Renormalization Group Approach to Density Functional Theory

Transcription

1 Renormalization Group Approach to Density Functional Theory Sandra Kemler, Jens Braun and Martin Pospiech DPG Frühjahrstagung Heidelberg [S. Kemler and J. Braun, 2013 J. Phys. G: Nucl. Part. Phys.] [S. Kemler, J. Braun and M. Pospiech, in preperation] SFB 634 March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 1

2 Contents Introduction Spin-1/2 Fermions in a Trap (1+1 d) Model Method Results Conclusion and Outlook March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 2

3 Hohenberg-Kohn Theorem one to one correspondence between the wave functions and the density of a system [Hohenberg and Kohn,1964] Ψ[ρgs ] Ô Ψ[ρgs ] = O[ρ gs ] energy functional: E V [ρ] = E HK [ρ] + d 3 x V (x)ρ(x) theorem provides no recipe for the computation of E HK take global ansatz for the energy density functional solve Kohn-Sham equations selfconsistently March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 3

4 Hohenberg-Kohn Theorem one to one correspondence between the wave functions and the density of a system [Hohenberg and Kohn,1964] Ψ[ρgs ] Ô Ψ[ρgs ] = O[ρ gs ] energy functional: E V [ρ] = E HK [ρ] + d 3 x V (x)ρ(x) theorem provides no recipe for the computation of E HK goal: calculate energy functional from microscopic interactions March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 3

5 1+1 dimensional Model: Spin-1/2 Fermions in a Trap ultimate goal: study realistic nuclei using short-range repulsive and long-range attractive interaction here: study simplified model of trapped fermions with a contact interaction: U σσ (x, x ) = u 0 δ(x x )(1 δ σ,σ ) u 0 > 0: repulsive interaction 1+1 d model is experimentally accessible ( ultracold gases, cf. e.g. group of S. Jochim) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 4

6 1+1 dimensional Model: Spin-1/2 Fermions in a Trap classical action: S = σ τ x σ,σ ψ σ (τ, x) [ τ x ω2 x 2 ] ψ σ (τ, x) τ x x ψ σ (τ, x)ψ σ (τ, x ) U σσ (x, x ) ψ σ (τ, x )ψ σ (τ, x) contact interaction: U σσ (x, x ) = u 0 δ(x x )(1 δ σ,σ ) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 5

7 1+1 dimensional Model: Spin-1/2 Fermions in a Trap idea: introduce control parameter λ: [ S λ = ψ σ (τ, x) τ x + 1 ] 2 ω2 x 2 ψ σ (τ, x) σ + λ 1 2 τ x σ,σ τ x x ψ σ (τ, x)ψ σ (τ, x ) U σσ (x, x ) ψ σ (τ, x )ψ σ (τ, x) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 5

8 1+1 dimensional Model: Spin-1/2 Fermions in a Trap partition function: Z λ [{J σ }] = Dψ Dψ e S λ+ σ τ x J σ(τ,x)ψ σ (τ,x)ψ σ (τ,x) e W λ[{j σ}] effective action (density functional): { Γ λ [{ρ σ }] = sup W λ [{J σ }] + {J σ} τ x J σ (τ, x)ψ σ (τ, x)ψ σ (τ, x) } density: ρ σ (τ, x) = δw λ[{j σ }] δj σ (τ, x) = ψ σ (τ, x)ψ σ (τ, x) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 6

9 Effective Action: Flow Equation for the Density Functional exact flow equation: ( ) 1 λ Γ λ [{ρ σ }] = 1 δ ρ 2 Γ 2 σ U σσ ρ σ + Tr U σσ λ δρ σ δρ σ σ,σ March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 7

10 Effective Action: Flow Equation for the Density Functional exact flow equation: ( ) 1 λ Γ λ [{ρ σ }] = 1 δ ρ 2 Γ 2 σ U σσ ρ σ + Tr U σσ λ δρ σ δρ σ σ,σ expand the density functional Γ λ about the ground state density ρ gs : Γ λ [{ρ σ }] = Γ (0) λ (ρ σ ρ gs,λ,σ ) Γ (2) λ,σ,σ (ρ σ ρ gs,λ,σ ) +... definition of the ground state: Γ (1) λ = δγ λ[{ρ σ }] δρ σ (τ, x) {ρ σ=ρ gs,σ} = 0 March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 7

11 Flow Equations Leading Order obtain flow equations by comparison of the coefficients: λ Γ (0) λ = 1 2 σ,σ λ ρ gs,λ,σ = [ ( ) ] 1 ρ gs,λ,σ U σ,σ ρ gs,λ,σ + Tr U σ,σ Γ (2) λ,σ,σ σ,σ ρ gs,λ,σ U σ,σ ( ) λ Γ (2) λ,σ,σ = U σ,σ +O Γ (3) λ, Γ(4) λ ( ) 1 ( Γ (2) λ,σ,σ +O calculate the ground state energy from the lowest expansion coefficient Γ (0) λ ( ) 1 Γ (2) λ,σ,σ relates to the density-density correlator ρλ,σ ρ λ,σ solve by discretizing space and time coordinates Γ (3) λ ) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 8

12 First Results 2 Fermions in a Trap Energy E gs E gs /ω pert. theory LO exact DFT-RG u 0 / ω exact result: [Busch et.al.,1997 Foundations of Physics] March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 9

13 First Results 2 Fermions in a Trap Density ρ gs, non-interacting exact u 0 =1 DFT-RG u 0 =1 0.4 ρ (x)/ ω x exact result: [Busch et.al.,1997 Foundations of Physics] March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 10

14 First Results 2 Fermions in a Trap ρ (x) ρ (y) 0.3 non-interacting exact u 0 =1 DFT-RG u 0 =1 <ρ (x)ρ (0)>/ω x exact result: [Busch et.al.,1997 Foundations of Physics] March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 11

15 First Results From Few to Many Body 8 Fermions: non-interacting DFT-RG u 0=1 0.8 non-interacting DFT-RG u 0= ρ (x)/ ω <ρ (x)ρ (0)>/ω x x study different systems (spin and/or mass imbalanced systems,... ) March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 12

16 Conclusion and Outlook computation of ground state properties via the density functional is possible systematic expansion of the density functional Γ λ in n-point functions strongly coupled systems can be described accurately study different systems (spin and/or mass imbalanced systems,... ) study effect of Γ (3) λ and Γ(4) λ in the equations for the 1+1 d fermionic system study self-bound system in 1+1 dimensions extending the method to 3+1 dimensional systems long term: nuclei with realistic microscopic interactions March 23rd 2015 DPG Heidelberg TU Darmstadt Institut für Kernphysik Sandra Kemler 13