1. Correlation Effects in Itinerant Magnets, Application of LDA+DMFT(Dynamical Mean Field Theory) and its static limit the LDA+U method. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University A.Lichtenstein M. Katsnelson and G. Kotliar Phys. Rev Lett. 87, 067205 (2001) I.Yang S. Savrasov and G. Kotliar Phys. Rev. Lett. 87, 216405 (2001) I. Yang Rutgers Ph.D Thesis (Dec-2001) Supported by the ONR

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Strong Correlation Problem Two limiting cases of the electronic structure of solids are understood:the high density limit and the limit of well separated atoms. Many materials have electron states that are in between these two limiting situations and require the development of new electronic structure methods to predict some of its properties (spectra, energy, transport,….) DMFT simplest many body technique which treats simultaneously the open shell atomic limit and the band limit.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Developed initially to treat correlation effects in model Hamiltonians. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Recently extended to perform first principles calculations. [V. Anisimov, A. Poteryaev, M. Korotin, Anokhin and G. Kotliar, J. Phys. Cond. Mat 9, 7359 (1997). S. Savrasov, G. Kotliar and E. Abrahams, Nature 410, 793 (2001). ] Unlike DFT, DMFT computes both free energies and one electron (photoemission ) spectra and many other physical quantities at finite temperatures.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Other aspects that require to treate correlations beyond LDA Magnetic anisotropy. L.S effect. LDA predicts the incorrect easy axis(100) for Nickel.(instead of the correct one (111) (Savrasov’s talk) LDA Fermi surface in Nickel has features which are not seen in DeHaas Van Alphen ( G. Lonzarich) High energy features in the photoemission spectra of Ni (6 ev satellite), 30% band narrowing, reduction of exchange splitting.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Spectral Density Functional ( Fukuda, Valiev and Fernando, Chitra and GK, Savrasov and GK ). DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the local density by Legendre transformation. Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a double Legendre transformton

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional Formal construction of a functional of the d spectral density DFT is useful because good approximations to the exact density functional  DFT  (r)] exist, e.g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Atom =Sum of local 2PI graphs build with local Coulomb interaction matrix, parametrized by Slater integrals F0, F2 and F4 and local G. Express  in terms of AIM model.

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and DMFT The light, SP electrons well described by LDA The heavier D electrons treat by model DMFT. LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term, Lichtenstein et.al.) Atomic physics parameters. U=F0 cost of double occupancy irrespectively of spin, J=F2+F4, Hunds energy favoring spin polarization.F2/F4=.6

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional, with  =  HF reduces to the LDA+U functional of Anisimov et.al. Simple extension to magnetic case.

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT If the self energy matrix is weakly k dependent is the physical self energy. Since is a matrix, DMFT changes the shape of the Fermi surface DMFT is absolutely necessary in the high temperature “local moment”regime. LDA+U with an effective U is OK at low energy. DMFT is needed to describe spectra with QP and Hubbard bands or satellites.

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures  has a Curie Weiss law with a moment much larger than the ordered moment. Magnetic anisotropy 

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  ( T=.9 Tc)/   ordered moment Fe 1.5 ( theory) 1.55 (expt) Ni.3 (theory).35 (expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction.3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe, RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Future directions Including short range correlations. Less local physics, C-DMFT. including the effects of long range and short range Coulomb interactions, E-DMFT and GW. Applications are just beginning. More complex systems…….. Alloys, systems containing f and d electrons.

21. Realistic Theories of Correlated Materials ITP, Santa-Barbara July 20 – December 20 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein http://www.itp.ucsb.edu/activities/future/

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC, NRG,ED….) Analytical Methods

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Construction is easily extended to states with broken translational spin and orbital order. Large number of techniques for solving DMFT equations for a review see A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional Kohn Sham eigenvalues, auxiliary quantities.

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting term (Lichtenstein et.al) subtracts average correlation

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis(100) for Nickel.(instead of the correct one (111) LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional of local Greens function G. Express in terms of Weiss field (semicircularDOS) Local self energy (Muller Hartman 89)

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Numerical Details 256 k points 10 5 - 10 6 QMC sweeps Analytic continuation via maximum entropy. ASA

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local Coulomb interaction matrix and local G