Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach

1. Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel Kotliar Physics Department Rutgers University In Electronic Structure and Computational Magnetism July 15-17 (2002)

2. Outline • Dynamical Mean Field Theory: a tool for treating correlations in model Hamiltonians. • Towards Realistic implementations of DMFT. • Applications to Fe and Ni. • Conclusions and outlook. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Acknowledgements • Collaborators and References: 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). • Useful Discussions K. Hathaway and G. Lonzarich • Support NSF and ONR THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Strong Correlation Problem • Two limiting cases of the electronic structure problem are well understood. The high density limit ( spectrum of one particle excitations forms bands) and the low density limit (spectrum of atomic like excitations, Hubbard bands). • Correlated compounds: electrons in partially filled shells. Not close to the well understood limits . Non perturbative regime. • Standard approaches (LDA, HF ) do not work well. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Motivations for going beyond density functional theory. • DFT is a theory for ground state properties. Its Kohn Sham spectra can be taken a starting point for perturbative (eg. GW ) calculations of the excitation spectra and transport. • This does not work for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. Character of the spectra (QP bands + Hubbard bands ) is not captured by LDA. • LDA –GGA is less accurate in determining some ground state properties in correlated materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. DMFT • DMFT simplest many body technique which describes correctly the open shell atomic limit and the band limit . Exact in the limit of large lattice coordination. • Band physics (i.e. kinetic energy) survive in the atomic limit (superexchange). Some aspects of atomic physics survive even in itinerant systems (J, U, Hubbard bands, satellites, L) • Computations of one electron spectra, transport properties… • Spectral density functional. Connects the one electron spectral function and the total energy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. Mean-Field : Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. DMFT Impurity cavity construction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. C-DMFT C:DMFT The lattice self energy is inferred from the cluster self energy. Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. Solving the DMFT equations • Wide variety of computational tools (QMC,ED….)Analytical Methods • Reviews: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. From model hamiltonians to realistic calculations. • DMFT as a method to be incorporated in electronic structure calculations. • Important in regimes where local moments are present (e.g. NiO above its Neel temperature) • Incorporate realistic structure and orbital degeneracy information in many body studies. • Combination of electronic structure(LDA,GGA,GW) and many body methods (DMFT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. Interface with electronic structure. • Derive model hamiltonians, solve by DMFT (or cluster extensions). Total energy? • Full many body aproach, treat light electrons byt GW or screend HF, heavy electrons by DMFT [GK and Chitra, GK and S. Savrasov, P.Sun and GK cond-matt 0205522] • Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. 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 • Calculations of U, Edc, study as a function of these parameters. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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16. Combine Dynamical Mean Field Theory with Electronic structure methods. • Single site DMFT made correct qualitative predictions. • Make realistic by: • Incorporating all the electrons. • Add realistic orbital structure. U, J….. • Add realistic crystal structure. • Allow the atoms to move. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Two roads for ab-initio calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Realistic Calculationsof the Electronic Structure of Correlated materials • Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. • Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). • Lichtenstein and Katsenelson PRB (1998) • Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. Combining LDA and DMFT • The light, SP (or SPD) electrons are extended, well described by LDA • The heavy, D (or F) electrons are localized,treat by DMFT. • LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles (Gunnarson and Anisimov, Mc Mahan et. Al. Hybertsen et.al) or viewed as parameters THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Density functional theory and Dynamical Mean Field Theory • DFT: Static mean field, electrons in an effective potential. • Functional of the density. • DMFT: Promote the local (or a few quasilocal Greens functions or observables) to the basic quantities of the theory. • Express the free energy as a functional of those quasilocal quantities. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar). • DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. GDFT[r(r)] • Introduce local orbitals, caR(r-R)orbitals, and local GF G(R,R)(i w) = • The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing G[r(r),G(R,R)(iw)] • A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) Full self consistent implementation. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. LDA+DMFT-outer loop relax Edc U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. Realistic DMFT loop THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. LDA+DMFT functional F Sum of local 2PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. Very Partial list of application of realistic DMFT to materials • QP bands in ruthenides: A. Liebsch et al (PRL 2000) • N phase of Pu: S. Savrasov et al (Nature 2001) • MIT in V2O3: K. Held et al (PRL 2001) • Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) • J-G transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al PRL (2000). • 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002) • ……………….. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. 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)] • Extension to realistic setting [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). ] Lichtenstein and Katsnelson [Phys.Rev. B 57, 6884(1998) ] • Unlike DFT, DMFT computes both free energies and one electron (photoemission ) spectra and many other physical quantities at finite temperatures. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. 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, caR(r-R)orbitals, and local GF • G(R,R)(i w) = • The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a double Legendre transformton THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

30. Outer loop relax Edc G0 Impurity Solver Hartree-Fock G,S U SCC DMFT LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. Outer loop relax Edc G0 Impurity Solver Imp. Solver: Hartree-Fock G,S U SCC DMFT LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

33. LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional , • with F= FHF reduces to the LDA+U functional of Anisimov et.al. • Crude approximation. Reasonable in ordered situations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. Applications of LDA+DMFT • Organics • Alpha-Gamma Cerium • V2O3 • Volume collapse in Pu • Photoemission of ruthenates • Doping driven Mott transition in LaSrTiO3 • Itinerant Ferromagnetism • Bucky Balls THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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37. Applications: Itinerant Ferromagnetism, Ni Fe • Compromise in the resources used for the solution of the one electron problem, and the many body problem. • Goal: obtain an overall approximate but consistent picture of how correlations affect physical properties. Estimate sensitivity on parameters. • Tc, spectra, susceptibility, [QMC- impurity solver] [ASA, relatively small number of k points] • Magnetic anisotropy [HF-impurity solver][full potential LMTO, large number of k points, non collinear magnetization] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. Case study Fe and Ni • Archetypical itinerant ferromagnets • LSDA predicts correct low T moment • Band picture holds at low T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

40. 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) ) • LDA Fermi surface in Nickel has features which are not seen in DeHaas Van Alphen ( G. Lonzarich) • Photoemission spectra of Ni : 6 ev satellite 30% band narrowing, reduction of exchange splitting. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. DMFT-QMC: Numerical Details • 256 k points • 105 - 106 QMC sweeps • Analytic continuation via maximum entropy. • Tight binding LMTO-ASA THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

45. Ni and Fe: theory vs exp meff / mB high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature Tc • Fe 1900 ( theory) 1043(expt) • Ni 700 (theory) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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47. Magnetic anisotropy • MAE is small (1 meV/Atom) • Ni: 2.8 meV/Atom easy axis 111 Fe: 1.4 meV/Atom easy axis 100 • Long standing problem Early papers • Van Vleck (PR 1937) • Brooks (PR 1940) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. LDA calculations • Trygg et.al (1995); SCF Total energy with large # of k-points; Wrong easy axis for Ni. • Other related works: Halilov et al. (1998) G. Schneider et al. (1997) Wang et al. (1993) Beiden et.al. (1998) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Method Full-potential multiple kappa LMTO method. Pauli treatment of relativistic effects. Non-collinear intraatomic magnetism included. Explore different Edc. (Details I Yang Ph.D thesis) Generalized relativistic LDA+U with occupancies nab,ss’ THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. Numerical Considerations Work of Trygg et.al. proves equivalence of special points and tetrahedra. Confirmed. (broadening 0.15 mRy.) Convergent Etot needs 15000 k’s. We use 28000k’s. Convergency checked to 100000 k’s. SUN E10K with 64 processors used. LDA results of Trygg et.al. reproduced: Ni 0.5 meV 001, exp. 2.8 meV 111, Fe 0.5 meV 001, exp. 1.4 meV 001. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS