Rigorous Results on the energy and structure of ground states of - PowerPoint PPT Presentation

Rigorous Results on the energy and structure of ground states of large many-body systems III. Specific many body systems Jan Philip Solovej Department of Mathematics University of Copenhagen Workshop on Large Many-Body Systems Warwick August 2004 1

List of Slides 3 The thermodynamic limit 4 Short range Fermi and Bose systems 5 Charged systems I: Atoms and molecules 6 Charged systems II: Matter 7 Charged systems II: One component plasma 8 Results on Atoms 9 Foldy’s use of Bogolubov aprox. to jellium like system 10 Results for the one-component plasma 2

The thermodynamic limit We now restrict to a box Ω = [0 , L ] 3 , i.e., consider the Hilbert space h = L 2 (Ω; C q ) and introduce suitable boundary condition for the Laplacian (Dirichlet, Neumann or periodic). | Ω | = L 3 , we write Ω → ∞ for L → ∞ . Allow the 1-particle potential V Ω to depend on the domain. N � � ( − 1 H N, Ω = 2 ∆ i + V Ω ( x i )) + W ( x i − x j ) . i =1 1 ≤ i<j ≤ N We say that the ground state energy E N (Ω) has a thermodynamic limit if for fixed ρ > 0 there is e ( ρ ) such that E N (Ω) → e ( ρ ) | Ω | as N → ∞ , Ω → ∞ with N/ | Ω | = ρ . Existence of thermodynamic limit requires stability of the 2nd kind . 3

Short range Fermi and Bose systems Now V = 0 and W ≥ 0 with finite range and scattering length : � �� � R 3 � � � |∇ φ | 2 + | φ | 2 = 1 W | φ | 2 a = lim 3 inf . � � R →∞ | x | <R | x | <R | x | <R � (no 1 / 2 in front of the kinetic energy; reduced mass). In particular, we may have a hard core with diameter a : W ( x ) = ∞ if | x | < a and 0 otherwise. In this case Hartree and Hartree-Fock are very bad. For both fermions and bosons the thermodynamic limits exist. THEOREM 1 (Low density Bose and Fermi gas ). For ρ → 0 e B ( ρ ) = 2 πaρ 2 + o ( ρ 2 ) Lieb-Yngvason e F ( ρ ) = C TF ρ 5 / 3 + 2 π (1 − q − 1 ) aρ 2 + o ( ρ 2 ) Lieb-Seiringer-Sol. (in prep.) A modification of Bogolubov is supposed to give next Bose term. 4

Charged systems I: Atoms and molecules Atoms and Molecules: Two types of particles N electrons and K nuclei. N N K Z k | x i − R k | − 1 + � � � � | x i − x j | − 1 − 1 H N,K = 2 ∆ i − i =1 i =1 k =1 1 ≤ i<j ≤ N � Z k Z ℓ | R k − R ℓ | − 1 + 1 ≤ k<ℓ ≤ K The electrons are fermions. The nuclei are classical particles at positions R 1 , . . . , R K ∈ R 3 and nuclear charges. Z 1 , . . . , Z K > 0. Since we look for the ground state we have ignored the kinetic energy of the nuclei. We simply look for the energetically best positions. From the point of view of stability this is the worst case. Recall: Stable of 1st kind by Sobolev. E F N,K = inf Ψ , � Ψ � =1 (Ψ , H N,K Ψ) > −∞ . inf R 1 ,...,R K 5

Charged systems II: Matter Fermionic Matter: As above, but confined to box Ω with Dirichlet b.c. and with Z 1 = . . . = Z K = Z . We consider the thermodynamic limit of E F N,K (Ω) with N/ | Ω | = ρ and N = ZK (neutrality). Bosonic Matter: We have two species of opposite charge. We will show that bosonic matter is unstable even if we keep the kinetic energy of both positive and negative particles: N � � z i z j | x i − x j | − 1 , − 1 H N = 2 ∆ i + z i = 1 or z i = − 1 i =1 1 ≤ i<j ≤ N E B N = inf Ψ , � Ψ � =1 (Ψ , H N Ψ) . inf z i = ± 1 6

Charged systems II: One component plasma One component plasma (jellium): This is again a charged system confined to a box Ω, but instead of nuclei we have a uniform background of density ρ . N � � � | x i − y | − 1 dy + | x i − x j | − 1 − 1 H N = 2 ∆ i − ρ Ω i =1 1 ≤ i<j ≤ N � � 2 ρ 2 | x − y | − 1 dxdy. + 1 Ω × Ω We consider thermodynamics for the neutral system N = ρ | Ω | . The neutral system has ground state energy E N . In the physics literature one finds a different formulation on a torus. We discuss this later. 7

Results on Atoms For neutral atoms N = Z E Z = c TF Z 7 / 3 E TF lim = 1 , Lieb-Simon Z E TF Z →∞ Z For neutral quantum density ρ ( x ) = Tr γ Ψ ( x, x ): Z − 2 ρ ( Z − 1 / 3 x ) → ρ TF Z =1 ( x ) , L 1 loc as Z → ∞ Improved asymptotics: = o ( Z 5 / 3 ) as Z → ∞ E F Z − E HF Bach, Fefferman-Seco Z Z = c TF Z 7 / 3 + C S Z 2 + C DS Z 5 / 3 + o ( Z 5 / 3 ) E F Siedentop-Weikards, Fefferman-Seco TF has minimizer iff N ≤ Z . HF has a minimizer for N < Z + 1 (Lieb-Simon) and no minimizer for N > Z + Q for some constant Q (Sol.). The quantum theory has stable ground state for N < Z + 1 (Zhislin) and no ground state for N > 2 Z + 1 (Lieb) or for N ≫ Z when Z large. 8

Foldy’s use of Bogolubov aprox. to jellium like system Bosons on Ω; translation invariant Hamiltonian periodic b.c (“jellium”) 4 π � 2 Ω − 1 � � 2 k 2 a ∗ a ∗ p a ∗ 1 k a k + 1 H = q a p + k a q − k k 2 p,q k k � =0 The sum is over momenta. After Bogolubov approximation ( ρ = N/ Ω) − k a − k ) + 4 πρ � 2 k 2 ( a ∗ k a k + a ∗ k 2 ( a ∗ k a k + a ∗ − k a − k + a ∗ k a ∗ − k + a ∗ − k a ∗ 1 1 k ) 2 k � =0 k + α k a − k ) ∗ + D k ( a ∗ � D k ( a ∗ k + α k a − k )( a ∗ − k + α k a k )( a ∗ − k + α k a k ) ∗ = k � =0 � 2 k 2 + 4 πρk − 2 ) − 4 k 4 + 4 πρ � k ([ a k , a ∗ k ] + [ a − k , a ∗ D k α 2 4 D k α 2 k = ( 1 1 − − k ]) , k � =0 Thus the ground state energy per volume for Ω → ∞ (thermod. limit) � − (2 π ) − 3 k dk = − Jρ 5 / 4 , 2 D k α 2 J explicit. 9

Results for the one-component plasma The termodynamic limits exist (Lieb-Narnhofer) for both fermionic ( e F ( ρ ) = lim Ω →∞ E F N / | Ω | ) and bosonic ( e B ( ρ )) neutral ( ρ = N/ | Ω | ) jellium. For large ρ we have e B ( ρ ) = − Jρ 5 / 4 + o ( ρ 5 / 4 ) Lieb-Sol ( ≥ ), Sol. ( ≤ ) and e F ( ρ ) = e HF ( ρ ) + o ( ρ 4 / 3 ) Graf-Sol. e HF ( ρ ) = C TF ρ 5 / 3 − C D ρ 4 / 3 + o ( ρ 4 / 3 ) Graf-Sol. based on Dirac . For small ρ e B ( ρ ) = c cl ρ 4 / 3 + o ( ρ 4 / 3 ) , e F ( ρ ) = c cl ρ 4 / 3 + o ( ρ 4 / 3 ) , where e classical ( ρ ) = c cl ρ 4 / 3 . Conjecture: Optimal classical configuration is a Body Centered Cubic lattice . 10