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 II. Approximate models Jan Philip Solovej Department of Mathematics University of Copenhagen Workshop on Large Many-Body Systems Warwick August 2004 1

List of Slides 3 Equivalence of canonical and grand canonical pictures 4 The bosonic Hartree approximation 5 The Hartree-Fock model 6 Properties of the Hartree-Fock model 7 Semiclassics 8 The Thomas-Fermi approximation 9 Quasi-free fermionic states 10 BCS theory 11 The Bogolubov approximation 2

Equivalence of canonical and grand canonical pictures If the system is stable of the 2nd kind we may define the grand canonical pressure for µ ∈ [ C, ∞ ) for some C > 0 P ( µ ) = � Ψ � =1 , Ψ ∈F (Ψ , ( H + µN )Ψ) . inf The parameter µ is called the chemical potential. In fact P is the Legendre transform of the energy function N �→ E N P ( µ ) = inf N ( E N + µN ) Thus if N �→ E N is convex we may reconstruct E N from P E N = sup µ ( P ( µ ) − µN ) . 3

The bosonic Hartree approximation In the following we will consider mainly the case where h = − 1 2 ∆ + V, W ij = W ( x i − x j ) In the bosonic Hartree approximation one restricts attention to wave functions of the non-interacting form Ψ = ψ ⊗ · · · ⊗ ψ . � �� � N Then E H N ( ψ ) := (Ψ , H N Ψ) = � � � � |∇ ψ | 2 + N 1 | ψ | 2 V + 1 | ψ ( x ) | 2 W ( x − y ) | ψ ( y ) | 2 dxdy. 2 N 2 N ( N − 1) E H ψ ∈ h , � ψ � =1 E H ( ψ ) ≥ E B N = inf N Or we may use the density ρ = N | ψ | 2 . If W is positive type (i.e., � W ≥ 0) then E H is convex and the minimizer ψ is unique (up to a constant phase). Is this a good approximation? Certainly not if W is a hard core. 4

The Hartree-Fock model In the Hartree-Fock approximation we do the same for fermions and restrict to Slater determinants Ψ = ψ 1 ∧ · · · ∧ ψ N . The energy expectation can be expressed entirely from the 1-particle density matrix γ , which is the projection onto the space spanned by ψ 1 , . . . , ψ N : � E HF ( γ ) (Ψ , H N Ψ) = Tr[ − 1 := 2 ∆ γ ] + ρ γ V � � + 1 ρ γ ( x ) W ( x − y ) ρ γ ( y ) dxdy 2 � � Tr C q | γ ( x, y ) | 2 W ( x − y ) dxdy − 1 2 The last two terms are called respectively the direct term and the exchange term. ρ γ is the density of γ N N � � ψ k ( x ) ψ k ( y ) ∗ , | ψ k ( x ) | 2 . γ ( x, y ) = ρ γ ( x ) = Tr C q γ ( x, x ) = k =1 k =1 5

Properties of the Hartree-Fock model E HF = inf {E HF ( γ ) | γ projection Tr γ = N } ≥ E F N N THEOREM 1 (Self-consistency, Bach-Lieb-Loss-Sol.) . If γ is an HF minimizer then γ is the unique projection onto the N “lowest” eigenvectors of the mean field operator � H MF = − 1 γ ( x, y ) ∗ W ( x − y ) φ ( y ) dy. 2 ∆ + V + ρ γ ∗ W − K γ , K γ φ ( x ) = The uniqueness means that there are no degeneracies. Put differently: There are no unfilled shells in Hartree-Fock theory . Minimizer are not necessarily unique. The approximation is again very bad for hard core. THEOREM 2 (Lieb’s variational principle). E HF = inf {E HF ( γ ) | 0 ≤ γ ≤ 1 , Tr γ = N } . N 6

Semiclassics We want next to approximate the fermionic energy by a functional of the density alone. We will ignore the exchange term. We make the semiclassical approximations for a non-interacting system � 1 dp = C d | V ( x ) | 3 / 2 ρ ( x ) = (2 π ) − 3 − , 1 2 p 2 + V ( x ) < 0 for the density and for the energy � � � � 2 p 2 + V ( x )) dpdx = − C cl ρ 5 / 3 + | V | 5 / 2 (2 π ) − 3 ( 1 = C TF ρV. − 1 2 p 2 + V ( x ) < 0 THEOREM 3 (semiclassics). � | V | 5 / 2 h → 0 (2 πh ) 3 Tr[ −| − h 2 ∆ + V | − ] = − C cl lim − . Here Tr[ −| − h 2 ∆ + V | − ] is the sum of the negative eigenvalues of − h 2 ∆ + V , i.e., the minimal fermionic energy. 7

The Thomas-Fermi approximation Motivated by semiclassics we define the Thomas-Fermi functional � � � � ρ 5 / 3 + E TF ( ρ ) := C TF ρV + 1 ρ ( x ) W ( x − y ) ρ ( y ) dxdy 2 � E TF = inf {E TF ( ρ ) | ρ ≥ 0 , ρ = N } . N If V, W tend to zero at infinity and W is positive type ( � W ≥ 0) then the minimzing ρ is unique and N �→ E TF is convex and non-increasing. There is N N TF (possibly= ∞ ) such that N �→ E TF is strictly decreasing for N ≤ N TF and c N c constant otherwise. For N < N TF a minimizing ρ exists Lieb-Simon). c In many cases one can prove that the TF approximation is good using semiclassical techniques. 8

Quasi-free fermionic states We shall now see that we may improve Hartree-Fock theory by considering a grand canonical generalization. We generalize from Slater determinants to ground states Ψ of general quadratic Hamiltonians � A αβ a ∗ α a β + B αβ a α a β − B αβ a ∗ α a ∗ β αβ (Slater determinants correspond to B = 0). If we introduce γ ij = (Ψ , a ∗ i a j Ψ) , α ij = (Ψ , a i a j Ψ) then γ 2 + αα ∗ = γ, [ γ, α ] = 0. In particular, 0 ≤ γ ≤ 1 . Slater: a ( ψ 1 ) ∗ · · · a ( ψ N ) ∗ | 0 � BCS: [ σ 1 + τ 1 a ( ψ 1 ) ∗ a ( ψ 2 ) ∗ ][ σ 2 + τ 2 a ( ψ 3 ) ∗ a ( ψ 4 ) ∗ ] · · · | 0 � , σ i , τ i ∈ C 9

BCS theory The BCS approximation to the pressure is P BCS ( µ ) := (Ψ , ( H + µN )Ψ) E HF ( γ ) + µ Tr γ = � � + 1 | Tr C q α ( x, y ) | W ( x − y ) dxdy 2 From Lieb’s variational principle we see that if W ≥ 0 the best choice is α = 0. Otherwise α � = 0 may be better. This has been analyzed in great detail by Bach-Lieb-Sol. for the Hubbard model: R 3 → Z 3 , − ∆ discrete, V = 0, W ( x ) = 0 unless x = 0. The BCS mean field operator is a quadratic Hamiltonian. 10

The Bogolubov approximation Quadratic Hamiltonians are also important in the Bogolubov approximation for bosons. Let us write the 2nd quantized Hamiltonian in an eigenbasis for the one-particle operator � � h α a ∗ α a α + 1 W αβµν a ∗ α a ∗ H = β a ν a µ 2 α αβµν The Bogolubov approximation is based on the assumption that α = 0 corresponds to a condensate. √ Bogolubov approximation: (1) a ∗ 0 , a 0 → N , (2) keep only quartic terms with at least two α, β, µ, ν being 0. The Hamiltonian becomes quadratic plus linear � � h α a ∗ α a α + 1 W α 0 β a ∗ α a β + W αβ 00 a ∗ α a ∗ 2 N β + . . . α αβ � =0 2 N 3 / 2 � + 1 W α 000 a ∗ α + . . . + 1 2 N 2 W 0000 . α 11