Fluctuating exchange theory of dynamical electron correlations and magnetism

Transcription

1 Fluctuating exchange theory of dynamical electron correlations and magnetism Václav Drchal Institute of Physics ASCR, Praha, Czech Republic Grant Agency of ASCR: project IAA11616 Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.1

2 INTRODUCTION DFT inappropriate for ground states of correlated systems such as transition metal oxides rare earths and their compounds actinides and their compounds DFT is also inappropriate for excited states (spectroscopies) correlation effects have to be included beyond the DFT Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.2

3 OUTLINE Multiband Hubbard Hamiltonian Solution of the many-body problem FLEX approximation Computational procedure Results and discussion Conclusions and outlook Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.3

4 ( ( ( +, ( ( ) MULTIBAND HUBBARD HAMILTONIAN $ $!#" '& '& '& &... energy bands * ) '& '& / -#. 3 3 & 2 1 & & ) & ) 1 2) ) 2 1 ) lattice site,... spinorbital Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.4

5 ) ) ( ) ( : 9 8 ) : GREEN FUNCTIONS... Heisenberg picture < )#; < )#; Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p. 4 ) 6 ) 7 ) 21 4 )

6 = B B G = = G F F G G F F = = = = PERTURBATION THEORY AT T= K A >@? E CD E CD A >@? A >@? A >@?... ground state of the non-interacting system (with Hamiltonian ). Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.6

7 = PERTURBATION THEORY AT T= K cont d Vocabulary for diagrams: ig ig -iu -iv... GF of noninteracting system... GF of interacting system... pair interaction... antisymmetrized pair interaction T eh -it eh... T-matrix Σ eh -iσ eh... selfenergy Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.7

8 FLEX N. Bickers and D. Scalapino, Ann. Phys. (N.Y.) 193, 26 (1989). Hartree-Fock electron-electron channel Σ HFA = + T ee = + T ee second order Σ ee = T ee + T ee Σ (2) = + Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.8

9 I H I H M FLEX: electron-hole channel antisymmetrized interaction GW (vertical channel) = - T v = + T v 3L & 3KJ RPA Σ v = T v - features T eh = + T eh Σ eh = T eh - conserving approximation infinite lifetime at Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.9

10 SELECTED REFERENCES DMFT A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). LDA+DMFT G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 86 (26). FLEX on real energy axis Yu. A. Babanov, V. Ye. Naish, O. B. Sokolov, and V. K. Finashkin: Fizika Metallov i Metallovedenije 32 (1971) , 3 (1973) , 3 (1973) tests of FLEX for multiband model V. Drchal et al., J. Phys. Cond. Matter 17 (2), FLEX+DMFT for actinide compounds, spin-orbit L. V. Pourovskii, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. 72, 1116 (2). Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.1

11 NR N P correlation effects for d-states only OUR APPROACH local approximation for selfenergy, i.e., NOQP R O NR NOQP R O NR reasonable for 3-dim systems, while for 2-dim and 1-dim systems is probably not sufficient perturbation theory: for simplicity FLEX (which depends only on one energy argument) sum up one or more classes of diagrams and derive equations for causal quantities GFs and for selfenergies the equations are transformed into equations for retarded quantities which have simpler analytic properties subtract contributions of first and second order to avoid double counting energy representation: energies along the axis of real energies assume Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.11

12 EQUATIONS FOR CAUSAL QUANTITIES for example e-e channel:... two-particle propagator... e-e T-matrix... e-e selfenergy Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.12

13 6 6 4 S T T S T EQUATIONS FOR RETARDED QUANTITIES... Hilbert transform Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.13

14 : : SELFCONSISTENCY selfenergy only for chemical potential is constructed from unperturbed, selfconsistency selfenergy is constructed from the perturbed GF DMFT selfconsistency: selfenergy GF defined as is constructed from the bath DMFT+LDA selfconsistency: as the DMFT, but the perturbed GF is used to calculate charge density used in the LDA Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.14

15 Y : Y : XW VU Z [ MANY-BODY THEORY & TB-LMTO METHOD 9 '& 9 '& Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.1

16 NUMERICAL IMPLEMENTATION cubic metals bcc (Fe) 1632 k-points in the IBZ fcc (Co, Ni, Cu) 248 k-points in the IBZ experimental lattice constant Ceperley-Alder exchange-correlation energy as parameterized by Perdew and Zunger many-body equations solved using Broyden method [Ry] J [Ry] bcc-fe.12.8 fcc-ni Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.16

17 ] \ ] fcc NICKEL ferromagnetic fcc Ni 3 2 ferromagnetic fcc Ni DMFT+FLEX U=.18, J=.12 Ry majority spin SELFENERGY majority spin DOS (states/ry) SELFENERGY minority spin minority spin ENERGY (Ry) ENERGY (Ry) Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.17

18 ] \ ] bcc IRON ferromagnetic bcc Fe 3 2 ferromagnetic bcc Fe DMFT+FLEX U=.12, J=.8 Ry majority spin SELFENERGY majority spin DOS (states/ry) SELFENERGY minority spin minority spin ENERGY (Ry) ENERGY (Ry) Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.18

19 bu a _ c ^U f ed ` ` ` RESULTS [ ] [ ] [ ] bcc-fe fcc-ni Ni: satellite at -6 ev broadening due to the Im infinite lifetime of quasiparticles at Fermi energy (Luttinger theorem) Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.19

20 CONCLUSIONS AND OUTLOOK FLEX good for weak interactions other systems such as DMS actinides:relativistic formulation necessary (other basis: ), jj-coupling search for other impurity solvers suitable for stronger interactions Workshop Frontiers in theoretical magnetism, Oct 4-6, 26, Praha p.2