Strongly Correlated Electron Materials : A Dynamical Mean Field Theory (DMFT) Perspective. Strongly Correlated Electron Materials : A Dynamical Mean Field Theory (DMFT) Perspective. Gabriel Kotliar and Center for Materials Theory $upport : NSF -DMR DOE-Basic Energy Sciences Cornell Ithaca NY November

Outline Introduction to strongly correlated materials. Brief overview of Dynamical Mean Field Theory. Application to heavy fermions: a case study of CeIrIn5 [with K. Haule and J. Shim, Science Express Nov 1 st (2007) ] Conclusions – and some thought about the 5f elemental metals.

.Interactions renormalize away (Landau). Band Theory: electrons as waves Electrons in a Solid:the Standard Model Quantitative Tools. Density Functional Theory Kohn Sham (1964) Rigid bands, optical transitions, thermodynamics, transport……… Static Mean Field Theory. 2

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy M VanShilfgaarde et. al. PRL 96, (2006) M VanShilfgaarde et. al. PRL 96, (2006) 3

Correlated Electron Systems Pose Basic Questions in CMT Correlated Electron Systems Pose Basic Questions in CMT FROM ATOMS TO SOLIDS How to describe electron from localized to itinerant ? How do the physical properties evolve ?

Strong Correlation Problem:where the standard model fails Fermi Liquid Theory works but parameters can’t be computed in perturbative theory. Fermi Liquid Theory does NOT work. Need new concepts to replace of rigid bands ! Partially filled d and f shells. Competition between kinetic and Coulomb interactions. Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. 4

Strongly correlated materials do “big” things Strongly correlated materials do “big” things Huge volume collapses Pu ……. Masses as large as 1000 m e (heavy fermions UPt3, CeIrIn5….. High Temperature Superconductivity. 150 K Ca 2 Ba 2 Cu 3 HgO 8. Large thermoelectric response in Na x Co 2 O 4 Large change in resistivity. MIT in TM oxides (V2O3, VO2, LaSrMnO3……..) ………………….. 5

Hubbard model   U/t   Doping  or chemical potential   Frustration t ij   T temperature 6

Mott-Hubbard Physics Real space picture High T : local moments Low T: spin orbital order H HH H H+H+H+H+ Excitations: Excitations: Excitations: adding (removing ) e, Upper Hubbard band. 7

Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). A(  ) 10

A. Georges, G. Kotliar (1992) A(  ) 11

Dynamical Mean Field Theory Weiss field is a function. Multiple scales in strongly correlated materials. Exact in the limit of large coordination (Metzner and Vollhardt 89), kinetic and interaction energy compete on equal footing. Immediate extension to real materials DFT+DMFT 12

DMFT Spectral Function Photoemission and correlations Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8

Evolution of the DOS. Theory and experiments 13

Summary: DMFT Self consistent Impurity problem, natural language to quantify localization/delocalization phenomena. Combines atomic physics and band theory Systematically improvable, cluster DMFT Implementation

 CeRhIn5: TN=3.8 K;   450 mJ/molK2  CeCoIn5: Tc=2.3 K;   1000 mJ/molK2;  CeIrIn5: Tc=0.4 K;   750 mJ/molK2 CeMIn 5 M=Co, Ir, Rh out of plane in-plane Ce In Ir

Ir atom is less correlated than Co or Rh (5d / 3d or 4d) CeIrIn 5 is more itinerant(coherent) than Co (further away from QCP) Why CeIrIn 5 ? Phase diagram of 115’s

Generalized Anderson Lattice Model 6 High temperature Ce-4f local moments Low temperature – Itinerant heavy bands C + ff +

S. Doniach, e-e- J k = V 2 /  f Kondo Exchange Kondo scale RKKY scale DONIACH PHASE DIAGRAM

Angle integrated photoemission Experimental resolution ~30meV Surface sensitivity at 122 ev, theory predicts 3meV broad band Expt Fujimori et al., PRB 73, (2006) P.R B 67, (2003). Theory: LDA+DMFT, impurity solvers SUNCA and CTQMC Shim Haule and GK (2007)

Very slow crossover! T*T* Slow crossover pointed out by NPF 2004 Buildup of coherence in single impurity case TKTK coherent spectral weight T scattering rate coherence peak Buildup of coherence Crossover around 50K

Consistency with the phenomenological approach of NPF Remarkable agreement with Y. Yang & D. Pines cond-mat/ ! +C

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:

Momentum resolved total spectra trA( ,k) Fujimori, PRB 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 Short lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

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 Expts: F. P. Mena, D. van der Marel, J. L. Sarrao, PRB 72, (2005). 16. K. S. Burch et al., PRB 75, (2007). 17. E. J. Singley, D. N. Basov, E. D. Bauer, M. B. Maple, PRB 65, (R) (2002).

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

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

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

Quantum Phase Transition: Kondo Breakdown vs SDW. SDW picture. Focus on order parameters. Neglect changes in the electronic structure.[Hertz, Morya] Kondo breakdown scenario. Drastic changes in the electronic structure. [Doniach] [ Coleman, Pepin, Paul, Senthil, Sachdev, Vojta, Si ] V> Vc Neglect Magnetic order V < Vc Magnetic Order.

DMFT: consider the underlying paramagnetic solution. Study finite T.

Kondo Breakdown as an Orbitally selective Mott Transition. [L. DeMedici, A. Georges GK and S. Biermann PRL (2005), C. Pepin (2006), L. DeLeo M. Civelli and GK ] Analogous situation to the Mott transition. Mott / Slater. f localization - Jump in the Fermi volume-Jump in DeHaas VanAlven frequencies. f Localized and f Itinerant phases have different compressibilities. Low but finite temperature aspects of the transition governed by a two impurity model.

Fermi surface changes under pressure in CeRhIn 5 –Fermi surface reconstruction at 2.34GPa. Sudden jump of dHva frequencies –Delocalization. Increase of electron FS frequencies. Localization decreases them. 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

 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 the electron FS with T

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

Conclusion- future directions Long wavelength vs short distance [ mean field ] physics in correlated materials. Further improvements and developments of DMFT [ CDMFT, electronic structure] Other systems. […..] System specific studies. Variety and universality in the localization delocalization phenomena. Towards a (Dynamical ) Mean Field Theory based theoretical spectroscopy.

Conclusions [115’s] DMFT in action: collective behavior of the hybridization field. Very slow crossover. Spectral evolution. Valence histograms. Theory/Experiment Spectroscopy. Multiple hybridization gaps in optics. Very different Ce-In hybridizations with In out of plane being larger. Kondo breakdown as an orbitally selective Mott transition. dhv orbits. Lessons for the 5f’s. Elemental actinides.

Thanks! $upport NSF-DMR. Collaborators: K. Haule, L. DeLeo, J. Shim, M. Civelli. K. Haule and J. Shim and GK, Science Express Nov 1st (2007). To appear in science.

after G. Lander, Science (2003) and Lashley et. al. PRB (2006). Mott Transition    Pu  Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)]

DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W 14

  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

Summary part 2 Modern understanding (DMFT) of the (orbitally selective) Mott transition across the actinde series (B. Johanssen 1970 ) sheds light on 5f physics. Important role of multiplets. Pu is non magnetic and mixed valent element mixture of f 6 and f 5 f electrons are localized in Cm f 7 Physics of 5f’s and 4f’s is similar but different. Main difference, the coherence scale in 5f’s much larger, resulting in a much larger coupling to the lattice. K. Haule and J. Shim Ref: Nature 446, 513, (2007)

Pu phases: A. Lawson Los Alamos Science 26, (2000) GGA LSDA predicts  Pu to be magnetic with a large moment ( ~ 5 Bohr). Experimentally Pu is not magnetic. [PRB (2005). Valence of Pu is controversial.

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory Experiment ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

Curie-Weiss Tc Photoemission of Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antif Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

What is the valence in the late actinides ?

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 the electron FS with T

 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 the electron FS with T

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 the electron FS with T

 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 the electron FS with T

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

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

Cluster DMFTlimitations of single site DMFT Cluster DMFT: removes limitations of single site DMFT No k dependence of the self energy. No d-wave superconductivity. No Peierls dimerization. No (R)valence bonds. Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier et. al. RMP (2005); Kotliar et..al. RMP (2006). 23

U/t=4. Two Site Cellular DMFTin the 1D Hubbard model Two Site Cellular DMFT ( G.. Kotliar et.al. PRL (2001)) in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69, (2004)T. D Stanescu and GK PRB (2006)24

High Temperature superconductors

LeTacon et.al. Nature Physics (2006) Raman Hg

Doping Driven Mott transiton at low temperature, in 2d (U=16 t=1, t’=-.3 ) Hubbard model Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k K.M. Shen et.al X2 CDMFT Nodal Region Antinodal Region Civelli et.al. PRL 95 (2005)

Conclusion DMFT conceptual framework to think about electrons in solids. Finite T Mott transition in 3d. Single site DMFT worked well! Ab-initio many body electronic structure of solids. Building theoretical spectroscopies. Frontier, cuprates, lower T, two dimensionality is a plaquette in a medium enough? Inhomogenous structure in correlated materials New renormalizaton group methods built around DMFT ? 28

Conclusion A Few References …… A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of. Modern Physics 68, 13 (1996). G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, , (2006). G. K and D. Vollhardt Physics Today, Vol 57, 53 (2004). 29

Outline The standard model of solids. Correlated electrons and Dynamical Mean Field Theory (DMFT). The temperature driven Mott transition. Mott transition across the actinide series. Future Directions, cuprate superconductors and Cluster DMFT……

+ Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) VanShilfgaarde (2005)

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) VanShilfgaarde (2005)

A. Georges, G. Kotliar (1992) A(  )

Raman Hg-1201 LeTacon et.al. Nature Physics (2006) Doping decreases Anti-Nodal Nodal B 1g B 2g

High Temperature superconductors

Mott transition: Mott transition: evolution of the electron from itinerant to localized ? How Matsuura et. al.(2000) L  -(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl Lefevre et.al. (2000)Limelette et al.,(2003) Kagawa et al. (2003) 9

Interaction with Experiments. Photoemission Three peak strucure. V2O3:Anomalous transfer of spectral weight M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) T=170 T=300 15

. Photoemission measurements and Theory V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) NiSxSe 1-x Matsuura Watanabe Kim Doniach Shen Thio Bennett (1998) Poteryaev et.al. (to be published) 16

Spinodals and Ising critical endpoint. Observation in V 2 O 3 : P. Limelette et.al. Science 302, 89 (2003) Spinodals and Ising critical endpoint. Observation in V 2 O 3 : P. Limelette et.al. Science 302, 89 (2003) Critical endpoint Spinodal Uc2 17

Spectral Function and Photoemission Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8

Georges Kotliar (1992) D MFT approximate quantum solid as atom in a medium 10

(GW) DFT+DMFT: determine H[k] and density and  self consitently from a functional and obtain total energies. 12 Spectra=- Im G(k,  ) Self consistency for V and 

Chitra and Kotliar PRB 62, (2000) PRB (2001)P.Sun and GK (2005) Zein et.al. PRL 96, (2006)). See also Bierman Aryasetiwan and Georges. Ir,>=|R,  > Gloc=G(R , R’  ’ )  R,R’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Electronic structure problem: compute and given structure

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

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

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

Ce In Ir Ce In Crystal structure of 115’s CeMIn5 M=Co, Ir, Rh 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

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