phys. ’14) Let γ (1) N,t be the reduced 1PDM of ψ N,t = e − iH N t/ε ψ N . Let ω N,t be the sol. of iε∂ t ω N,t = [ − ε 2 ∆ + V ∗ ρ t − X t , ω N,t ] , ω N, 0 ≡ ω N Then, for some constant c > 0 and for all t ∈ R : N,t − ω N,t | ≤ N 1 / 2 exp( c exp( c | t | )) � γ (1) tr | γ (1) N,t − ω N,t � HS ≤ exp( c exp( c | t | )) , Marcello Porta Mean field evolution of fermions October 8, 2016 6 / 12
Pure states Remarks 1 Result still holds replacing Hartree-Fock with Hartree: ω N,t = [ − ε 2 ∆ + V ∗ ρ t , � iε∂ t � ω N,t ] 2 Pseudorelativistic case [Benedikter-P-Schlein, J. Math. Phys. ’14] : � N � � � − ε 2 ∆ j + m 2 + N − 1 � iε∂ t ψ N,t = V ( x i − x j ) ψ N,t j =1 i<j with m = O (1). Under similar assumptions, we proved the emergence of the pseudorelativistic time-dependent HF equation: � − ε 2 ∆ + m 2 + V ∗ ρ t − X t , ω N,t ] . iε∂ t ω N,t = [ 3 Commutator estimates ≡ semiclassical structure. Implied by � x − y � � x + y � ω N ( x ; y ) ≃ Nϕ ξ for suitable ϕ , ξ . ε 2 true for the semiclassical approximation of the HF ground state. 4 Similar result: Petrat-Pickl ’14 . Marcello Porta Mean field evolution of fermions October 8, 2016 7 / 12
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