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reduced density-matrix functional theory: Energies and forces for materials with strong correlations Robert Schade 1 , Ebad Kamil 2 , Peter Bl¨ ochl 1 , Thomas Pruschke 2 1 Institute for Theoretical Physics, Clausthal University of Technology, Germany 2 Institute for Theoretical Physics, Georg-August-University G¨ ottingen, Germany Overview Goals • fast ground-state calculations for correlated electron systems • integration into the infrastructure of existing density-functional theory calculations for structure relaxations and ab-initio molecular dynamics Framework • variational formulation based on reduced density-matrix func- tional theory (rDMFT) • direct evaluation of the density-matrix functional with a DMFT-like local approximation Ensemble rDMFT Reduced ensemble density-matrix functional theory Gilbert [1975], Levy [1979], Lieb [1983] • the main quantity is the one-particle reduced density- matrix ˆ ρ (1) =ˆ c † α ˆ c β • grand canonical potential Ω β,µ ( ˆ h) Ω ˆ W β,µ ˆ h = − 1 β ln Tr e −β ˆ h+ ˆ W −µ ˆ N = min ρ (1) α,β h α,β ρ (1) β,α one−particle terms + F ˆ W β ρ (1) density − matrix functional interaction energy−T·entropy • the density-matrix functional F ˆ W β (ρ (1) ) can be calcu- lated/approximated in several ways: 1. via Legendre-Fenchel (Lieb [1983]) transform of the grand canonical potential with respect to the one-particle operator ˆ h F ˆ W β ρ (1) = max h α,β Ω ˆ W β ˆ h − Tr ˆ hρ (1) 2. via constrained search (Levy [1979]) over an ensemble of many-particle wave functions |Ψ i 〉: F ˆ W β ρ (1) = min P i ,|Ψ i 〉 stat h α,β ,Λ i,j ,λ i P i 〈Ψ i | ˆ W |Ψ i 〉 interaction energy + β −1 i P i ln (P i ) −T ·entropy + α,β h α,β i P i 〈Ψ i | ˆ c † α ˆ c β |Ψ i 〉− ρ (1) β,α density−matrix constraints − i,j Λ j,i ( 〈Ψ i |Ψ j 〉− δ i,j ) orthogonality constraints −λ i P i − 1 probability constraint 3. via a relation to the Luttinger-Ward functional and Greens functions [Bl¨ochl, Pruschke, Potthoff (2013)] F ˆ W β (ρ (1) )= 1 β Tr ρ (1) ln(ρ (1) )+( − ρ (1) ) ln( − ρ (1) ) entropy of a non−interacting electron gas + stat h ′ ,G,Σ Φ LW β (G, ˆ W ) − 1 β ν Tr ln − G(iω ν )i ( h ′ − h + Σ(iω ν ) ) + ( h ′ − h + Σ(iω ν ) ) G(iω ν ) − G(iω ν ) − G(iω ν ) (h ′ − h) h = µ + 1 β ln − ρ (1) ρ (1) G(iω ν )= ( (iω ν + µ) − h ) −1 4. via parametrized approximations like Mueller-functional (M¨ uller [1984]) resp. Power-functional (Sharma et al. [2008]) P. E. Bl¨ochl, C. F. J. Walther, and T. Pruschke. Method to include explicit correlations into density- functional calculations based on density-matrix functional theory. Phys. Rev. B, 84:205101, Nov 2011. P. E. Bl¨ochl, T. Pruschke, and M. Potthoff. Density-matrix functionals from green’s functions. Phys. Rev. B, 88:205139, Nov 2013. doi: 10.1103/PhysRevB.88.205139. T. L. Gilbert. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B, 12:2111– 2120, Sep 1975. E. Kamil, R. Schade, T. Pruschke, and P. E. Bl¨ochl. Reduced density-matrix functionals applied to the Hubbard dimer. ArXiv e-prints, accepted for pub. in PRB, 1509.01985, Sept. 2015. M. Levy. Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. Proceedings of the National Academy of Science, 76:6062–6065, Dec. 1979. A. M. K. M¨ uller. Explicit approximate relation between reduced two- and one-particle density matrices. Phys. Lett., 105A:446, Aug 1984. doi: 10.1016/0375-9601(84)91034-X. S. Sharma, J. K. Dewhurst, N. N. Lathiotakis, and E. K. U. Gross. Reduced density matrix functional for many-electron systems. Phys. Rev. B, 78:201103, Nov 2008. doi: 10.1103/Phys- RevB.78.201103. W. T¨ows and G. M. Pastor. Lattice density functional theory of the single-impurity anderson model: Development and applications. Phys. Rev. B, 83:235101, Jun 2011. doi: 10.1103/Phys- RevB.83.235101. DFT+rDMFT [Bl¨ochl,Walther,Pruschke2011] E 0 (N ) ≈ min |φ n 〉,f n ∈[0,1] stat µ,Λ nm E DFT [{|φ n 〉}, {f n }] + F ˆ W HF [ρ (1) ] − F ˆ W DFT,DC [ρ (1) ] hybrid functional + F ˆ ˜ W [˜ ρ (1) ] − F ˆ ˜ W HF [˜ ρ (1) ] high−level correction − µ n f n − N particle−number constraint − i,j Λ i,j (〈φ i |φ j 〉− δ i,j ) orthogonality constraints with ρ (1) α,β = n 〈χ α |φ n 〉f n 〈φ n |χ β 〉 Local approximation [Bl¨ochl,Walther,Pruschke 2011] Define cluster of interacting orbitals C R and restrict interaction to these clusters ˆ W ≈ R ˆ W R ˆ W R = 1 2 a,b,c,d∈C R U a,b,d,c ˆ c † a ˆ c † b ˆ c c ˆ c d F ˆ W β [ρ (1) ] ≈ F ∑ R ˆ W R β [ρ (1) ] ≈ R F ˆ W R β [ρ (1) ]. Adaptive cluster approximation [Schade, Bl¨ ochl 2016] Main idea: transform density-matrix with a unitary transformation before doing a cluster approximation neglect coupling between interacting orbitals (impu- rity)+effective bath and remaining system in the transformed density matrix ρ (1) T (N = N A · (M + 1), N A interacting orbitals): F ˆ W R [ρ (1) ] ≈ F ˆ W R ACA(M ) [ρ (1) ]= F ˆ W R [T † N ρ (1) T T N ] ⇒ reduced sensitivity to truncation of one-particle basis example: histogram of D 0/T,α,β = ∂F W center [ρ 0/T ] ∂ρ 0/T,αβ of half-filled 3-by-3 Hubbard cluster with U/t =5 −10 0 10 20 30 40 0.001 0.01 0.1 1 2 |D 0,αβ |/t, −|D T,αβ |/t ⇒ rapid convergence to the exact result with increasing truncation level M The ACA can be seen as a generalization of the two-level- approximation by T¨ ows and Pastor [2011] to • multi-band and multi-site interactions (TLA only one-band). • arbitrary truncation parameters M (TLA only M = 1). Corrected adaptive cluster approximation: Additional correction based on parametrized approximate functional F ˆ W ≈ [ρ] to mediate effect of truncation for small M : F ˆ W R cACA(M ) [ρ (1) ]= F ˆ W R ACA(M ) [ρ (1) ] − F ˆ W R ≈ [T † N ρ (1) T T N ]+ F ˆ W R ≈ [ρ (1) T ] Application to a finite single-orbital SIAM: ˆ H = σ ǫ f ˆ f † σ ˆ f σ + U ˆ n f,↑ ˆ n f,↓ + σ,i ǫ i ˆ n i,σ + V N b i,σ ˆ f † σ ˆ c i,σ + cc. (N b = 11, N e = 12, ǫ i = −2t cos(2πn/N b )) Interaction strength dependence (V/t =0.4, ǫ f =0, t> 0): −14.3 −14.2 −14.1 −14 −13.9 0 0.05 0.1 0.15 0.75 1 1.25 1.5 0 2 4 6 8 0 0.5 1 E 0 /t W 0 /t n f U/t m f −20 −10 0 −10 −5 0 5 0 0.01 0.02 0.03 0 2 4 6 8 10 3 ΔE/t 10 3 ΔW/t Δn f U/t Impurity onsite-energy dependence (V/t =0.4, U/t =8, t> 0): −27 −20 −13 0 4 8 0 1 2 −10 −5 0 5 0 0.5 1 0 0.025 0.05 E 0 /t W 0 /t n f ǫ f /t m f ΔE 0 /t −30 −20 −10 0 −300 −200 −100 0 −0.025 0 0.025 −10 −5 0 5 10 3 ΔE/t 10 3 ΔW/t Δn f ǫ f /t Bandwidth dependence (ǫ f /V = −1, U/V =5, V> 0): −300 −200 −100 0 0 1 2 3 4 0.7 1 1.3 1.6 1.9 0 10 20 0 0.5 1 0 0.1 0.2 0.3 E 0 /V W 0 /V n f t/V m f ΔE 0 /V −80 −60 −40 −20 0 −40 −20 0 0 0.01 0.02 0.03 0.04 0 10 20 0 0.02 0.04 0 1 2 10 3 ΔE/V 10 3 ΔW/V Δn f t/V Application to a half-filled 24-site Hubbard ring: • impurity size in the local approximation N A • truncation parameter in the ACA M (N = N A · (M + 1)) Convergence with truncation parameter for exact ground state density matrix (single site local approx.): 0.28 0.29 0.3 0.31 0.32 1 2 3 4 5 6 F ˆ W 1 (c)ACA(M ) [ρ 0 ]/t M Exact results and local approximation with ACA for total energy E 0 , interaction energy W 0 and next-neighbor density matrix elements ρ 12 : −1 −0.5 0 N A =1 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0 10 20 E 0 /(Nt) exact M=1 M=2 M=3 W 0 /(Nt) ρ 12 U/t −1 −0.5 0 N A =2 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0 10 20 E 0 /(Nt) exact M=1 M=2 W 0 /(Nt) ρ 12 U/t HF exact HF ↓ ACA M=3 ↓ cACA M=2 ↓ cACA M=1 ↓ ACA M=2 ACA M=1 HF exact/HF HF HF exact ACA M=1 ACA M=2 cACA M=2 ↑ cACA M=1 HF exact/HF HF exact HF ↓ cACA M=2 ← ACA M=1 ← ACA M=2 ←← cACA M=1 ↑ ACA ↓ cACA
