1. Bonn, 2008 Band structure of strongly correlated materials from the Dynamical Mean Field perspective K Haule Rutgers University Collaborators : J.H. Shim & Gabriel Kotliar, S. Savrasov

2. Outline Dynamical Mean Field Theory in combination with band structure  LDA+DMFT results for 115 materials (CeIrIn 5 )  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 References: J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007). J.H. Shim, KH, and G. Kotliar, Nature 446, 513 (2007).

3. 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 M. Van Schilfgarde

4. 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. Strong correlation – Standard theory fails

5. V2O3V2O3 Ni 2-x Se x  organics Universality of the Mott transition First order MIT Critical point Crossover: bad insulator to bad metal 1B HB model (DMFT): Bad insulator Bad metal 1B HB model (plaquette):

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

7. atom solid Hund’s rule, SO coupling, CFS DMFT + electronic structure method (G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006). 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 

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

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

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

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

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

14. CeIn 3 CeCoIn 5 CeRhIn 5 CeIrIn 5 PuCoG 5 Na Tc[K]0.2K2.3K2.1K0.4K18.3Kn/a T crossover ~50K ~370K C v /T[mJ/molK^2]10003004007501001 Phase diagram of CeIn 3 and 115’s N.D. Mathur et al., Nature (1998) CeIn 3 CeCoIn 5 CeRhIn 5 CeIrIn 5 CeCoIn 5 CeXIn 5 layering T crossover α T c

15. Ce In Ir Ce In Crystal structure of 115’s CeIn 3 layer IrIn 2 layer Tetragonal crystal structure 4 in plane In neighbors 8 out of plane in neighbors 3.27au 3.3 au

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

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

18. (e Temperature dependence of the local Ce-4f spectra 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 At 300K, only Hubbard bands J. H. Shim, KH, and G. Kotliar Science 318, 1618 (2007). Broken symmetry (neglecting strong correlations) can give Hubbard bands, but not both Hubbard bands And quasiparticles!

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

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

21. ARPES Fujimori, 2006 (T=10K) 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

22. Angle integrated photoemission vs DMFT ARPES Fujimori, 2006 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

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

24. DMFT qp bandsLDA bands DMFT qp bands Quasiparticle bands three bands, Z j=5/2 ~1/200

25. Momentum resolved total spectra A( ,k) Fujimori, 2003 LDA+DMFT at 10K ARPES, HE I, 15K LDA f-bands [-0.5eV, 0.8eV] almost disappear, only In-p bands remain Most of weight transferred into the UHB 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 Large lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

26. Optical conductivity Typical heavy fermion at low T: Narrow Drude peak (narrow q.p. band) Hybridization gap k  Interband transitions across hybridization gap -> mid IR peak CeCoIn 5 no visible Drude peak no sharp hybridization gap F.P. Mena & D.Van der Marel, 2005 E.J. Singley & D.N Basov, 2002 second mid IR peak at 600 cm -1 first mid-IR peak at 250 cm -1

27. 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 Optical conductivity in LDA+DMFT

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

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

31. Fermi surface changes under pressure in CeRhIn 5  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 Shishido, (2005) localized itinerant 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. Electron fermi surfaces at (z=0) LDA+DMFT (10 K) LDA LDA+DMFT (400 K)  XM X X X M MM 22 22 Slight decrease of th e electron FS with T

33.  RA R R R A AA 33 a 33 LDA+DMFT (10 K) LDA LDA+DMFT (400 K) Electron fermi surfaces at (z=  ) No a in DMFT! No a in Experiment! Slight decrease of th e electron FS with T

34. LDA+DMFT (10 K) LDA LDA+DMFT (400 K)  XM X X X M MM c 22 22 11 11 Electron fermi surfaces at (z=0) Slight decrease of th e electron FS with T

35.  RA R R R A AA c 22 22 LDA+DMFT (10 K) LDA LDA+DMFT (400 K) Electron fermi surfaces at (z=  ) No c in DMFT! No c in Experiment! Slight decrease of th e electron FS with T

36. LDA+DMFT (10 K) LDA LDA+DMFT (400 K)  XM X X X M MM g h Hole fermi surfaces at z=0 g h Big change-> from small hole like to large electron like 11

37. Localization – delocalization transition in Lanthanides and Actinides Delocalized Localized

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

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 hole in the f shell One more electron in the f shell  No magnetic moments,  large mass  Large specific heat,  Many phases, small or large volume  Magnetic moments! (Curie-Weiss law at high T,  Orders antiferromagnetically at low T)  Small effective mass (small specific heat coefficient)  Large volume

41. Standard theory of solids:  DFT: All Cm, Am, Pu are magnetic in LSDA/GGA LDA: Pu(m~5   ), Am (m~6   ) Cm (m~4   ) Exp: Pu (m=0), Am (m=0) Cm (m~7.9   )  Non magnetic LDA/GGA predicts volume up to 30% off.  In atomic limit, Am non-magnetic, but Pu magnetic with spin ~ 5  B  Can LDA+DMFT account for anomalous properties of actinides?  Can it predict which material is magnetic and which is not? 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

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

43. 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 Magnetization of Cm:

44. 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) One dominant atomic state – ground state of the atom Pu partly f 5 partly f 6 Probabilities: 5 electrons 80% 6 electrons 20% 4 electrons <1% J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).

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

46. Photoemission and valence in Pu |ground state > = |a f 5 (spd) 3 >+ |b f 6 (spd) 2 > f 5 f 6 f 5 ->f 4 f 6 ->f 7 Af()Af() approximate decomposition

47. 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 pr esence 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 Conclusions

48. Thank you!

49.  Gradual decrease of electron FS  Most of FS parts show similar trend  Big change might be expected in the  plane – small hole like FS pockets (g,h) merge into electron FS  1 (present in LDA-f-core but not in LDA)  Fermi surface a and c do not appear in DMFT results Increasing temperature from 10K to 300K: Fermi surfaces

50. ARPES of CeIrIn 5 Fujimori et al. (2006)

51. Ce 4f partial spectral functions LDA+DMFT (10K) LDA+DMFT (400K) Blue lines : LDA bands

52. Hole fermi surface at z=   RA R R R A AA No Fermi surfaces LDA+DMFT (400 K) LDA+DMFT (10 K) LDA

53. dHva freq. and effective mass

54. 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  SUNCA Allows correct treatment of multiplets Very precise at high and intermediate frequencies and high to intermediate temperatures Complementary to CTQMC (imaginary axis -> low energy)