Methods for electronic structure calculations with dynamical mean

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

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

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

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 R. Arita

LDA+DMFT Application to various correlated materials (reviews) Held et al 03, Kotliar et al 06, etc o o o Transition metal oxides n La. Ti. O 3 n V 2 O 3, VO 2 n (Sr, Ca)VO 3 n Li. V 2 O 4 n (Sr, Ca)2 Ru. O 4 n Nax. Co. O 2 n Cuprates n Manganites n … Transition metals n Fe, Ni Heussler alloys o o Organic compounds n BEDT-TTF n TMTSF Fullerenes Nanostructure materials n Zeolites f-electron systems n Rare earths: Ce n Actinides: Pu n … R. Arita

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

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

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

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

QMC for the Anderson impurity model ( Hirsch-Fye 86 ) Integrate out the conduction bands 0<t 1, t 2<b=1/T, b=LDt Calculate G 0(t 1, t 2) G{s}(t 1, t 2), w{s} … Updating: numerical effort ~L 2 R. Arita

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

Projective QMC and its application to DMFT calculation

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

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

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

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

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) Application to LDA+DMFT at T→ 0 R. Arita

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

Li. V 2 O 4: 3 d heavy Fermion system l eta m et ren I he nco Crossover at T*~20 K FL(T 2 law) g(T→ 0)~190 m. J/Vmol・K 2 ・ resistivity: r =r 0+AT 2 with an enhanced A ・ specific heat coefficient: anomalously large g(T→ 0)~190 m. J/V mol・K 2 cf) Ce. Ru 2 Si 2 ~350 m. J/Ce mol・K 2 UPt 3 ~420 m. J/U mol・K 2 (Kadowaki-Woods relation satisfied) CW law at HT S=1/2 per V ion ・ c: broad maximum (Wilson ratio～ 1. 8) T* (Urano et al. PRL 85, 1052(2000)) heavy mass quasiparticles (m*～ 25 m. LDA) R. Arita

Li. V 2 O 4: 3 d heavy Fermion system Photo. Emission Spectroscopy (Shimoyamada et al. PRL 96 026403(2006)) LDA+DMFT(HF-QMC) (Nekrasov et al, PRB 67 085111 (2003)) T=750 K A sharp peak appears for T<26 K w=4 me. V, D~10 me. V LDA+DMFT(PQMC) R. Arita

Results U U’ U’-J (Hund coupling = Ising) U=3. 6, U’=2. 4, J=0. 6 a 1 g PQMC T=300 K T=1200 K T=300 K eg p R. Arita

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

Results: G(t) & A(w) R. Arita

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))

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

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 Held-Vollhardt, 98 Ising-type couling: Ferromagnetic instability overestimated Sakai, RA, Held, Aoki 06 R. Arita

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

PSE + Hirsch-Fye QMC Large U, U’, J Sakai, RA, Held, Aoki PRB 74 155102 (2006) <k> becomes large L needed It is not a good idea to treat all U, U’, J terms as V H 0+HU+HU‘+HIsing≡ H 0+H 1 → 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+HF Nk Nk PSE only 0 60 120 0 40 80 R. Arita

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’+2 J, U’=4 t We have to consider sn=± 1 for every tn, PSE+HF: (Sakai et al 04) t For small HJ, small number of tn have sn≠ 0 Lower T, large J can be explored Expansion with respect to m-b. HJ : m~ <b. HJ> →negative sign problem relaxed R. Arita

Application to LDA+DMFT calculation for Sr 2 Ru. O 4 Ising-type Hund, b=70 SU(2) Hund + pair hopping, b=40 U=1. 2, U’=0. 8, J=0. 2 [e. V] 1 dxy dyz/zx 0 -3 [Liebsch-Lichtenstein, PRL 84, 1591 (2000)] -2 -1 0 1 Energy [e. V] SU(2) symmetric 3 -band LDA+DMFT R. Arita

Continuous time QMC Weak coupling expansion: Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005) 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) R. Arita

Applications LDA+DMFT study for V 2 O 3 (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) R. Arita

Continuous time QMC (2) Hybridization expansion: Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006） Impurity-bath hybridization Matrix size b=100 (~5 b. U) (~0. 5 b. U) Numerical effort decreases with increasing U Allows access to low T, even at large U U R. Arita

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