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  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. 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/T3 • 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 ofvariousperturbation series expansionsfor Z (Sakai et al 06, Rubtsov et al 05, Werner et al 07)

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

  5. Transition metal oxides LaTiO3 V2O3, VO2 (Sr,Ca)VO3 LiV2O4 (Sr,Ca)2RuO4 NaxCoO2 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 … LDA+DMFT Application to various correlated materials (reviews) Held et al 03, Kotliar et al 06, etc

  6. Supplementing LDA with local Coulomb interactions LDA+DMFT • Downfolding: LDA → effective low-energy Hamiltonian Expand Ψ+ w.r.t. a localized basis Φilm :

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

  8. 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 … Solvers for the DMFT impurity model

  9. Auxiliary-field QMC • Suzuki-Trotter decomposition • Hubbard-Stratonovich transformation for Hint • Many-particle system = (free one-particle system + auxiliary field) Monte Carlo sampling

  10. QMC for the Anderson impurity model ( Hirsch-Fye 86 ) Integrate out the conduction bands Calculate 0<t1,t2<b=1/T, b=LDt G0(t1,t2) G{s}(t1,t2), w{s} G{s}(t1,t2), w{s} … Updating: numerical effort ~L2

  11. norm Problems & Recent developments • Numerically expensive for low T: numerical effort ~ 1/T3 • 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) Zs can be negative: Norm can be small → <A>=0/0

  12. Projective QMC and its application to DMFT calculation

  13. Projective QMC Projective QMC Feldbacher et al, PRL 93 136405(2004) Conventional QMC • Thermal fluctuations • effort: ~1/T3 Interaction Ising fields no interaction t 0 q →∞ →∞

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

  15. Application of PQMC to DMFT (1) • DMFT self-consistent loop Maximum Entropy Method PQMC (T=0) (T=0) Problem: How to obtain S(iw)? G(t)→FT→G(iw)? No only G(t),t<qP obtained by PQMC qP Calculate G only for t<qP Large t: Extrapolation by Maximum Entropy Method

  16. Application of PQMC to DMFT (2) Single band Hubbard model I M HF-QMC b=40 b=16 insulating metallic

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

  18. Application of PQMC to LDA+DMFT for LiV2O4 RA-Held-Lukoyanov-Anisimov PRL 98 166402 (2007)

  19. LiV2O4: 3d heavy Fermion system Incoherenet metal Crossover at T*~20K ・ resistivity: r =r0+AT2 with an enhanced A ・ specific heat coefficient: anomalously large g(T→0)~190mJ/V mol・K2 (Kadowaki-Woods relation satisfied) ・ c: broad maximum (Wilson ratio~1.8) FL(T2 law) g(T→0)~190mJ/Vmol・K2 cf) CeRu2Si2 ~350mJ/Ce mol・K2 UPt3 ~420mJ/U mol・K2 CW law at HT S=1/2 per V ion T* heavy mass quasiparticles (m*~25mLDA) (Urano et al. PRL85, 1052(2000))

  20. LiV2O4: 3d heavy Fermion system PhotoEmission Spectroscopy LDA+DMFT(HF-QMC) (Nekrasov et al, PRB 67 085111 (2003)) (Shimoyamada et al. PRL 96 026403(2006)) T=750K A sharp peak appears for T<26K w=4meV, D~10meV LDA+DMFT(PQMC)

  21. U’ U’-J U (Hund coupling = Ising) PQMC T=300K T=1200K T=300K Results U=3.6, U’=2.4, J=0.6 a1g egp

  22. FAQ Why can we discuss A(w→0) without calculating G(t→∞) explicitly? T→0 Large T A(w) A(w) w0 w0 0 0 G(t) G(t) ~exp(-w0t0) Slow-decay component 0 0 t t

  23. Results: G(t) & A(w)

  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. HJ :usually neglected -J -J QMC for multi-orbital systems • sign problem • difficult to treat for multi-orbital systems ⇒ Non-trivial Suzuki-Trotter decomposition?

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

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

  28. PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006) Large U,U’,J <k> becomes large Large L needed It is not a good idea to treat all U,U’,J termsas V H0+HU+HU‘+HIsing≡ H0+H1 → standard HF HJ→ PSE (<k> is small for HJ) 2-band Hubbard model, n=1.9, b=8, U=4.4, U‘=4, J=0.2, W=2 PSE only PSE+HF Nk Nk 0 40 80 0 60 120

  29. t t PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006) • Sign problem: less severe Conventional HF: Wide region of norm >0.01 2-band, n=2,W=2, U=U’+2J, U’=4 We have to consider sn=±1 for every tn, PSE+HF: (Sakai et al 04) For small HJ, small number of tn have sn≠0 Lower T, large Jcan be explored • Expansion with respect to m-bHJ : • ~ <bHJ>→negative sign problem relaxed

  30. Application to LDA+DMFT calculation for Sr2RuO4 • Ising-type Hund, b=70 • SU(2) Hund + pair hopping, b=40 U=1.2, U’=0.8, J=0.2 [eV] 1 dxy dyz/zx 0 -3 -2 -1 0 1 Energy [eV] [Liebsch-Lichtenstein, PRL 84,1591 (2000)] SU(2) symmetric 3-band LDA+DMFT

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

  32. Applications LDA+DMFT study for V2O3 (Ising type of Hund coupling) Poteryaev et al, cond-mat/0701263 Correlated Adatom Trimer on a Metal Surface Savkin et al, PRL 94 026402 (2005)

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

  34. Summary • QMC: A powerful tool for LDA+DMFT, but • low Tnot 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 • …