Dynamical Mean Field Theory and Numerical Renormalization Group at Finite Temperature: Prospects and Challenges

Transcription

1 Dynamical Mean Field Theory and Numerical Renormalization Group at Finite Temperature: Prospects and Challenges Frithjof B. Anders Institut für Theoretische Physik Universität Bremen Göttingen, December Collaborators: C. Grenzebach, G. Czycholl, Bremen Th. Pruschke, R. Peters, Göttingen

2 Dynamical Mean Field Theory and Numerical Renormalization Group at Finite Temperature: Prospects and Challenges Frithjof B. Anders Institut für Theoretische Physik Universität Bremen Göttingen, December Collaborators: C. Grenzebach, G. Czycholl, Bremen Th. Pruschke, R. Peters, Göttingen

3 Outline 1 Introduction to the Dynamical Mean Field Theory (DMFT) 2 NRG as DMFT impurity solver Numercial renormalization group (NRG) Spectral functions at finite temperatures 3 Results for model Hamiltonians Falicov-Kimball Model: f-green s Function Hubbard model: metall-insulator transition The Periodic SU(N) Anderson model 4 Conclusion

4 Contents 1 Introduction to the Dynamical Mean Field Theory (DMFT) 2 NRG as DMFT impurity solver Numercial renormalization group (NRG) Spectral functions at finite temperatures 3 Results for model Hamiltonians Falicov-Kimball Model: f-green s Function Hubbard model: metall-insulator transition The Periodic SU(N) Anderson model 4 Conclusion

5 Metal-Isolator Transition in V 2 O 3 Transition metal oxids: insulator = metal correlation driven pressure reduces U/W on-site U W How do describe this phase transition quantitatively? D. B. McWhan et al, PRB 7, 1920 (1973)

6 Metal-Isolator Transition in V 2 O 3 Transition metal oxids: insulator = metal correlation driven pressure reduces U/W on-site U W How do describe this phase transition quantitatively? D. B. McWhan et al, PRB 7, 1920 (1973)

7 Specific heat of the heavy fermion alloy Ce x La 1 x Cu 6 γ 1J/K 2 mol specific heat C m : local dynamics incomplete filled f -shell: heavy-fermion properties U/W large Onuki et al 1987

8 Paradigm model: the Hubbard model n i ˆn i n i ˆn i U/W > 1: mean-field theory insufficient H = t <i,j>σ c iσ c jσ + U i ˆn i ˆn i

9 Correlated Electron Systems Combination of The challenge independent quasi-particle propagation: large local Coulomb repulsion quantitative description of metal/insulator magnetic (FM/AFM) orbital superconducting structural phase transitions

10 = lattice problem Mapping the lattice problem onto an effective site problem (SIAM) plus dynamical bath (DMFT) Kuramoto 85, Grewe 87, Brand,Mielsch 89, Jarrell 92, Kotliar, Georges 92,...

11 ~ G(z) dynamical bath = lattice problem Mapping the lattice problem onto an effective site problem (SIAM) plus dynamical bath (DMFT) Kuramoto 85, Grewe 87, Brand,Mielsch 89, Jarrell 92, Kotliar, Georges 92,...

12 Σ σ (ω, k) Σ σ (ω) The local approximation: DMFT exact for d (Metzner, Vollhardt, Müller-Hartmann 89) SCC: G latt (z) = G SIAM loc plus bath G(z): (simple) G 1 latt (z) + Σ(z) = G 1 (z) solution of the effective site: (difficult)

13 Σ σ (ω, k) Σ σ (ω) The local approximation: DMFT exact for d (Metzner, Vollhardt, Müller-Hartmann 89) SCC: G latt (z) = G SIAM loc plus bath G(z): (simple) G 1 latt (z) + Σ(z) = G 1 (z) solution of the effective site: (difficult)

14 Σ σ (ω, k) Σ σ (ω) The local approximation: DMFT exact for d (Metzner, Vollhardt, Müller-Hartmann 89) SCC: G latt (z) = G SIAM loc plus bath G(z): (simple) G 1 latt (z) + Σ(z) = G 1 (z) solution of the effective site: (difficult)

15 The challenge: the impurity solver non-crossing approximation plus extensions (NCA,PNCA,ENCA,CTMA,...) quantum Monte Carlo (QMC) exact diagonalization (ED) iterative perturbation theory (ITP) modified perturbation theory (MPT) local moment approximation (LMA) density matrix renormalization group (DMRG) numerical renormalization group (NRG)

16 The challenge: the impurity solver non-crossing approximation plus extensions (NCA,PNCA,ENCA,CTMA,...) quantum Monte Carlo (QMC) exact diagonalization (ED) iterative perturbation theory (ITP) modified perturbation theory (MPT) local moment approximation (LMA) density matrix renormalization group (DMRG) numerical renormalization group (NRG)

17 Contents 1 Introduction to the Dynamical Mean Field Theory (DMFT) 2 NRG as DMFT impurity solver Numercial renormalization group (NRG) Spectral functions at finite temperatures 3 Results for model Hamiltonians Falicov-Kimball Model: f-green s Function Hubbard model: metall-insulator transition The Periodic SU(N) Anderson model 4 Conclusion

18 Numercial renormalization group (NRG) Impurity Wilson s NRG (1975) Bath continuum ξ 1 impurity: arbitrary complex local atom discretization of the bath continuum ρ σ (ω) = Im G σ (ω) Householder transformation: w mσ [ ρ σ (ω)], ξ mσ [ ρ(ω)] H bath = m=0,σ σ f 0σ = dε ρ σ (ω) c εσ ( )] [w mσ f mσf mσ + ξ mσ f mσf m+1σ + f m+1σ f mσ ξ m, w m Λ m/2 : multi-precision package (GMP) needed

19 Numercial renormalization group (NRG) Wilson s NRG (1975) 0 z (z+1) Λ Λ Λ 0 Λ (z+1) Λ z Λ 0 impurity: arbitrary complex local atom discretization of the bath continuum ρ σ (ω) = Im G σ (ω) Householder transformation: w mσ [ ρ σ (ω)], ξ mσ [ ρ(ω)] H bath = m=0,σ σ f 0σ = dε ρ σ (ω) c εσ ( )] [w mσ f mσf mσ + ξ mσ f mσf m+1σ + f m+1σ f mσ ξ m, w m Λ m/2 : multi-precision package (GMP) needed

20 Numercial renormalization group (NRG) Wilson s NRG (1975) 0 z (z+1) Λ Λ Λ 0 Λ (z+1) Λ z Λ 0 impurity: arbitrary complex local atom discretization of the bath continuum ρ σ (ω) = Im G σ (ω) Householder transformation: w mσ [ ρ σ (ω)], ξ mσ [ ρ(ω)] H bath = m=0,σ σ f 0σ = dε ρ σ (ω) c εσ ( )] [w mσ f mσf mσ + ξ mσ f mσf m+1σ + f m+1σ f mσ ξ m, w m Λ m/2 : multi-precision package (GMP) needed

21 Numercial renormalization group (NRG) Wilson s NRG (1975) 0 z (z+1) Λ Λ Λ 0 Λ (z+1) Λ z Λ 0 impurity: arbitrary complex local atom discretization of the bath continuum ρ σ (ω) = Im G σ (ω) Householder transformation: w mσ [ ρ σ (ω)], ξ mσ [ ρ(ω)] H bath = m=0,σ σ f 0σ = dε ρ σ (ω) c εσ ( )] [w mσ f mσf mσ + ξ mσ f mσf m+1σ + f m+1σ f mσ ξ m, w m Λ m/2 : multi-precision package (GMP) needed

22 Numercial renormalization group (NRG) Wilson s NRG (1975) Impurity N ξ 1 ξ 2 ξ 3 ξ Ν Λ Ν/2 discretization of the bath continuum switching on iteratively the couplings ξ m Λ m/2 recursion relation (RG transformation) H N+1 = ΛH N + ) ξ N (f Nσ f N+1σ + f N+1σ f Nσ σ RG: elimination of the high energy states temperature: T m Λ m/2 stop at chain length N, when desired T N reached

23 Numercial renormalization group (NRG) Wilson s NRG (1975) Impurity N ξ 1 ξ 2 ξ 3 ξ Ν Λ Ν/2 discretization of the bath continuum switching on iteratively the couplings ξ m Λ m/2 recursion relation (RG transformation) H N+1 = ΛH N + ) ξ N (f Nσ f N+1σ + f N+1σ f Nσ σ RG: elimination of the high energy states temperature: T m Λ m/2 stop at chain length N, when desired T N reached

24 Numercial renormalization group (NRG) Wilson s NRG (1975) Impurity N ξ 1 ξ 2 ξ 3 ξ Ν Λ Ν/2 discretization of the bath continuum switching on iteratively the couplings ξ m Λ m/2 recursion relation (RG transformation) H N+1 = ΛH N + ) ξ N (f Nσ f N+1σ + f N+1σ f Nσ σ RG: elimination of the high energy states temperature: T m Λ m/2 stop at chain length N, when desired T N reached

25 Numercial renormalization group (NRG) Wilson s NRG (1975) Impurity N ξ 1 ξ 2 ξ 3 ξ Ν Λ Ν/2 discretization of the bath continuum switching on iteratively the couplings ξ m Λ m/2 recursion relation (RG transformation) H N+1 = ΛH N + ) ξ N (f Nσ f N+1σ + f N+1σ f Nσ σ RG: elimination of the high energy states temperature: T m Λ m/2 stop at chain length N, when desired T N reached

26 Numercial renormalization group (NRG) Wilson s NRG (1975) Impurity N ξ 1 ξ 2 ξ 3 ξ Ν Λ Ν/2 discretization of the bath continuum switching on iteratively the couplings ξ m Λ m/2 recursion relation (RG transformation) H N+1 = ΛH N + ) ξ N (f Nσ f N+1σ + f N+1σ f Nσ σ RG: elimination of the high energy states temperature: T m Λ m/2 stop at chain length N, when desired T N reached

27 Spectral functions at finite temperatures Spectral functions at finite temperatures Assumption: solve the Wilson chain exactly, i.e H N n = E n n Then: Lehmann representation of ρ(ω) (text book) ρ A,B (ω) = ( e βe n + e βem) A nm B mn δ(ω + E n E m ) Z n,m The challenge 1 discrete spectrum = continous ρ(ω) 2 how do we gather the information from different iterations? 3 how do we guarantee the sum-rules? n σ (T ) = dω ρ σ (ω) f (ω)

28 Spectral functions at finite temperatures Spectral functions at finite temperatures Assumption: solve the Wilson chain exactly, i.e H N n = E n n Then: Lehmann representation of ρ(ω) (text book) ρ A,B (ω) = ( e βe n + e βem) A nm B mn δ(ω + E n E m ) Z n,m The challenge 1 discrete spectrum = continous ρ(ω) 2 how do we gather the information from different iterations? 3 how do we guarantee the sum-rules? n σ (T ) = dω ρ σ (ω) f (ω)

29 Spectral functions at finite temperatures Spectral functions at finite temperatures Assumption: solve the Wilson chain exactly, i.e H N n = E n n Then: Lehmann representation of ρ(ω) (text book) ρ A,B (ω) = ( e βe n + e βem) A nm B mn δ(ω + E n E m ) Z n,m The challenge 1 discrete spectrum = continous ρ(ω) 2 how do we gather the information from different iterations? 3 how do we guarantee the sum-rules? n σ (T ) = dω ρ σ (ω) f (ω)

30 Spectral functions at finite temperatures Spectral functions at finite temperatures Assumption: solve the Wilson chain exactly, i.e H N n = E n n Then: Lehmann representation of ρ(ω) (text book) ρ A,B (ω) = ( e βe n + e βem) A nm B mn δ(ω + E n E m ) Z n,m The challenge 1 discrete spectrum = continous ρ(ω) 2 how do we gather the information from different iterations? 3 how do we guarantee the sum-rules? n σ (T ) = dω ρ σ (ω) f (ω)

31 Spectral functions at finite temperatures Spectral functions at finite temperatures Assumption: solve the Wilson chain exactly, i.e H N n = E n n Then: Lehmann representation of ρ(ω) (text book) ρ A,B (ω) = ( e βe n + e βem) A nm B mn δ(ω + E n E m ) Z n,m The challenge 1 discrete spectrum = continous ρ(ω) 2 how do we gather the information from different iterations? 3 how do we guarantee the sum-rules? n σ (T ) = dω ρ σ (ω) f (ω)

32 Spectral functions at finite temperatures The answer 1 continous ρ(ω) = broadening of δ(ω ω n ): δ(ω ω n ) = B(ω, ω n ) = e b2 /4 bω n π e [ ln( ω ωnm )]2 b 2 Using the equation of motion (Bulla, Hewson, Pruschke 1998) yields with H = H 0 + V z G A,B (z) = {A, B} + G [A,H],B (z) Σ A,B (z) = G [A,V ],B(z) G A,B (z) [A,V ],B (z) GA,B NRG (z) G NRG

33 Spectral functions at finite temperatures The answer 1 continous ρ(ω) = broadening of δ(ω ω n ): δ(ω ω n ) = B(ω, ω n ) = e b2 /4 bω n π e [ ln( ω ωnm )]2 b 2 Using the equation of motion (Bulla, Hewson, Pruschke 1998) yields with H = H 0 + V z G A,B (z) = {A, B} + G [A,H],B (z) Σ A,B (z) = G [A,V ],B(z) G A,B (z) [A,V ],B (z) GA,B NRG (z) G NRG

34 Spectral functions at finite temperatures The answer 1 continous ρ(ω) = broadening of δ(ω ω n ): δ(ω ω n ) = B(ω, ω n ) = e b2 /4 bω n π e [ ln( ω ωnm )]2 b 2 Using the equation of motion (Bulla, Hewson, Pruschke 1998) yields with H = H 0 + V z G A,B (z) = {A, B} + G [A,H],B (z) Σ A,B (z) = G [A,V ],B(z) G A,B (z) [A,V ],B (z) GA,B NRG (z) G NRG

35 Spectral functions at finite temperatures The answer 1 continous ρ(ω) = broadening of δ(ω ω n ): δ(ω ω n ) = B(ω, ω n ) = e b2 /4 bω n π e [ ln( ω ωnm )]2 b 2 Using the equation of motion (Bulla, Hewson, Pruschke 1998) yields with H = H 0 + V z G A,B (z) = {A, B} + G [A,H],B (z) Σ A,B (z) = G [A,V ],B(z) G A,B (z) [A,V ],B (z) GA,B NRG (z) G NRG 2 patching excitations from diff. iterations (Costi et al 94,...)

36 Spectral functions at finite temperatures The answer 1 continous ρ(ω) = broadening of δ(ω ω n ): δ(ω ω n ) = B(ω, ω n ) = e b2 /4 bω n π e [ ln( ω ωnm )]2 b 2 Using the equation of motion (Bulla, Hewson, Pruschke 1998) yields with H = H 0 + V z G A,B (z) = {A, B} + G [A,H],B (z) Σ A,B (z) = G [A,V ],B(z) G A,B (z) [A,V ],B (z) GA,B NRG (z) G NRG 2 using a complete basis set of the NRG chain

37 Spectral functions at finite temperatures The complete basis set of the NRG Impurity N Environment e e> l,e,1> l,e,2> l,e,3> complete basis: { e } = { α imp, α 1, α 2, α 3, α 4,, α N }

38 Spectral functions at finite temperatures The complete basis set of the NRG Impurity N ξ Environmente 1 e> k,e,1> k,e,1> complete basis: { e } = { k, e; 1 } l,e,1> l,e,2> l,e,3>

39 Spectral functions at finite temperatures The complete basis set of the NRG Impurity N ξ Environment e 1 ξ 2 e> l,e,2> k,e,2> l,e,2> l,e,3> complete basis: { e } = { k, e; 2 } + { l, e; 2 }

40 Spectral functions at finite temperatures The complete basis set of the NRG Impurity N ξ 1 ξ 2 ξ 3 e e> l,e,2> l,e,3> k,e,3> complete basis: { e } = { k, e; 3 } + 3 m=2 { l, e; m }

41 Spectral functions at finite temperatures The complete basis set of the NRG Impurity N ξ ξ ξ ξ Ν e> l,e,2> complete basis: { e } = N m=2 { l, e; m } l,e,3> l,e,n>

42 Spectral functions at finite temperatures Sum rule conserving NRG Green functions G A,B (z) = N m=m min + N m=m min A l,k (m)ρ red k,k (m)b k,l(m) z + E l E k l l k,k B l,k (m)ρ red k,k (m)a k,l(m) z + E k E l k,k reduced density matrix (Feynman 72, White 92, Hofstetter 2000) ρ red k,k (m) = e k, e; m ˆρ k, e; m, Weichelbaum, von Delft: cond-mat/ Peters, Pruschke, FBA, cond-mat/ , PRB in press

43 Spectral functions at finite temperatures advantages of the NRG solver non-perturbative arbitrary local interaction strength arbitrary low temperatures inside into the nature of possible phase transitions challenges of the NRG solver CPU time scales exponential with the number of baths frequency resolution limited to ω > T weak Λ and broadening dependency of the results: violation of causality in Σ(ω) due to numerical errors possible

44 Spectral functions at finite temperatures advantages of the NRG solver non-perturbative arbitrary local interaction strength arbitrary low temperatures inside into the nature of possible phase transitions challenges of the NRG solver CPU time scales exponential with the number of baths frequency resolution limited to ω > T weak Λ and broadening dependency of the results: violation of causality in Σ(ω) due to numerical errors possible

45 Contents 1 Introduction to the Dynamical Mean Field Theory (DMFT) 2 NRG as DMFT impurity solver Numercial renormalization group (NRG) Spectral functions at finite temperatures 3 Results for model Hamiltonians Falicov-Kimball Model: f-green s Function Hubbard model: metall-insulator transition The Periodic SU(N) Anderson model 4 Conclusion

46 Falicov-Kimball Model: f-green s Function Falicov-Kimball model H = t <i,j> c i c j + i ε f f i f i + U i n f i n c i exact solution for the c-electrons in d ([H, n f i ] = 0) simple: (Brand, Mielsch, 1989) no closed expression for the f -Green function, difficult reason: [H, n f i ] = 0 X-ray threshhold problem (Georges, 1992) missing in the evaluation of real time Keldysh-equations by Freericks and Zlatic

47 Falicov-Kimball Model: f-green s Function Falicov-Kimball model H = t <i,j> c i c j + i ε f f i f i + U i n f i n c i exact solution for the c-electrons in d ([H, n f i ] = 0) simple: (Brand, Mielsch, 1989) no closed expression for the f -Green function, difficult reason: [H, n f i ] = 0 X-ray threshhold problem (Georges, 1992) missing in the evaluation of real time Keldysh-equations by Freericks and Zlatic

48 Falicov-Kimball Model: f-green s Function Falicov-Kimball model H = t <i,j> c i c j + i ε f f i f i + U i n f i n c i exact solution for the c-electrons in d ([H, n f i ] = 0) simple: (Brand, Mielsch, 1989) no closed expression for the f -Green function, difficult reason: [H, n f i ] = 0 X-ray threshhold problem (Georges, 1992) missing in the evaluation of real time Keldysh-equations by Freericks and Zlatic

49 Falicov-Kimball Model: f-green s Function Falicov-Kimball model: NRG solution for the f -spectrum ρ f (ω) α U (FBA and G. Czycholl, PRB 2005) ρ(ω) ω α α < ω

50 Falicov-Kimball Model: f-green s Function Falicov-Kimball model: NRG solution for the f -spectrum ρ f (ω) α (FBA and G. Czycholl, PRB 2005) Vρ eff (0) ρ(ω) ω α α < ω

51 Hubbard model: metall-insulator transition Hubbard model: Mott-Hubbard insolator transition H = t <i,j>σ c iσ c jσ + U i ˆn i ˆn i Hubbard 1963, Gutzwiller 1963 scaling of t for d : Metzner, Vollhardt (1989) effective site, DMFT approaches Pruschke 1988 (PhD) (LNCA) Jarrell 1992 metall-insulator transition: Georges and Kotliar 1992 (ITP), Bulla 1999 (NRG) ferromagnetism: Ulmke et al 1994, Pruschke, Zitzler 2002 multiband: Held, Vollhardt,...

52 Hubbard model: metall-insulator transition One-band Hubbard model: Mott-Hubbard insolator transition ρ(ω) T = 0 U = 2 U = 3.5 U = 4 U = 6 ρ ω T/W IPT U c1, NRG U c2, NRG U c1, QMC U c2, QMC U/W Bulla, Costi, Vollhardt, PRB 64, (2001)

53 Hubbard model: metall-insulator transition One-band Hubbard model: Mott-Hubbard insolator transition ρ(ω) T = 0.12 U = 1 U = 2 U = 3 U = 3.5 U = 4 U = ω T/W IPT U c1, NRG U c2, NRG U c1, QMC U c2, QMC U/W Bulla, Costi, Vollhardt, PRB 64, (2001)

54 Hubbard model: metall-insulator transition One-band Hubbard model: Mott-Hubbard insolator transition ρ(ω) U = 3.5 T = 10 T = 0.5 T = T = T = ω T/W IPT U c1, NRG U c2, NRG U c1, QMC U c2, QMC U/W Bulla, Costi, Vollhardt, PRB 64, (2001)

55 Hubbard model: metall-insulator transition One-band Hubbard model: Mott-Hubbard insolator transition ρ(ω) U = 3.5 T = T = ω T/W IPT U c1, NRG U c2, NRG U c1, QMC U c2, QMC U/W Bulla, Costi, Vollhardt, PRB 64, (2001)

56 Hubbard model: metall-insulator transition One-band Hubbard model: Mott-Hubbard insolator transition R(T) [µωcm] U = 4 U = 3.5 U = T T/W IPT U c1, NRG U c2, NRG U c1, QMC U c2, QMC U/W Bulla, Costi, Vollhardt, PRB 64, (2001)

57 The Periodic SU(N) Anderson model The Periodic SU(N) Anderson model H = i + im N ε f mf i,m f i,m + U N i n m ( ) V m f i,m c i,m + c i,m f i,m m=1 ˆn f i,mˆn f i,n t <i,j>m c im c jm CEF Levels two f -shell doublets broken SU(4) symmetry ε f 1,2 = ε f ε f 3,4 = ε f + cef

58 The Periodic SU(N) Anderson model The Periodic SU(N) Anderson model H = i + im N ε f mf i,m f i,m + U N i n m ( ) V m f i,m c i,m + c i,m f i,m m=1 ˆn f i,mˆn f i,n t <i,j>m c im c jm CEF Levels two f -shell doublets broken SU(4) symmetry ε f 1,2 = ε f ε f 3,4 = ε f + cef

59 The Periodic SU(N) Anderson model renormalized band structure ω ε Grenzebach, FBA, Czycholl, Pruschke, PRB 74, (2006)

60 The Periodic SU(N) Anderson model Transport properties of the periodic SU(2) Anderson model ρ(t) [µωcm] U = 12 U = 11 U = 10 U = 9 U = 8 U = 7 U = T Grenzebach, FBA, Czycholl, Pruschke, PRB 74, (2006)

61 The Periodic SU(N) Anderson model Transport properties of the periodic SU(2) Anderson model ρ(t) ρ(t 0 /100) + 5 [µωcm] U = 12 U = 11 U = 10 U = 9 U = 8 U = 7 U = T/T 0 Grenzebach, FBA, Czycholl, Pruschke, PRB 74, (2006)

62 The Periodic SU(N) Anderson model Novel results: the PAM with two CEF doublets

63 The Periodic SU(N) Anderson model Novel results: the PAM with two CEF doublets 5 medium at T=1*10-5 Γ m (ω) ω Γ 1,2 Γ 3, ω

64 Contents 1 Introduction to the Dynamical Mean Field Theory (DMFT) 2 NRG as DMFT impurity solver Numercial renormalization group (NRG) Spectral functions at finite temperatures 3 Results for model Hamiltonians Falicov-Kimball Model: f-green s Function Hubbard model: metall-insulator transition The Periodic SU(N) Anderson model 4 Conclusion

65 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands

66 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands

67 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands

68 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands

69 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands

70 Conclusion The NRG accurate impurity solver for DMFT useful of phase transitions (FM/AFM, orbital ordering) applied to three different model finite temperature and T 0 accessible more realistic models possible with new GF algorithm Drawback spectral resolution limited to ω > T good enough for transport? scaling exponentially with # of bands