1. Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments

2. Thanks to … S. Sakai (Dept. Applied Phys. Univ. Tokyo) H. Aoki (Dept. Phys. Univ. Tokyo) K. Held (Max Planck Inst. Stuttgart) A. V. Lukoyanov (Ural State Technical Univ.) V. I. Anisimov (Inst. Metal Phys, Ekaterinburg)

3. R.Arita Outline  Introduction LDA+DMFT Various solvers for DMFT  IPT, NCA, ED, NRG, DDMRG, QMC, …  Conventional QMC (Hirsch-Fye 86) Algorithm Problems  numerically expensive for low T: numerical effort ~ 1/T 3  sign problem in multi-orbital systems: difficult to treat spin flip terms  New QMC algorithms Projective QMC for T→0 calculations (Feldbacher et al 04, Application: Arita et al 07) Application of various perturbation series expansions for Z (Sakai et al 06, Rubtsov et al 05, Werner et al 07)  Introduction LDA+DMFT Various solvers for DMFT  IPT, NCA, ED, NRG, DDMRG, QMC, …  Conventional QMC (Hirsch-Fye 86) Algorithm Problems  numerically expensive for low T: numerical effort ~ 1/T 3  sign problem in multi-orbital systems: difficult to treat spin flip terms  New QMC algorithms Projective QMC for T→0 calculations (Feldbacher et al 04, Application: Arita et al 07) Application of various perturbation series expansions for Z (Sakai et al 06, Rubtsov et al 05, Werner et al 07)

4. R.Arita LDA+DMFT Computational scheme for correlated electron materials Model HamiltoniansDFT/LDA systematic many-body approach input parameters unknown material specific, ab initio fails for strong correlations Anisimov et al 97, Lichtenstein, Katsnelson 98 Dr Aryasetiawan July 25, Prof. Savrasov July 27

5. R.Arita LDA+DMFT  Transition metal oxides LaTiO 3 V 2 O 3, VO 2 (Sr,Ca)VO 3 LiV 2 O 4 (Sr,Ca) 2 RuO 4 Na x CoO 2 Cuprates Manganites …  Transition metals Fe, Ni  Heussler alloys  Organic compounds BEDT-TTF TMTSF  Fullerenes  Nanostructure materials Zeolites  f-electron systems Rare earths: Ce Actinides: Pu … Application to various correlated materials (reviews) Held et al 03, Kotliar et al 06, etc

6. R.Arita  Supplementing LDA with local Coulomb interactions LDA+DMFT Expand Ψ + w.r.t. a localized basis Φ ilm :  Downfolding: LDA → effective low-energy Hamiltonian

7. R.Arita Lattice model: DOS Self Energy LDA+DMFT  Solve model by DMFT Metzner & Vollhardt 89, Georges & Kotliar 92 Effective impurity model: Hybridization F Self Energy Self-consistency: F

8. R.Arita Solvers for the DMFT impurity model  Iterated perturbation theory Perturbation expansion in U  Non-crossing approximation Perturbation expansion in V  Exact diagonalization for small number of host sites Max # of orbitals <2  Numerical renormalization group (logarithmic discretization of host spectrum) Max # of orbitals <2  Dynamical density matrix renormalization group  Quantum Monte Carlo  …

9. R.Arita  Suzuki-Trotter decomposition  Hubbard-Stratonovich transformation for H int  Many-particle system = (free one-particle system + auxiliary field) Monte Carlo sampling Auxiliary-field QMC

10. R.Arita     ,  =L  QMC for the Anderson impurity model ( Hirsch-Fye 86 ) Integrate out the conduction bands G {s} (     ), w {s} Calculate G 0 (     )G {s} (     ), w {s} … Updating: numerical effort ~L 2

11. R.Arita  Numerically expensive for low T: numerical effort ~ 1/T 3 Projective QMC (Feldbacher et al 04) : A new route to T→0  Sign problem in multi-orbital systems: difficult to treat spin flip terms Application of various perturbation series expansions (Rombouts et al, 99) : less severe sign problem  Combination with HF algorithm (Sakai et al, 06)  Continuous time QMC weak coupling expansion (Rubtsov et al, 05) hybridization expansion (Werner et al, 06) norm Z s can be negative: Norm can be small → =0/0 Problems & Recent developments

12. Projective QMC and its application to DMFT calculation

13. R.Arita Projective QMC 0 Interaction Ising fields no interaction   →∞ Feldbacher et al, PRL 93 136405(2004) Conventional QMC Projective QMC Thermal fluctuations effort: ~1/T 3

14. R.Arita Projective QMC Interaction U only in red part for sufficiently large P: Accurate information on G for light red part -  /2  /2  /2+ 

15. R.Arita Application of PQMC to DMFT (1) PQMC Maximum Entropy Method (T=0) DMFT self-consistent loop Problem: How to obtain  (i  )? G(  )→FT→G(i  )? No only G(  ),  P obtained by PQMC PP Calculate G only for  P Large  Extrapolation by Maximum Entropy Method

16. R.Arita M I HF-QMC insulatingmetallic  =16  =40 Application of PQMC to DMFT (2) Single band Hubbard model

17. R.Arita Metallic solution obtained for  =16 (same numerical effort as HF-QMC with  =16) M I PQMC  =16  =40 Application of PQMC to DMFT (2) Single band Hubbard model Application to LDA+DMFT at T→0

18. Application of PQMC to LDA+DMFT for LiV 2 O 4 RA-Held-Lukoyanov-Anisimov PRL 98 166402 (2007)

19. R.Arita Crossover at T*~20K ・ resistivity:  =   + AT 2 with an enhanced A ・ specific heat coefficient: anomalously large  ( T →0)~190mJ/V mol ・ K 2 (Kadowaki-Woods relation satisfied) ・  : broad maximum (Wilson ratio ～ 1.8) cf) CeRu 2 Si 2 ~350mJ/Ce mol ・ K 2 UPt 3 ~420mJ/U mol ・ K 2 heavy mass quasiparticles (m* ～ 25m LDA ) T*T* (Urano et al. PRL85, 1052(2000)) LiV 2 O 4 : 3d heavy Fermion system Incoherenet metal FL(T 2 law)  (T → 0)~ 190mJ/Vmol ・ K 2 CW law at HT S=1/2 per V ion

20. R.Arita (Shimoyamada et al. PRL 96 026403(2006)) PhotoEmission Spectroscopy A sharp peak appears for T<26K  =4meV,  10meV LiV 2 O 4 : 3d heavy Fermion system LDA+DMFT(HF-QMC) (Nekrasov et al, PRB 67 085111 (2003)) T=750K LDA+DMFT(PQMC)

21. R.Arita Results U=3.6, U’=2.4, J=0.6 U U’ U’-J(Hund coupling = Ising) a 1g egeg T=1200K T=300K PQMC T=300K

22. R.Arita 0 0 FAQ A(  ) T→0 00 G(  ) 0  ～ exp (-   0 ) G(  ) 0  Slow-decay component 00 Large T Why can we discuss A(  →0) without calculating G(  →∞) explicitly?

23. R.Arita Results: G(  ) & A(  )

24. Application of perturbation series expansions to QMC ・ Combination with Hirsch-Fye’s algorithm (Sakai, RA, Held, Aoki PRB 74 155102 (2006)) ・ Continuous time QMC weak coupling expansion (Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005)) hybridization expansion (Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006))

25. R.Arita -J sign problem QMC for multi-orbital systems difficult to treat for multi-orbital systems H J : usually neglected ⇒ Non-trivial Suzuki-Trotter decomposition?

26. R.Arita Held-Vollhardt, 98 n=1.25, Bethe-lattice, W=4, U=9, U’=5, J=2 (Ising) J Ising-type vs Heisenberg-type interaction DMFT study for ferromagnetism in the 2-band Hubbard model Ising-type couling: Ferromagnetic instability overestimated Sakai, RA, Held, Aoki 06

27. R.Arita PSE with respect to  V (V: interaction term) (Rombouts et al, 99) Same Algorithm as Hirsch-Fye extention to m>2 straightforward: For spin flip & pair hopping term: PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

28. R.Arita Large U,U’,J becomes large Large L needed 04080 NkNk NkNk 060120 PSE onlyPSE+HF 2-band Hubbard model, n=1.9,  =8, U=4.4, U‘=4, J=0.2, W=2 It is not a good idea to treat all U,U’,J terms as V H 0 +H U +H U‘ +H Ising ≡ H 0 +H 1 → standard HF H J → PSE ( is small for H J ) PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

29. R.Arita Wide region of norm >0.01 Lower T, large J can be explored 2-band, n=2, W=2, U=U’+2J, U’=4 PSE+HF: Conventional HF: (Sakai et al 04) We have to consider s n =±1 for every  n,   For small H J, small number of  n have s n ≠0 Expansion with respect to  H J :  H J  →negative sign problem relaxed Sign problem: less severe PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

30. R.Arita U=1.2, U’=0.8, J=0.2 [eV] Ising-type Hund,  =70 SU(2) Hund + pair hopping,  =40 [Liebsch-Lichtenstein, PRL 84,1591 (2000)] Application to LDA+DMFT calculation for Sr 2 RuO 4 d xy d yz/zx 1 0 -301-2 Energy [eV] SU(2) symmetric 3-band LDA+DMFT

31. R.Arita Continuous time QMC Weak coupling expansion: Perform a random walk in the space of K={k, (arguments of integrals)} (cf. K={auxiliary spins} for Hirsch-Fye scheme) Non-local in time & space Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005)

32. R.Arita Applications Poteryaev et al, cond-mat/0701263 LDA+DMFT study for V 2 O 3 (Ising type of Hund coupling) Correlated Adatom Trimer on a Metal Surface Savkin et al, PRL 94 026402 (2005)

33. R.Arita Hybridization expansion: Continuous time QMC (2) Numerical effort decreases with increasing U Allows access to low T, even at large U Impurity-bath hybridization (~5  U) (~0.5  U) U Matrix size  =100 Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006 ）

34. R.Arita Summary  QMC: A powerful tool for LDA+DMFT, but low T not accessible sign problem in multi-orbital systems …  Recent developments Access to low T, strong coupling, multi-orbital systems  Projective QMC for T→0 calculations  Application of various perturbation series expansions for Z  Future Problems Spatial fluctuations (cluster extensions) Coupling to bosonic baths …  QMC: A powerful tool for LDA+DMFT, but low T not accessible sign problem in multi-orbital systems …  Recent developments Access to low T, strong coupling, multi-orbital systems  Projective QMC for T→0 calculations  Application of various perturbation series expansions for Z  Future Problems Spatial fluctuations (cluster extensions) Coupling to bosonic baths …