Thermodynamic limit for a system of interacting fermions in a random medium. Pieces one-dimensional model

Transcription

1 Thermodynamic limit for a system of interacting fermions in a random medium. Pieces one-dimensional model Nikolaj Veniaminov (in collaboration with Frédéric Klopp) CEREMADE, University of Paris IX Dauphine 5th Meeting of the GDR Quantum Dynamics Laboratory Paul Painlevé, University of Lille 1 February 6, 2013 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

2 Outline 1 Model One particle system Multiparticle system and thermodynamic limit Motivation 2 Results: approximating ground state Energy asymptotic expansion One- and two-particle density matrices 3 Discussion and perspectives N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

3 Simple one-dimensional random model Pieces or Luttinger-Sy model N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

4 Simple one-dimensional random model Pieces or Luttinger-Sy model On R, consider Poisson point process dµ(ω) of intensity 1 dµ(ω) = k Z δ xk (ω) N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

5 Simple one-dimensional random model Pieces or Luttinger-Sy model On R, consider Poisson point process dµ(ω) of intensity 1 dµ(ω) = k Z δ xk (ω) For Λ = [ L/2, L/2], on L 2 (Λ), define H ω (L) = k Z d2 dx 2 D [x k,x k+1 ] Λ N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

6 Simple one-dimensional random model Pieces or Luttinger-Sy model On R, consider Poisson point process dµ(ω) of intensity 1 dµ(ω) = δ xk (ω) k Z For Λ = [ L/2, L/2], on L 2 (Λ), define H ω (L) = k Z Integrated density of states d2 dx 2 D [x k,x k+1 ] Λ {eigenvalues of H ω (L) in (, E]} N(E) := lim L + L = exp( l E ) 1 exp( l E ) 1 E 0 where l E := π. E N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

7 Multiparticle system On n j=1 L 2 (Λ) = L 2 (Λ n ), consider the free operator H 0 ω(l, n) = n 1 }. {{.. 1 } H ω (L) } 1. {{.. 1 }. i 1 times n i times i=1 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

8 Multiparticle system On n j=1 L 2 (Λ) = L 2 (Λ n ), consider the free operator (n fermions) H 0 ω(l, n) = n 1 }. {{.. 1 } H ω (L) } 1. {{.. 1 }. i 1 times n i times i=1 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

9 Multiparticle system On n j=1 L 2 (Λ) = L 2 (Λ n ), consider the free operator (n fermions) H 0 ω(l, n) = n 1 }. {{.. 1 } H ω (L) } 1. {{.. 1 }. i 1 times n i times i=1 Pick U : R R + not identically vanishing, even, bounded and define H U ω (L, n) = H 0 ω(l, n) + W n, where W n (x 1,, x n ) := i<j U(x i x j ). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

10 Multiparticle system On n j=1 L 2 (Λ) = L 2 (Λ n ), consider the free operator (n fermions) H 0 ω(l, n) = n 1 }. {{.. 1 } H ω (L) } 1. {{.. 1 }. i 1 times n i times i=1 Pick U : R R + not identically vanishing, even, bounded and define H U ω (L, n) = H 0 ω(l, n) + W n, where W n (x 1,, x n ) := i<j U(x i x j ). U decays sufficiently fast at infinity: x 4 U(x)dx < +, R x 2 U(x) < const. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

11 Multiparticle system On n j=1 L 2 (Λ) = L 2 (Λ n ), consider the free operator (n fermions) H 0 ω(l, n) = n 1 }. {{.. 1 } H ω (L) } 1. {{.. 1 }. i 1 times n i times i=1 Pick U : R R + not identically vanishing, even, bounded and define H U ω (L, n) = H 0 ω(l, n) + W n, where W n (x 1,, x n ) := i<j U(x i x j ). U decays sufficiently fast at infinity: x 4 U(x)dx < +, R x 2 U(x) < const. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

12 Thermodynamic limit Let E U ω (L, n) be the ground state energy of H ω (L, n). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

13 Thermodynamic limit Let E U ω (L, n) be the ground state energy of H ω (L, n). Let Ψ U ω (L, n) be the associated eigenfunction. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

14 Thermodynamic limit Problem Let E U ω (L, n) be the ground state energy of H ω (L, n). Let Ψ U ω (L, n) be the associated eigenfunction. Describe E U ω (L, n) and Ψ U ω (L, n) in the limit L +, n L ρ > 0. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

15 Existence for ground state energy per particle Theorem Under the above assumptions, the following limits exist in L 2 ω E 0 (ρ) := E lim ω(l, 0 n) L + n n/l ρ and E U Eω U (L, n) (ρ) := lim L + n n/l ρ and they are independent of ω. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

16 Existence for ground state energy per particle Theorem Under the above assumptions, the following limits exist in L 2 ω E 0 (ρ) := E lim ω(l, 0 n) L + n n/l ρ and E U Eω U (L, n) (ρ) := lim L + n n/l ρ and they are independent of ω. Moreover, E 0 (ρ) = 1 ρ Eρ ( Eρ )) E dn(e) = E ρ (1 + O = π 2 log ρ 2 ( 1 + O ( log ρ 1)), where the Fermi energy E ρ is the unique solution to N(E ρ ) = ρ. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

17 Why pieces model is so good? N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

18 Why pieces model is so good? the eigenfunctions are localized (on a scale log Λ ) N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

19 Why pieces model is so good? the eigenfunctions are localized (on a scale log Λ ) the localization centers and the eigenvalues satisfy Poisson statistics N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

20 Why pieces model is so good? the eigenfunctions are localized (on a scale log Λ ) the localization centers and the eigenvalues satisfy Poisson statistics the model exhibits Lifshitz tail asymptotics N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

21 Why pieces model is so good? the eigenfunctions are localized (on a scale log Λ ) the localization centers and the eigenvalues satisfy Poisson statistics the model exhibits Lifshitz tail asymptotics eigenfunctions and eigenvalues are known explicitly N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

22 Why pieces model is so good? the eigenfunctions are localized (on a scale log Λ ) the localization centers and the eigenvalues satisfy Poisson statistics the model exhibits Lifshitz tail asymptotics eigenfunctions and eigenvalues are known explicitly tunneling effect is missing N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

23 A system of two fermions Lemma Consider two interacting (via the pair potential U) fermions in [0, l], i.e., on L 2 ([0, l]) L 2 ([0, l]), consider the Hamiltonian 2 + U(x 1 x 2 ). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

24 A system of two fermions Lemma Consider two interacting (via the pair potential U) fermions in [0, l], i.e., on L 2 ([0, l]) L 2 ([0, l]), consider the Hamiltonian 2 + U(x 1 x 2 ). Then, E U (l, 2), the ground state energy of this system admits the following expansion for large l: E U (l, 2) = 5π2 l 2 + β l 3 + o ( l 3). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

25 A system of two fermions Lemma Consider two interacting (via the pair potential U) fermions in [0, l], i.e., on L 2 ([0, l]) L 2 ([0, l]), consider the Hamiltonian 2 + U(x 1 x 2 ). Then, E U (l, 2), the ground state energy of this system admits the following expansion for large l: Moreover, β = 5π2 4 E U (l, 2) = 5π2 l 2 + β l 3 + o ( l 3). u U(u), ( Id + 1 ) 1 2 U1/2 ( 1 ) 1 U 1/2 u U(u). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

26 Ground state energy asymptotic expansion Theorem For sufficiently small ρ, E U (ρ) = E 0 (ρ) + π2 β ( log ρ 3 ρ + o ρ log ρ 3), where ( β = 1 exp β ) 8π 2. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

27 Reduced density matrices: definition Let Ψ L 2 (Λ n ) be a normalized n-fermion wave function. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

28 Reduced density matrices: definition Let Ψ L 2 (Λ n ) be a normalized n-fermion wave function. One-particle density matrix is an operator on L 2 (Λ) with kernel γ Ψ (x, y) = n Ψ(x, x)ψ (y, x)d x. Λ n 1 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

29 Reduced density matrices: definition Let Ψ L 2 (Λ n ) be a normalized n-fermion wave function. One-particle density matrix is an operator on L 2 (Λ) with kernel γ Ψ (x, y) = n Ψ(x, x)ψ (y, x)d x. Λ n 1 Two-particle density matrix is an operator on L 2 (Λ 2 ) with kernel γ (2) Ψ (x 1, x 2, y 1, y 2 n(n 1) ) = Ψ(x 1, x 2, x)ψ (y 1, y 2, x)d x. 2 Λ n 2 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

30 Reduced density matrices: definition Let Ψ L 2 (Λ n ) be a normalized n-fermion wave function. One-particle density matrix is an operator on L 2 (Λ) with kernel γ Ψ (x, y) = n Ψ(x, x)ψ (y, x)d x. Λ n 1 Two-particle density matrix is an operator on L 2 (Λ 2 ) with kernel γ (2) Ψ (x 1, x 2, y 1, y 2 n(n 1) ) = Ψ(x 1, x 2, x)ψ (y 1, y 2, x)d x. 2 Λ n 2 Both γ Ψ and γ (2) Ψ are positive trace class operators satisfying Tr γ Ψ = n and Tr γ (2) Ψ n(n 1) =. 2 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

31 Free ground state reduced density matrices γ Ψ 0 ω (Λ,n) = + + R (1) γ ϕ 1 γ [xk+1,x k ] ϕ 2 [xk+1,x k ] l Eρ x k+1 x k 3l Eρ 2l Eρ x k+1 x k 3l Eρ where for an interval I, we let ϕ j I be the j-th normalized eigenvector of D I, the operator R (1) is trace class and R (1) 1 Cρ 2 n. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

32 Free ground state reduced density matrices γ Ψ 0 ω (Λ,n) = where l Eρ x k+1 x k 3l Eρ γ ϕ 1 [xk+1,x k ] + 2l Eρ x k+1 x k 3l Eρ γ ϕ 2 [xk+1,x k ] + R (1) for an interval I, we let ϕ j I be the j-th normalized eigenvector of D I, the operator R (1) is trace class and R (1) 1 Cρ 2 n. For the two-particle density matrix, we obtain where γ (2) Ψ 0 ω(λ,n) = 1 2 (Id Ex) [ ] γ Ψ 0 ω (Λ,n) γ Ψ 0 ω (Λ,n) + R (2) Ex is the exchange operator on a two-particle space: Ex f g = g f the operator R (2) is trace class and R (2) 1 Cn. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

33 Interacting ground state: optimal approximation Let ζ 1 I be the ground state of D I 2 + U acting on L 2 (I 2 ). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

34 Interacting ground state: optimal approximation Let ζi 1 be the ground state of D + U acting on L 2 I (I 2 ). Define 2 γ Ψ opt = γ Λ,n ϕ 1 [xk+1,x k ] l Eρ ρβ x k+1 x k 2l Eρ log(1 β ) + 2l Eρ log(1 β ) x k+1 x k γ ζ 1 [xk+1,x k ], N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

35 Interacting ground state: optimal approximation Let ζi 1 be the ground state of D + U acting on L 2 I (I 2 ). Define 2 γ Ψ opt = γ Λ,n ϕ 1 [xk+1,x k ] Theorem l Eρ ρβ x k+1 x k 2l Eρ log(1 β ) + There exists ρ 0 > 0 such that, for ρ (0, ρ 0 ), one has lim sup L + n/l ρ 1 n 2 lim sup L + n/l ρ 1 n γ Ψ U ω (Λ,n) γ Ψ opt γ(2) Ψ U ω(λ,n) 1 [ 2 (Id Ex) 2l Eρ log(1 β ) x k+1 x k γ ζ 1 [xk+1,x k ] Λ,n γ Ψ opt Λ,n = o(ρ), 1 Λ,n] γ Ψ opt = o(ρ). 1, N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

36 Quantification of the influence of interactions Influence of interactions on the ground state is essentially described by ( ) γ Ψ 0 ω (Λ,n) γ Ψ opt = γ Λ,n ϕ 1 + γ [xk+1,x k ] ϕ 2 γ [xk+1,x k ] ζ 1 [xk+1,x k ] 2l Eρ log(1 β ) x k+1 x k l Eρ ρβ x k+1 x k l Eρ γ ϕ 1 [xk+1,x k ] + 2l Eρ x k+1 x k 2l Eρ log(1 β ) γ ϕ 2 [xk+1,x k ] + R (1) N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

37 Quantification of the influence of interactions Influence of interactions on the ground state is essentially described by ( ) γ Ψ 0 ω (Λ,n) γ Ψ opt = γ Λ,n ϕ 1 + γ [xk+1,x k ] ϕ 2 γ [xk+1,x k ] ζ 1 [xk+1,x k ] 2l Eρ log(1 β ) x k+1 x k l Eρ ρβ x k+1 x k l Eρ γ ϕ 1 [xk+1,x k ] In particular, + 2l Eρ x k+1 x k 2l Eρ log(1 β ) 1 lim γ L + n Ψ 0 ω (Λ,n) γ Ψ U ω (Λ,n) = 2β ρ + o(ρ), 1 n/l ρ γ ϕ 2 [xk+1,x k ] + R (1) and lim L + n/l ρ 1 γ (2) n 2 Ψ 0 ω γ(2) (Λ,n) Ψ U ω(λ,n) = 2β ρ + o(ρ). 1 N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

38 Some interesting questions N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

39 Some interesting questions 1 We ask for x 4 U(x) L 1 (R), however β (second order coefficient in two fermions problem) needs only x 2 U(x) L 1 (R). N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

40 Some interesting questions 1 We ask for x 4 U(x) L 1 (R), however β (second order coefficient in two fermions problem) needs only x 2 U(x) L 1 (R). 2 How the picture changes if only x 2 U(x) L 1 (R) and x 3 U(x) / L 1 (R)? Conjecture: interactions at a distance are more important than local interactions in the same piece. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

41 Some interesting questions 1 We ask for x 4 U(x) L 1 (R), however β (second order coefficient in two fermions problem) needs only x 2 U(x) L 1 (R). 2 How the picture changes if only x 2 U(x) L 1 (R) and x 3 U(x) / L 1 (R)? Conjecture: interactions at a distance are more important than local interactions in the same piece. 3 What happens beyond this threshold? Conjecture: interactions are more important than kinetic energy term. N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18

42 The end Thank you for your attention! N. Veniaminov (CEREMADE, Dauphine) Interacting Fermions in Random Media Lille, February 6, / 18