Hartree-Fock dynamics for weakly interacting fermions Benjamin - PowerPoint PPT Presentation

Hartree-Fock dynamics for weakly interacting fermions Benjamin Schlein, University of Zurich Princeton, February 19, 2014 Joint work with Niels Benedikter and Marcello Porta 1

Bosonic systems: described by N N � � H N = − ∆ x j + λ V ( x i − x j ) j =1 i<j acting on Hilbert space L 2 s ( R 3 N ) of symmetric wave functions. Mean field regime : large number of weak collisions. Realized when N ≫ 1, | λ | ≪ 1, Nλ ≃ 1. Study Schr¨ odinger evolution   N N � � − ∆ x j + 1  ψ N,t  V ( x i − x j ) i∂ t ψ N,t = N j =1 i<j ground state approximated by ϕ ⊗ N , with ϕ Trapped bosons: determined by Hartree theory. For this reason, it makes sense to study evolution of approxi- mately factorized initial data 2

Dynamics: factorization approximately preserved ψ N,t ≃ ϕ ⊗ N t where ϕ t solves Hartree equation i∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | 2 ) ϕ t , with ϕ t =0 = ϕ One-particle reduced density: defined by kernel � γ (1) N,t ( x ; y ) = N dx 2 . . . dx N ψ N,t ( x, x 2 , . . . , x N ) ψ N,t ( y, x 2 , . . . , x N ) Theorem: under appropriate assumptions on potential � � � � � γ (1) � ≤ Ce K | t | � � N,t − N | ϕ t �� ϕ t | Tr Rigorous works : Hepp, Ginibre-Velo, Spohn, Erd˝ os-Yau, Rodnianski-S., Fr¨ ohlich-Knowles-Schwarz, Knowles-Pickl, Grillakis-Machedon-Margetis, Lewin-Nam-S. , . . . 3

Fermionic systems: described by Hamiltonian N � � H N = − ∆ x j + λ V ( x i − x j ) j =1 i<j Scaling: kinetic energy is of order N 5 / 3 ⇒ take λ = N − 1 / 3 Velocities are order N 1 / 3 ⇒ consider times of order N − 1 / 3 ;   N N � � 1 iN 1 / 3 ∂ t ψ N,t =  ψ N,t  − ∆ x j + V ( x i − x j ) N 1 / 3 j =1 i<j Set ε = N − 1 / 3 . We find   N N � − ε 2 ∆ x j + 1 �  ψ N,t  iε∂ t ψ N,t = V ( x i − x j ) N j =1 i<j 4

Hartree-Fock theory: consider trapped fermions, with N N � � ( − ε 2 ∆ x j + V ext ( x j )) + 1 H N = V ( x i − x j ) N j =1 i<j Ground state ≃ Slater determinant, with reduced density ω N minimizing the Hartree-Fock energy E HF ( ω N ) = Tr( − ε 2 ∆ + V ext ) ω N � � ω N ( x, x ) ω N ( y, y ) − | ω N ( x, y ) | 2 � + 1 dxdyV ( x − y ) 2 N Goal : show that evolution of Slater determinant is approximately a Slater determinant, with reduced density ω N,t s.t. � � − ε 2 ∆ + ( V ∗ ρ t ) − X t , ω N,t iε∂ t ω N,t = However : cannot be true for arbitrary initial Slater determinants. 5

consider system of free fermions in Semiclassical structure: box Λ, with volume one. Ground state: is a Slater determinant � � e ik · ( x − y ) ≃ ε − 3 | k |≤ c dk e ik · ( x − y ) /ε ω N ( x, y ) = k ∈ Z 3 : | k |≤ cN 1 / 3 Consequence: ω N ( x, y ) ≃ ε − 3 ϕ (( x − y ) /ε ) concentrates close to diagonal. General trapping potential: we expect (linear combination of) � x − y � � x + y � ω N ( x, y ) ≃ ε − 3 ϕ g ε 2 Conclusion: Slater determinants like ω N satisfy � Tr | [ x, ω N ] | ≤ CNε Tr | [ ε ∇ , ω N ] | ≤ CNε 6

Thomas-Fermi theory: reduced density of ground state of N N � � ( − ε 2 ∆ x j + V ext ( x j )) + 1 H N = V ( x i − x j ) N j =1 i<j approximated by � 1 dp M ( p, ( x + y ) / 2) e ip · ( x − y ) /ε ω N ( x, y ) = Op M ( x, y ) = (2 πε ) 3 with phase-space density M ( p, q ) = χ ( | p | ≤ c ρ 1 / 3 TF ( x )). Thomas-Fermi density: ρ TF minimizes � � � E TF ( ρ ) = 3 dxV ext ( x ) ρ ( x )+1 dxρ 5 / 3 ( x )+ dxdy V ( x − y ) ρ ( x ) ρ ( y ) . 5 γ 2 Semiclassics: since [ x, ω N ] = iε Op ∇ p M , [ ε ∇ , ω N ] = ε Op ∇ q M , � � ε ρ 2 / 3 Tr | [ x, ω N ] | ≃ dpdq |∇ p M ( p, q ) | = CNε TF ( x ) dx ≤ CNε (2 πε ) 3 � � ε Tr | ε ∇ , ω N ] | ≃ dpdq |∇ q M ( p, q ) | = CNε |∇ ρ TF ( x ) | dx ≤ CNε (2 πε ) 3 7

Fock space: we introduce � L 2 a ( R 3 n , dx 1 . . . dx n ) F = n ≥ 0 Creation and annihilation operators: for f ∈ L 2 ( R 3 ) we define a ∗ ( f ) und a ( f ), satisfying the CAR { a ( f ) , a ∗ ( g ) } = � f, g � , { a ( f ) , a ( g ) } = { a ∗ ( f ) , a ∗ ( g ) } = 0 We also introduce operator valued distributions a ∗ x , a x so that � � a ∗ ( f ) = dx f ( x ) a ∗ and a ( f ) = dx f ( x ) a x x Hamilton operator: Using these distributions, we define � � x ∇ x a x + 1 H N = ε 2 dx ∇ x a ∗ dxdyV ( x − y ) a ∗ x a ∗ y a y a x 2 N 8

Bogoliubov transformations: let N � ω N = | f j �� f j | j =1 be orthogonal projection onto L 2 ( R 3 ) with Tr ω N = N . Let { f j } j ∈ N be an orthonormal basis of L 2 ( R 3 ). Unitary implementor: find unitary map R ω N on F such that R ω N Ω = a ∗ ( f 1 ) . . . a ∗ ( f N )Ω and � if j ≤ N a ( f j ) R ∗ ω N a ∗ ( f j ) R ω N = a ∗ ( f j ) if j > N For general g ∈ L 2 ( R 3 ), we have (with u N = 1 − ω N ) R ω N a ∗ ( g ) R ω ∗ N = a ∗ ( u N g ) + a ( ω N g ) 9

Theorem: let V : R 3 → R s.t. � V ( p ) | (1 + p 2 ) dp < ∞ | � Initial data: let ω N be family of projections with Tr ω N = N and Tr | [ x, ω N ] | ≤ CNε and Tr | [ ε ∇ , ω N ] | ≤ CNε Let ξ N be a sequence in F , with � ξ N , N ξ N � ≤ C . Time evolution: consider ψ N,t = e − i H N t/ε R ν N ξ N we have Convergence in Hilbert-Schmidt norm: � γ (1) N,t − ω N,t � HS ≤ C exp( c 1 exp( c 2 | t | )) Convergence in trace norm: if additionally � ξ N , N 2 ξ N � ≤ C and d Γ( ω N,t ) ξ N = 0, we have � � � � � ≤ CN 1 / 6 exp ( c 1 exp ( c 2 | t | )) � γ (1) � � N,t − ω N,t Tr 10

Extension: weaker bounds also hold if � ξ N , N ξ N � ≤ CN α , for 0 ≤ α < 1 . Corollary: let ψ N ∈ L 2 a ( R 3 N ) be s.t. � � � � � γ (1) � ≤ CN α � � Tr − ω N N for 0 ≤ α < 1 and for orthogonal projection ω N with Tr ω N = N , satisfying semiclassical bounds. Then ψ N,t = e − iH N t/ε ψ N is such that � γ (1) N,t − ω N,t � HS ≤ CN α exp( c 1 exp( c 2 | t | )) Proof: set ξ N = R ∗ ω N ψ N and observe that � � � � � γ (1) � ≤ CN α � � � ξ N , N ξ N � ≤ Tr − ω N N 11

Remarks: Higher order densities: similar bounds can be proven for γ ( k ) N,t , for any fixed k ∈ N . Hartree-Fock versus Hartree: Exchange term in Hartree-Fock equation is of smaller order. Bounds continue to hold for � � − ε 2 ∆ + ( V ∗ � iε∂ t � ω N,t = ρ t ) , � ω N,t Hartree-Fock equation still depend on N . Vlasov dynamics: Let � � � 1 x + εy 2 , x − εy e iv · y W N,t ( x, v ) = dy ω N,t (2 πε ) 3 2 Then W N,t → W ∞ ,t as N → ∞ , where ∂ t W ∞ ,t + v · ∇ x W ∞ ,t + ∇ ( V ∗ ρ t ) · ∇ v W ∞ ,t = 0 is classical Vlasov equation. 12

Previous works: Narnhofer-Sewell (1980) proved convergence towards Vlasov dynamics for smooth potentials. Spohn (1982) extended convergence to bounded potentials. os-S.-Yau (2003) proved convergence to Hartree but Elgart-Erd˝ only for short times and analytic potentials. Bardos-Golse-Gottlieb-Mauser (2002) and Fr¨ ohlich-Knowles (2010) showed convergence to Hartree-Fock dynamics, but with different scaling (no semiclassical limit). Bach (1992) and Graf-Solovej (1994) proved that Hartree-Fock theory approximates ground state energy of systems of matter, up to relative error o ( ε 2 ). 13

Fluctuation dynamics: we define ξ N,t s.t. e − i H N t/ε R ω N ξ N = R ω N,t ξ N,t Equivalently ξ N,t = U N ( t ) ξ N with unitary evolution U N ( t ) = R ∗ ω N,t e − i H N t/ε R ω N We want to compute γ (1) N,t ( x, y ) = � e − i H N t/ε R ω N ψ N , a ∗ x a y e − i H N t/ε R ω N ψ N � = � R ω N,t ξ N,t , a ∗ x a y R ω N,t ξ N,t � � � � � � � a ∗ ( u N,t,x ) + a ( ω N,t,x a ( u N,t,y ) + a ∗ ( ω N,t,y ) = ξ N,t , ξ N,t = ω N,t ( x, y ) + normally ordered terms Conclusion: need to control � ξ N,t , N ξ N,t � = � ξ N , U ∗ N ( t ) NU N ( t ) ξ N � uniformly in N . 14

Growth of fluctuations: we compute iε ∂ t � U N ( t ) ξ N , N U N ( t ) ξ N � � R ω N ξ N , e i H N t/ε � � � e − i H N t/ε R ω N ξ N N − 2 d Γ( ω N,t ) + N = iε ∂ t � � � e − i H N t/ε R ω N ξ N , = − 2 [ H N , d Γ( ω N,t )] + d Γ( iε ∂ t ω N,t ) � × e − i H N t/ε R ω N ξ N Identity for derivative: We obtain iε ∂ t � U N ( t ) ξ N , N U N ( t ) ξ N � � = Re 1 dxdyV ( x − y ) N � � a ∗ ( u N,t,y ) a ∗ ( ω N,t,y ) a ∗ ( ω N,t,x ) a ( ω N,t,x ) × U N ( t ) ξ N , + a ∗ ( u N,t,x ) a ( u N,t,x ) a ( ω N,t,y ) a ( u N,t,y ) � � + a ( u N,t,x ) a ( ω N,t,x ) a ( ω N,t,y ) a ( u N,t,y ) U N ( t ) ξ N Consider for example, the last contribution � � � 1 dxdyV ( x − y ) a ( u N,t,x ) a ( ω N,t,x ) a ( ω N,t,y ) a ( u N,t,y ) N 15