Band structure of strongly correlated materials from the Dynamical Mean Field perspective

1. K Haule Rutgers University Collaborators : J.H. Shim & Gabriel Kotliar, S. Savrasov Band structure of strongly correlated materials from the Dynamical Mean Field perspective

2. Dynamical Mean Field Theory in combination with band structure LDA+DMFT results for 115 materials (CeIrIn5) Local Ce 4f - spectra and comparison to AIPES) Momentum resolved spectra and comparison to ARPES Optical conductivity Two hybridization gaps and its connection to optics Fermi surface in DMFT Actinides Absence of magnetism in Pu and magnetic ordering in Cm explained by DMFT Valence of correlates solids, example of Pu Outline • References: • J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007). • J.H. Shim, KH, and G. Kotliar, Nature 446, 513 (2007).

3. M. Van Schilfgarde Standard theory of solids Band Theory: electrons as waves: Rigid band picture: En(k) versus k Landau Fermi Liquid Theory applicable Very powerful quantitative tools: LDA,LSDA,GW • Predictions: • total energies, • stability of crystal phases • optical transitions

4. Strong correlation – Standard theory fails • Fermi Liquid Theory does NOT work . Need new concepts to replace rigid bands picture! • Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). • Non perturbative problem.

5. Universality of the Mott transition Crossover: bad insulator to bad metal Critical point First order MIT Ni2-xSex k organics V2O3 Bad metal Bad insulator 1B HB model (DMFT): 1B HB model (plaquette):

6. How to computed spectroscopic quantities (single particle spectra, optical conductivity phonon dispersion…) from first principles? How to relate various experiments into a unifying picture. New concepts, new techniques….. DMFT maybe simplest approach to meet this challenge Basic questions to address

7. D atom solid Hund’s rule, SO coupling, CFS DMFT + electronic structure method Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated orbitals (s,p): use LDA or GW For correlated orbitals (f or d): add all local diagrams by solving QIM (G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).

8. LDA functional ALL local diagrams Generalized Q. impurity problem! LDA+DMFT (G. Kotliar et.al., RMP 2006). observable of interestis the "local“Green's functions (spectral function) Exact functional of the local Green’s function exists, its form unknown! Currently Feasible approximations: LDA+DMFT: Variation gives st. eq.:

9. DMFT + electronic structure method Dyson equation correlated orbitals hybridization other “light” orbitals obtained by DFT Ce(4f) obtained by “impurity solution” Includes the collective excitations of the system all bands are affected: have lifetime fractional weight Self-energy is local in localized basis, in eigenbasis it is momentum dependent!

10. An exact impurity solver, continuous time QMC - expansion in terms of hybridization K.H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005). General impurity problem Diagrammatic expansion in terms of hybridization D +Metropolis sampling over the diagrams • Exact method: samples all diagrams! • Allows correct treatment of multiplets

11. Analytic impurity solvers (summing certain types of diagrams), expansion in terms of hybridization K.H. Phys. Rev. B 64, 155111 (2001) Fully dressed atomic propagators hybridization D • Allows correct treatment of multiplets • Very precise at high and intermediate • frequencies and high to intermediate • temperatures Complementary to CTQMC (imaginary axis -> low energy)

12. DMFT “Bands” are not a good concept in DMFT! instead of “bands” Frequency dependent complex object lifetime effects quasiparticle “band” does not carry weight 1 Spectral function is a good concept

13. high T low T DMFT is not a single impurity calculation Auxiliary impurity problem: temperature dependent: Weiss field High-temperature D given mostly by LDA low T: Impurity hybridization affected by the emerging coherence of the lattice (collective phenomena) DMFT SCC: Feedback effect on D makes the crossover from incoherent to coherent state very slow!

14. Phase diagram of CeIn3 and 115’s CeXIn5 CeIn3 layering CeCoIn5 CeIrIn5 CeCoIn5 CeRhIn5 N.D. Mathur et al., Nature (1998) Tcrossoverα Tc

15. Ir In Ce In Ce In Crystal structure of 115’s Tetragonal crystal structure IrIn2 layer 3.27au 4 in plane In neighbors 3.3 au CeIn3 layer IrIn2 layer 8 out of plane in neighbors

16. ALM in DMFT Schweitzer& Czycholl,1991 Crossover scale ~50K • High temperature • Ce-4f local moments • Low temperature – • Itinerant heavy bands Coherence crossover in experiment out of plane in-plane

17. ? A(w) w k Issues for the system specific study • How does the crossover from localized moments • to itinerant q.p. happen? • How does the spectral • weight redistribute? • Where in momentum space q.p. appear? • What is the momentum • dispersion of q.p.? • How does the hybridization gap look like in momentum space?

18. (e Temperature dependence of the localCe-4f spectra • At 300K, only Hubbard bands • At low T, very narrow q.p. peak • (width ~3meV) • SO coupling splits q.p.: +-0.28eV • Redistribution of weight up to very high • frequency SO Broken symmetry (neglecting strong correlations) can give Hubbard bands, but not both Hubbard bands And quasiparticles! J. H. Shim, KH, and G. Kotliar Science 318, 1618 (2007).

19. Buildup of coherence in single impurity case Very slow crossover! coherent spectral weight TK T T* Buildup of coherence coherence peak scattering rate Slow crossover pointed out by S. Nakatsuji, D. Pines, and Z. Fisk Phys. Rev. Lett. 92, 016401 (2004) Crossover around 50K

20. Anomalous Hall coefficient Consistency with the phenomenological approach of NPF Fraction of itinerant heavy fluid m* of the heavy fluid Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789!

21. Angle integrated photoemission vs DMFT Very good agreement, but hard to see resonance in experiment: resonance very asymmetric in Ce ARPES is surface sensitive at 122eV ARPES Fujimori, 2006 (T=10K)

22. Angle integrated photoemission vs DMFT • Nice agreement for the • Hubbard band position • SO split qp peak • Hard to see narrow resonance • in ARPES since very little weight • of q.p. is below Ef Lower Hubbard band ARPES Fujimori, 2006

23. Momentum resolved Ce-4f spectra Af(w,k) Hybridization gap q.p. band Fingerprint of spd’s due to hybridization scattering rate~100meV SO Not much weight T=10K T=300K

24. DMFT qp bands LDA bands LDA bands DMFT qp bands Quasiparticle bands three bands, Zj=5/2~1/200

25. Momentum resolved total spectra A(w,k) Most of weight transferred into the UHB LDA f-bands [-0.5eV, 0.8eV] almost disappear, only In-p bands remain Very heavy qp at Ef, hard to see in total spectra Below -0.5eV: almost rigid downshift Unlike in LDA+U, no new band at -2.5eV ARPES, HE I, 15K LDA+DMFT at 10K Fujimori, 2003 Large lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

26. w k first mid-IR peak at 250 cm-1 CeCoIn5 Optical conductivity F.P. Mena & D.Van der Marel, 2005 Typical heavy fermion at low T: no visible Drude peak no sharp hybridization gap Narrow Drude peak (narrow q.p. band) Hybridization gap second mid IR peak at 600 cm-1 Interband transitions across hybridization gap -> mid IR peak E.J. Singley & D.N Basov, 2002

27. Optical conductivity in LDA+DMFT • At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) • At 10K: • very narrow Drude peak • First MI peak at 0.03eV~250cm-1 • Second MI peak at 0.07eV~600cm-1

28. 10K In eV Ce In Multiple hybridization gaps non-f spectra 300K • Larger gap due to hybridization with out of plane In • Smaller gap due to hybridization with in-plane In

29. Fermi surfaces of CeM In5 within LDA Localized 4f: LaRhIn5, CeRhIn5 Shishido et al. (2002) Itinerant 4f : CeCoIn5, CeIrIn5 Haga et al. (2001)

30. LDA (with f’s in valence) is reasonable for CeIrIn5 de Haas-van Alphen experiments Experiment LDA Haga et al. (2001)

31. Fermi surface reconstruction at 2.34GPa Sudden jump of dHva frequencies Fermi surface is very similar on both sides, slight increase of electron FS frequencies Reconstruction happens at the point of maximal Tc Fermi surface changes under pressure in CeRhIn5 localized itinerant Shishido, (2005) We can not yet address FS change with pressure  We can study FS change with Temperature - At high T, Ce-4f electrons are excluded from the FS At low T, they are included in the FS

32. Slight decrease of the electron FS with T M X M G X X M M X Electron fermi surfaces at (z=0) LDA+DMFT (400 K) LDA LDA+DMFT (10 K) a2 a2

33. Slight decrease of the electron FS with T No a in DMFT! No a in Experiment! A R A Z R R A A R Electron fermi surfaces at (z=p) LDA+DMFT (400 K) LDA LDA+DMFT (10 K) a3 a3 a

34. Slight decrease of the electron FS with T M X M G X X M M X Electron fermi surfaces at (z=0) LDA+DMFT (400 K) LDA+DMFT (10 K) LDA b1 b1 b2 b2 c

35. No c in DMFT! No c in Experiment! Slight decrease of the electron FS with T A R A Z R R A A R Electron fermi surfaces at (z=p) LDA+DMFT (400 K) LDA+DMFT (10 K) LDA b2 b2 c

36. M X M G X X M M X Hole fermi surfaces at z=0 Big change-> from small hole like to large electron like LDA+DMFT (400 K) LDA+DMFT (10 K) LDA e1 g h h g

37. Localization – delocalization transitionin Lanthanides and Actinides Localized Delocalized

38. Electrical resistivity & specific heat Heavy ferm. in an element Itinerant closed shell Am J. C. Lashley et al. PRB 72 054416 (2005)

39. NO Magnetic moments in Pu! Pauli-like from melting to lowest T No curie Weiss up to 600K

40. Curium versus Plutonium nf=6 -> J=0 closed shell (j-j: 6 e- in 5/2 shell) (LS: L=3,S=3,J=0) One more electron in the f shell One hole in the f shell • Magnetic moments! (Curie-Weiss law at high T, • Orders antiferromagnetically at low T) • Small effective mass (small specific heat coefficient) • Large volume • No magnetic moments, • large mass • Large specific heat, • Many phases, small or large volume

41. Standard theory of solids: • DFT: • All Cm, Am, Pu are magnetic in LSDA/GGA LDA: Pu(m~5mB), Am (m~6mB) Cm (m~4mB) Exp: Pu (m=0), Am (m=0) Cm (m~7.9mB) • Non magnetic LDA/GGA predicts volume up to 30% off. • In atomic limit, Am non-magnetic, but Pu magnetic with spin ~5mB • Many proposals to explain why Pu is non magnetic: • Mixed level model (O. Eriksson, A.V. Balatsky, and J.M. Wills) (5f)4 conf. +1itt. • LDA+U, LDA+U+FLEX (Shick, Anisimov, Purovskii) (5f)6 conf. • Cannot account for anomalous transport and thermodynamics • Can LDA+DMFT account for anomalous properties of actinides? • Can it predict which material is magnetic and which is not?

42. Starting from magnetic solution, Curium develops antiferromagnetic long range order below Tc above Tc has large moment (~7.9mB close to LS coupling) Plutonium dynamically restores symmetry -> becomes paramagnetic J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).

43. Magnetization of Cm: Multiplet structure crucial for correct Tk in Pu (~800K) and reasonable Tc in Cm (~100K) Without F2,F4,F6: Curium comes out paramagnetic heavy fermion Plutonium weakly correlated metal

44. Pu partly f5 partly f6 Valence histograms Density matrix projected to the atomic eigenstates of the f-shell (Probability for atomic configurations) f electron fluctuates between these atomic states on the time scale t~h/Tk (femtoseconds) • Probabilities: • 5 electrons 80% • 6 electrons 20% • 4 electrons <1% One dominant atomic state – ground state of the atom J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).

45. Fingerprint of atomic multiplets - splitting of Kondo peak Gouder , Havela PRB 2002, 2003

46. Photoemission and valence in Pu |ground state > = |a f5(spd)3>+ |b f6 (spd)2> approximate decomposition Af(w) f5<->f6 f6->f7 f5->f4

47. Conclusions • DMFT can describe crossover from local moment regime to heavy fermion state in heavy fermions. The crossover is very slow. • Width of heavy quasiparticle bands is predicted to be only ~3meV. We predict a set of three heavy bands with their dispersion. • Mid-IR peak of the optical conductivity in 115’s is split due to presence of two type’s of hybridization • Ce moment is more coupled to out-of-plane In then in-plane In which explains the sensitivity of 115’s to substitution of transition metal ion • DMFT predicts Pu to be nonmagnetic (heavy fermion like) and Cm to be magnetic

48. Thank you!

49. Fermi surfaces Increasing temperature from 10K to 300K: • Gradual decrease of electron FS • Most of FS parts show similar trend • Big change might be expected in the G plane – small hole like FS pockets (g,h) merge into electron FS e1 (present in LDA-f-core but not in LDA) • Fermi surface a and c do not appear in DMFT results

50. ARPES of CeIrIn5 Fujimori et al. (2006)