Realistic Materials Simulations Using Dynamical Mean Field Theory

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1 Realistic Materials Simulations sing Dynamical Mean Field Theory Elias Assmann AG Held, Institut für Festkörperphysik, T Wien VSC ser Workshop, Feb Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

2 Drastic approximations ˆ The Theory of Everything is well-known! Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

3 Drastic approximations ˆ The Theory of Everything is well-known... in solid state physics. N H ψ(r 1,..., r }{{ N ) = 1 2 Z α } 2 i r i=1 α,i i R α ψ = Eψ 2 r i =j i r j N electrons Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

4 Drastic approximations ˆ The Theory of Everything is well-known... in solid state physics. N H ψ(r 1,..., r }{{ N ) = 1 2 Z α } 2 i r i=1 α,i i R α ψ = Eψ 2 r i =j i r j N electrons ˆ nfortunately, it is also completely intractable because N is so large ( /cm 3 ). Drastic approximations are called for. Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

5 Drastic approximations ˆ The Theory of Everything is well-known... in solid state physics. N H ψ(r 1,..., r }{{ N ) = 1 2 Z α } 2 i r i=1 α,i i R α ψ = Eψ 2 r i =j i r j N electrons ˆ nfortunately, it is also completely intractable because N is so large ( /cm 3 ). Drastic approximations are called for. Traditionally, two distinct approaches: density-functional theory ( LDA ) apply approximations to full Hamiltonian single-particle model realistic calculations fails when correlations (interactions) among electrons are strong y model Hamiltonians simplify Hamiltonian, hoping to retain essential physics many-body model abstract, simplest possible model strong correlations y dynamical mean-field theory (DMFT) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

6 LDA+DMFT merges the two approaches LDA ˆ realistic calculations ˆ fails for strong correlations model Hamiltonians ˆ simplified, abstract model ˆ full treatment of correlations LDA+DMFT ˆ merger of the two approaches realistic calculation with most important (local) correlation effects included Held, Adv. Phys. 56, 829 (2007), Kotliar et al., Rev. Mod. Phys. 78, 865 (2006) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

7 From lattice models to DMFT t lattice model ˆ lattice with infinitely many sites ˆ electrons hop between sites ˆ and interact when they meet Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

8 From lattice models to DMFT t Σ lattice model ˆ lattice with infinitely many sites ˆ electrons hop between sites ˆ and interact when they meet impurity model ˆ one interacting site ˆ effective bath models other sites ˆ electrons hop between site and bath Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

9 From lattice models to DMFT t DMFT Σ lattice model ˆ lattice with infinitely many sites ˆ electrons hop between sites ˆ and interact when they meet dynamical mean-field theory ˆ maps lattice model to impurity model ˆ (ω): self-energy describes interactions at other sites impurity model ˆ one interacting site ˆ effective bath models other sites ˆ electrons hop between site and bath Georges et al., Rev. Mod. Phys. 68, 13 (1996), Kotliar and Vollhardt, Phys. Today 57, 53 (2004) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

10 DMFT as a computational method ˆ DMFT maps a lattice model to an impurity model ˆ find correct impurity model: self-consistency loop (O(10) iterations) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

11 DMFT as a computational method ˆ DMFT maps a lattice model to an impurity model ˆ find correct impurity model: self-consistency loop (O(10) iterations) solving the impurity model quantum-mechanical many-body problem Hirsch-Fye : map to sum of single-particle problems. det. O(10 5 ) terms. }{{} O(10 3 ) Σ 2 #time steps #orbitals2 terms summed with (quantum) Monte Carlo small matrices [#orbitals #time steps MB] but many (O(10 9 )) operations ˆ efficient matrix updates: M 1 (M + a b) 1 [Sherman-Morrison formula] easy parallelization: multiple Monte Carlo chains not very much communication needed Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

12 Merging LDA + DMFT DMFT model parameters ˆ hopping t input from LDA ˆ interaction free parameter or computable in constrained LDA Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

13 Merging LDA + DMFT DMFT model parameters ˆ hopping t input from LDA ˆ interaction free parameter or computable in constrained LDA Wannier projection ˆ LDA basis is delocalized ( plane waves e ik r ), but DMFT needs localized basis Wannier functions (essentially Fourier transform) program: Wannier90 ˆ interface from WIEN2k: Wien2Wannier ˆ AG Held: Philipp Wissgott Wien2Wannier: Kuneš et al., Comp. Phys. Commun. 181, 1888 (2010) Vasp2Wannier: Franchini et al., arxiv: (2011) LaVO 3 t 2g Wanniers Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

14 New developments at T Wien continuous-time quantum Monte Carlo (CTQMC) ˆ big improvement on Hirsch-Fye method (computationally similar) ˆ idea: expand around exact solution of local problem ˆ treat up to 7 bands, complicated interactions ˆ AG Held: Nico Parragh, Giorgio Sangiovanni Läuchli and Werner, Phys. Rev. B 80, (2009) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

15 New developments at T Wien continuous-time quantum Monte Carlo (CTQMC) ˆ big improvement on Hirsch-Fye method (computationally similar) ˆ idea: expand around exact solution of local problem ˆ treat up to 7 bands, complicated interactions ˆ AG Held: Nico Parragh, Giorgio Sangiovanni Läuchli and Werner, Phys. Rev. B 80, (2009) dynamical vertex approximation (DΓA) ˆ generalization of DMFT: ˆ self-energy (ω) two-particle vertex (ν, ν, ω) ˆ accounts for (some) non-local correlations ˆ computational needs: larger matrices ˆ AG Held: Georg Rohringer, Alessandro Toschi Toschi et al., Phys. Rev. B 75, (2007) Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop / 8

16 New developments at T Wien II Challenge: describe systems with more and more inequivalent constituents LDA+DMFT with inequivalent sites Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop /8

17 New developments at T Wien II Challenge: describe systems with more and more inequivalent constituents LDA+DMFT with inequivalent sites complicated unit cells inequivalent atoms monoclinic LaVO3 Elias Assmann (IFP T Wien) LDA+DMFT VSC Workshop /8

18 New developments at T Wien II Challenge: describe systems with more and more inequivalent constituents LDA+DMFT with inequivalent sites complicated unit cells inequivalent atoms heterostructures inequivalent layers monoclinic LaVO3 Elias Assmann (IFP T Wien) LaVO3 SrTiO3 LDA+DMFT VSC Workshop /8