\ Gabriel Kotliar Stony Brook March 13th

Strongly Correlated Electron Materials: Some DMFT Concepts and Applications
\ Gabriel Kotliar Stony Brook March 13th Paul Dirac: The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. "The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe..The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. At each level of complexity entirely new properties appear. The whole becomes not merely more, but very different from the sum of its parts."(PW. Anderson1972) Strongly correlate electron systems, are materials which keep suprirising our imagination. Completely Unexpectedly, and thru an almost random search in the space of materials, fishing the Fermi sea, Materials scientits kept discovering treasures, including heavy fermions, colossal magentio-resistance and high temperaturesupeconductors based on copper oxide and more recently iron pnictides. In this talk I will tell you about a theoreticla Development, that helps understand and one day will help predict materials with extrordinary propretiesl “Approximate practical methods of applying quantum mechanics should be developed which can lead to an explanation of the main features of complex atomic systems without too much computation” Paul Dirac (1929) 1

“Standard Model of Solids “ Band Theory
“Standard Model of Solids “ Band Theory. Fermi Liquid Theory (Landau 1957). Density Functional Theory (Kohn Sham 1964) energy functional of the density. Reference Frame for Weakly Correlated Systems. Starting point for perturbation theory in the screened Coulomb interactions (Hedin 1965) + [ ] Framework is reiigorous and predictive. BANDS AT THE FERMI LEVEL-- Metal No bands at the fermi level insulator. Wave metals are NOT very resistive. Thermodynamics, cv chi hall coefficient etc.. As free electrons. KEY Phys. Rev. Lett. 93, (2004). Many other properties can be computed, transport, optics, phonons, etc… 2

Cuprate Experimental Phase diagram
Anomalously small conductivities 3 Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Anomalous resistivities
Sr2RuO4 C. Urano et. al. PRL 85, 1052 (2000) 4

Probing Electronic Structure:Photoemission
b)Strong correlation: fermi liquid parameters can’t be evaluated in perturbation theory or fermi liquid theory does not work. Weak correlations A(k, w) A(k, w) e w Weakly Correlated Systems, a precise correspondence in energy and momentum. Adding the electron to the material, we put it in some band the energy omega= ek Angle integrated spectra Many other spectroscopic tools to “see” correlated electrons ! 5 5

How to Make Correlated materials ?
Put open shell in a cage Oxygen transition metal ion Transition metal ions Rare earth ions Cage : e.g 6 oxygen atoms (octahedra) or other ligands/geometry Actinides Transition metal (open shell ) Build crystal with this building block or build layers separated by spacers LixCoO2, NaxCoO2 Battery materials Thermoelectrics VO2 Room temperature MIT La1-xSrxMnO3 Colossal Magnetoresistance La1-xSrxCuO4 High temperature superconductor 6

How to find interesting correlated materials ?
7 + Edisonian approach Serendipity An aptitude for making desirable discoveries by accident The Edisonian approach to innovation is characterized by trial and error discovery rather than a systematic theoretical approach. (e.g. carbon microphone, basis of telephone) The historical record indicates that Edison's approach was much more complex, that he made use of available theories and resorted to trial and error only when no adequate theory existed The method works ! Resulted in fascinating compounds . Correlated electron materials do “big things “ . Large volume collapses, ultra strong magnets, heavy fermions, ………. , high temperature superconductivity …… New phenomenal every few years…….. Copper oxides -- studying jahn teller ferroelectrics finding superconductivity near Mott insulators But the serendipity part is is a bit slow…. … 6

A. Georges and G. Kotliar PRB 45, 6479 (1992).
Mean Field Theories Replace a many body problem by a single site problem in an effective medium reference frame DMFT Effective medium: quantifieds the notion of “ metallicity” or itineracy DMFT self consistency : medium to reproduce the exact (best ) local spectral function of the problem. A. Georges and G. Kotliar PRB 45, 6479 (1992). 8

Phase diagram :frustrated Hubbard model, integer filling M. Rozenberg G. Kotliar H. Kajuter G. Thomas PRL75, 105 (1995) Mott transition Coherence Incoherence Crossover Transfer of spectralweight T/W Quasiparticles +Hubbard bands Different way of thinking was generated by the study of the Mott transition at integer filling. Universality and system specificity. . Bridge atomic physic and band physics. Crossovers with changing degrees of freedom. Spectral functions 9

High temperature universality and V2O3
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL . (2003) T=170 T=300 Critical endpoint Spinodal Uc2 P. Limelette et.al. Science 302, 89 (2003) 10

Localization Delocalization in Actinides
Mott Transition d Pu a a Modern understanding of this phenomenaDMFT. 11

DMFT Phonons in fcc d-Pu
Notice the agreement. Usefulness of theory. Notice the discreapancy. Scientfici opportunity. ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003) 12

DMFT concept: Solids are Made out of Atoms.
f shell in a medium . Valence Histogram 13 Plutonium has an unusual form of MIXED VALENCE with clear spectral fingerprints. Shim, Khaule Kotliar, Nature, 446, (2007). 8

Photoemission Havela et. al. Phys. Rev. B 68, 085101 (2003)
Pu is non magnetic – Cm is magnetic TN ~ 150 K. K.Haule J. Shim and GK Nature 446, 513 (2007) 14

Conclusions Pu Delta Plutonium sits at the edge of the localization delocalization transition. Origin of all the anomalies. The dominant valence is 5f5 but there is strong admixture of 5f6 which quenches the magnetic moment. Mixed valent element. Clear fingerprint in the presence of “quasiparticle multiplets”, low energy peaks which inherit the multiplet structure present at high energy in the atom. Need both real space and momentum space concepts to describe its physics.

Cuprates : fundamental questions Mechanism of Superconductivity
Quasiparticles glued by spin fluctuations. Predicted d-wave symmetry of order parameter . D. Scalapino , D. Pines K > -K> SC emerges from doping RVB paired spins in Mott insulator. P. W. Anderson, Science 235, 1196 (1987) Predicted d-wave symmetry of order parameter and the pseudogap at low doping. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) 15

Hubbard model : plaquette in a medium.
Lichtenstein and Kastnelson PRB (2000) 16

Momentum Space Picture: High T
Link DMFT. Normal state Real Space Picture. Ferrero et. al. (2010) (similar to plaquette Haule and GK) (2006) Singlet formation. S (singlet),T (triplet) N=2 singlet, triplet E (empty) N=0 1+ states with 1 electron in + orb Momentum Space Picture: High T Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reduced. 17

Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t
Superexchange Mechanism? . K. Haule and GK Phys. Rev. B 76, (2007). Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t How is the energy distributed in q and w ? D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). Expts; Dai et.al. 18

Building phase diagram magnetization at T=0 vs d.
Single site Two site 19

Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev)
Cedric Weber, Kristjan Haule, Gabriel Kotliar Nature Physics 10, 1038 (2010). Comanac et. al. Nature PhysicsNature Phys. 4, 287290 (2008). 20

Origin of magnetism :Comparing the AF and the “underlying PM state “
<KE>sdw -<KE>pm Weber Haule and GK Nature Physics 10, 1038 (2010). NCCO magnetizes to lower its double occupancy ! Slater. LSCO gains kinetic energy when it magnetizes. [Mott ] NCCO pays kinetic energy [Slater ] Can be traced to the structure: absence of apical oxygens reduces the charge transfer energy 21

Cuprates Superconductors
Plaquette DMFT reasonable reference frame to think about the qualitative physics of cuprates, starting from high temperatures. High Tc materials. are near the single site DMFT Mott boundary. LSCO more correlated than NCCO, role of apical oxygens. High temperature superconductivity occurs in the region where neither wave/itinerant nor localized/ particle picture fully applies. 22

Realistic DMFT as a conceptual tool and a computational tool
DMFT (simple yet accurate ? ) reference frame to think about electrons in solids and compute their properties. Compare different “states” of the system for the same value of parameters.  Understand Mechanism for ordering , magnetic, superconducting, exotic, ………. Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation Learning --> Design ? ] Qualitative and quantitative system specific results gives us confidence in the method. Many examples (sp, 3d,4d, 5d, 4f, 5f…) New arenas Interfaces, junctions heterostructures, artificial materials containing correlated electrons 23

“Matthias’s Rules” for High Tc
Metals. Must have d electrons (not just s s-p, nor f). Stay away from oxides. High symmetry is good, cubic is best. Nb3Sn Certain electron concentrations are favored (look for peak in density of states at Fermi level) Stay away from theorists “ Do not follow my rules “ 23

Needed :New set of Matthias Rules for the 21st Century
High symmetry is good, but layered structures offer more flexibility. Charged planes are good. Search for tetragonal symmetry. Avoid excessive itineracy or localization. Too much localization, i.e. magnets are bad. Too itinerant, low Tc. Goldilocks optimum at the intersection of solid state physics and chemistry. Collaborative efforts between experiment and theory. Expterimental group of Meigan Aronson Stony Brook and Rutgers. 25

Thanks!! for your attention!
$upport : NSF -DMR , DOE-Basic Energy Sciences, DOE-CMSN, AOSR - MURI, NSF-materials world network. Reference: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. Mod. Phys. 78, (2006) J. Shim K. Haule C Marianetti C. Weber S. Savrasov 26

Cuprates : fundamental questions Mechanism of Superconductivity
Quasiparticles glued by spin fluctuations. Predicted d-wave symmetry of order parameter . D. Scalapino , D. Pines K > -K> SC emerges from doping RVB paired spins in Mott insulator. [ P. W. Anderson, Science 235, 1196 (1987) Predicted d-wave symmetry of order paramteter and the pseudogap at low doping. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) 15

Optical Conductivity Schlesinger t.al (1993)
Shining light on correlated electrons. Optical conductivity Failure of the Standard Model: Anomalous Spectral Weight Transfer = Neff (T, )depends on T Optical Conductivity Schlesinger t.al (1993) Very Non local transfer of spectral weight in FeSi D. Van der Marel et.al (2005) [ 1 ev 800 cm-1] Weight does not recover up to 5 ev. s Other probes for correlated electrons X-rays, neutrons, electrons, the kitchen sink, theory ………. 6

Slave boson MFT. D wave symmetry of the SC OP
G. Kotliar and J. Liu PRB 38,5412 (1988) 18 D wave symmetry of the SC OP SC order and Tc decrease as x decreases. Low doping . pseudogap with D wave symmetry . VF is weakly dependent on doping, . Coherence incoherence crossover on the overdoped side. Related T=0 approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi , A. Paramenkanti PRL 87, (2001)

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved. Paul Dirac

"The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe..The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. At each level of complexity entirely new properties appear. Psychology is not applied biology, nor is biology applied chemistry. We can now see that the whole becomes not merely more, but very different from the sum of its parts."(Anderson 1972)

PWAnderson:Nature Physics 2, 138 (2006)
“It has been my (published) opinion for years that the cause of high-temperature superconductivity is no mystery. We now have a workable theory — not just for calculating the broad outlines (the transition temperature Tc, energy-gap shape, effect of doping, pseudogap temperature) but details of the anomalous phenomenology.” don’t “A crude version of this theory was published in 1988 by Zhang and co-authors (Supercond. Sci. Technol. 1, 36–38; 1988), based partly on my earlier ideas, and in a similar paper, Kotliar and Liu came to the same conclusions independently (Phys. Rev. B 38, 5142–5145; 1988). But the successes weren't then recognized because experiments were too primitive.” theory was Cluster (link/plaquette)DMFT, promising new avenue 19

4f’s heavy fermions, 115’s, CeMIn5 M=Co, Ir, Rh
 CeRhIn5: TN=3.8 K;   450 mJ/molK2  CeCoIn5: Tc=2.3 K;   mJ/molK2;  CeIrIn5: Tc=0.4 K;   750 mJ/molK2 Expts: F. P. Mena et.al, PRB 72, (2005). K. S. Burch et al., PRB 75, (2007). E. J. Singley, et, al PRB 65, (R) (2002). 12

J. Shim K Haule and GK Science (2007)
Structure Property Relation: Ce115’s Optics and Multiple hybridization gaps non-f spectra eV 10K 300K Ce In Out of plane In controls the physical properties -> we want to play with it. Larger gap due to hybridization with out of plane In Smaller gap due to hybridization with in-plane In 13

Mean-Field : Classical vs Quantum
Classical case Quantum case Hard!!! Easy!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) Pruschke et. al Adv. Phys. (1995) Georges et. al RMP (1996) IPT: Georges Kotliar (1992). . QMC: M. Jarrell, (1992), T. Pruschke D. Cox and M. Jarrell (1993), ED:Caffarel Krauth and Rozenberg (1994) Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999) ,……………………………………... Animate, and expand. A. Georges, G. Kotliar (1992)

Non local transfer of spectral weight in underdoped cuprates
Photoemission spectra near the antinodal direction in a Bi2212 underdoped sample. Campuzano et.al

DMFT concepts and tools in electronic structure.
LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). DMFT atom in a medium described DMFT Bands in a frequency dependent potential Numerous technical advances in the implementations of these ideas, advanced impurity solvers, optimal definitions of projectors, advanced basis sets ……… APPROXIMATE COMPUTATION OF ALL OBSERVABLES, OPTICS PHOTOEMISSION , TRANSPORT, ETC. . STUDY POSSIBLE “STATES” 9

alpa->delta volume collapse transition
F0=4,F2=6.1 Gouder , Havela PRB 2002, 2003 Photoemission 19

Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev)
24 Cedric Weber, Kristjan Haule, Gabriel Kotliar Nature Physics 10, 1038 (2010). Comanac et. al. Nature PhysicsNature Phys. 4, 287290 (2008). 23

Structure Property Relation in Correlated Systems: c axis optics in YBCO.
Compare with experiments Long sought goal: link structure to property. Dream becoming reality with the development of LDA+ cluster DMFT. Here we start with the structure of YBCO in the underdoped regime, we start from the structure, we derived electronic structure via LDA+ 2-site cluster DMFT which reduces the problem to 2-sites in a medium. The electronic spectral function near the fermi surface (spectra) shows characteristic arcs . The cycle continues the evaluatino of the optical conductivity which compares favorably with experiments.(by the UCSD group). Reference : M. Ferrero O. Parcollet G. Kotliar .A.Georges and D. Basov Phys. Rev. B 82, (2010) C axis optical conductvity Spectra

With decreasing doping gap increases, coherence peaks
Ratio AS/AN Ratio more universal, more symmetric With decreasing doping gap increases, coherence peaks less sharp->Non BCS McElroy,.. JC Davis, PRL 94, (2005) Exp:Bi2212 with STM Alternative explanation Fang, et.al. PRL vol 96, (2006). Sanibel 2008

Early studies Hubbard model : plaquette in a medium.
Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lett. 91, DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 56, 563. 20

Hubbard Model DMFT Collective field describing the localization delocalization phenomena Can be sublattice dependent, spin dependent, superconducting.. … 8 Require that the effective medium produces the best possible local spectral function. DMFT self consistency condition A. Georges and G. Kotliar PRB 45, 6479 (1992).

Mean-Field : Classical vs Quantum
Classical case Quantum case Hard!!! Easy!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) Pruschke et. al Adv. Phys. (1995) Georges et. al RMP (1996) IPT: Georges Kotliar (1992). . QMC: M. Jarrell, (1992), T. Pruschke D. Cox and M. Jarrell (1993), ED:Caffarel Krauth and Rozenberg (1994) Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999) ,……………………………………... Animate, and expand. A. Georges, G. Kotliar (1992)

Why do we need a MFT of correlated materials ?
Few exact solutions available. Need for simplification, understanding, design. Separate essential ingredients [e.g. phonons, orbitals, structure etc. ] responsible for an effects. Separate long wavelength non linearities (fluctuations, collective modes, defects) from local physics. Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation  Design] Compare different “states” of the system for the same value of parameters.  Understand the Mechanism .

DMFT Strategy and Ideas
Breaks problems in two parts a) study of mean field states from b ) evaluation of their energies. Second step, and detailed comparison experiments s within realistic implementions of electronic structure, e.g. LDA+DMFT. Qualitative lessons can be drawn from a) applied to simple models. Tools to think about correlated materials, e.g. Weiss fields, valence histograms, impurity model reference frames. etc. Locality assumption exact at high T.

Difficulties Technical Issues
2x2 cluster DMFT equations are considerably harder to solve and to interpret than single site DMFT. Uniqueness: no unique formulation of cluster DMFT Landscape of DMFT Solutions Problem Even within the same scheme at low T. In some region of parameters of the Hamiltonian, there are at low temperature many many solutions to the DMFT equations with different broken symmetries and ever increasing unite cells.

RVB phase diagram of the Cuprate Superconductors. Superexchange.
Tc controlled by J. Trvb, onset of spin pairing. < b>, TBE , coherence temperature, formation of QP.. Superconducting dome. Pseudogap evolves into SC Problems: a) poor description of the incoherent part b) MFT too uniform c) other states i.e. AF. Restricted form of the electron self energy. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi , A. Paramenkanti PRL 87, (2001)

Optimal doping: Coherence scale seems to vanish
underdoped scattering at Tc optimally overdoped Tc

4f’s heavy fermions, 115’s, CeMIn5 M=Co, Ir, Rh
 CeRhIn5: TN=3.8 K;   450 mJ/molK2  CeCoIn5: Tc=2.3 K;   mJ/molK2;  CeIrIn5: Tc=0.4 K;   750 mJ/molK2 out of plane in-plane 21

Theory : arXiv:1007.2867 Magnetism and Charge Dynamics in Iron Pnictides
Z. P. Yin, K. Haule, G. Kotliar arXiv:

Zhiping Yin, Rutgers University
X’ Z Y’ X’ Z Y’ The in-plane optical conductivity is way much better than in c-axis. 4/26/2017 arXiv: Zhiping Yin, Rutgers University

Z. P. Yin, KH, G. Kotliar Nature Physics in press .
QUALITATIVE INSIGHTS: a) Strongly Frequency Dependent Spin and Orbital Exchange Splitting b) Spin splitting large at high frequency. Orbital splitting large at low frequency. c) Qualitative difference between BaFe2As2 and Oxides. Z. P. Yin, KH, G. Kotliar Nature Physics in press .

DOS and valence histogram
There is transfer of spectral weight to high energies, spectral weight is conserved. But the DOS is featuresless no satellites, and resembles the LDA! Strong Correlations without Hubbard bands. Big difference between oxides and pnictides important role of As.

Evolution of the correlations in Hunds metals
LDA+DMFT LDA Static Magnetic Moment is Determined by strength of correlatoins AND by the shape of the Fermi Surface

On First Principles Approaches to Materials Science
Auguste Compte (1830). “ Every attempt to emply mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry “ Paul Dirac (1929) “The underlying laws necessary for the mathematical theory of the whole chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble “ “Approximate practical methods of applying quantum mechanics should be developed which can lead to an explanation of the main features of complex atomic systems without too much computation”

In the Rise of Complexity 1953-2002, PW Anderson writes:
“ …John Slater, already in 1953 , was obsessed with what I have rudely called the Great Solid State Dream machine. He envisioned that the new electronic computer could be applied to the task of automatically providing the electronic structure of any desired solid; and he literally believed that he would have the answers to any conceivable question . The latter idea was wrongheaded; but the former has gradually become a reality with the rise of what is now known as LDA. It is not enough appreciated that Slater himself provided the key element in that method. “

Phil Anderson on the rise of complexity 1953-2002
But there are many cases where it fails spectacularly: essentially all of the interesting class of substances with magnetic inner shell atoms, most of which exhibit what has been called the Mott Phenomenon- a dominance of the local repulsion among inner shell electrons. “…A new version of the Dream Machine has recently been invented which is quite successful in most of these cases-DMFT, dynamical mean field theory cooked up by Georges and Kotliar where the assumption of locality of the self energy in time is abandoned… “ Phil Anderson on the rise of complexity

Correlated electrons are not well described by either the itinerant (wave) picture, or the localized (particle) picture. Difficult non perturbative problem. Theoretical Approaches Phenomenology Study of exactly soluble (by numerical and analytical means) 1-d Hamiltonians Approximate methods for models in 2d-3d Guessing effective low energy theories. Dynamical Mean Field Theory . [ Exact in the Metzner Vollhard limit of infinite dimensions – now can bridge between structure and property of materials ]

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G
LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++ Spectra=- Im G(k,w) 12

Conceptual Underpinning Diagrams: PT in W and G.
Introduce projector Gloc Wloc : Chitra and Kotliar Phys. Rev. B 62, (2000) and Phys. Rev.B (2001).

Test notion of locality in LMTO basis set in various materials.
Proof of Principle Implementation Full implementation in the context of a a one orbital lattice model. P Sun and G. Kotliar Phys. Rev. B 66, (2002). Propose GW+DMFT . P.Sun and GK PRL (2004). Test various levels of self consistencyin Gnonloc Pinonloc Test notion of locality in LMTO basis set in various materials. N. Zeyn S. Savrasov and G. Kotliar PRL 96, , 2006 N Zeyn S. Savrasov and G. K PRL 96, (2006) GW self energy for Si Beyond GW Still, summing all diagramas with dynamical U and obtaining the GW starting point is extremely expensive. So this is still a point of principle rather than a practical tool. Coordination Sphere Coordination Sphere

LDA+DMFT as an approximation to the general scheme
U is parametrized in terms of Slater integrals F0 F2 F4 …. Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, (2008) Total energy is derived from a functional of the density and Gloc CHARGE SELF CONSISTENT LDA+DMFT S. Savrasov GK (2002) Savrasov, Kotliar, Abrahams, Nature ( 2001) 12

LDA+DMFT Self-Consistency loop [Savrasov Kotliar 2002] Derived from the functional.
Edc U DMFT REVIEW : G. Kotliar S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865 (2006).

Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev)
Bistability of a material near the Mott transition. Model realization of the Johanssen ideas. Central for understanding the physics of Pu.. New paradigm for thinking, about materials. N, Zein Following Aryasetiwan Imada Georges Kotliar Bierman and Lichtenstein. PRB (2004) Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu.

DMFT Phonons in fcc d-Pu
Notice the agreement. Usefulness of theory. Notice the discreapancy. Scientfici opportunity. C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

Photoemission Havela et. al. Phys. Rev. B 68, 085101 (2003)
Pu is non magnetic – Cm is magnetic TN ~ 150 K. K.Haule J. Shim and GK Nature 446, 513 (2007)

Hubbard Green DMFT Collective field describing the localization delocalization phenomena DMFT self consistency condition A. Georges and G. Kotliar PRB 45, 6479 (1992).

Superexchange Mechanism . K. Haule and GK Phys. Rev. B 76, (2007). Compare “normal “ and SC state at the same tempearture! Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 31

Neutron spectroscopy with LDA+DMFT
Theory: H. Park , K. Haule and GK Experiments: L Harriger H. Luo M. Liu T. Perring C Frost H. Ju M. Norman and Pengcheng Dai : arXiv:

CUPRATES

Return to models, Hubbard, t-J
Kinetic Energy Exchange Energy Plaquette DMFT: Lichtenstein and Kastnelson PRB (2000) T. Maier, et. al. 2001, Europhys. Lett. 56, 563. Sordi et.al. . arXiv: Civelli et. al. Phys. Rev. Lett. 100, (2008) Haule and Kotliar Phys. Rev. B 76, (2007) Bath 1 2 Real Space Link DMFT Ferrero et. al. Europhys. Lett. 85, (2009) Stanescu and Phillips P RB,69, (2004). Momentum Space 26

Early studies of plaquette and link DMFT of Hubbard
Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lett. 91, DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 56, 563.

Electron and Hole Doped Cuprates : Similar but Yet Different, why. C
Electron and Hole Doped Cuprates : Similar but Yet Different, why? C. Weber et.al. Nature Physics 2010 Apical oxygen NCCO : Robust AF Phase Comensurate Magnetism Lower Tc T^2 resistivity. Non monotonic angle dependence of SC order parameter ……… Review: Armitage Fournier Green (2009) 18

DMFT studies of copper oxides
Good agreement with many experiments follow from a simple [ plaquette/link /site ] reference frame. In general, better modeling with DMFT (more sites, more orbitals etc ) better results. Power of mean field theory : comparing “normal “ and magnetic states, comparing “normal” and superconducting states Strength of correlations (as quantified by single site DMFT) the most fundamental difference between NCCO and LSCO compounds. NCCO ( D < Dc2 )and LSCO (D > Dc2)straddle the Zaanen Sawatsky Allen localization delocalization boundary. Traced to the absence of apical oxygen in NCCO (structure property relation). 25

Iron Pnictides

Iron Pnictides- Chalcogenides
(Fe++)(Se__) Ba++(Fe++)2(As---)2 FeSe1-0.08, 1.48GPa), Mizuguchi et.al., arXiv:

Similarity/differences with cuprates
Iron pnictides (electron, hole, isovalent doping) AFM: Antiferromagnetic metal SC: Superconductor T: Tetragonal O: Orthorombic PG hole electron isovalent In high Tc’s carrier doping seems essential. In pnictides, isovalent (chemical pressure) works as well. It is just important to kill magnetism.

Basic Questions Relevant degreens of freedom, effective hamiltonians
Strength of the correlations . Localized vs itinerant Fe d electrons Mechanism of the superconductivity and magnetism……. New arena to test the LDA+DMFT methodology [ with and without experimental informantion!]

High temperature universality and V2O3
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL . (2003) T=170 T=300 Spinodal Uc2 Critical endpoint P. Limelette et.al. Science 302, 89 (2003) 20

Coherence Incoherence Crossover
LaO1-0.1F0.1FeAs Hubbard U is not the “relevant” parameter. The Hund’s coupling brings correlations! Specific heat within LDA+DMFT for LaO1-0.1F0.1FeAs at U=4eV n LDA value For J=0 there is negligible mass enhancement at U~W! K. Haule and G. Kotliar cond-mat arXiv: ,

wc=3000cm-1 ~ ev M. M. Qazilbash,1,, J. J. Hamlin,1 R. E. Baumbach,1 Lijun Zhang,2 D. J. Singh,2 M. B. Maple,1 and D. N. Basov1 Nature Physics 5, 647 (2009)

Photoemission reveals now Z ~ .3

Freq. dep. U matrix well parametrized by F0 F2 F4
F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2 Z =1/2 for x2- y2 and z2 , Z =1/3 f xz; yz zx orbitals.

F0 = 4:9 eV, F2 = 6:4 eV F4 = 4:3 eV., nc=6.2 LDA+DMFT calculations Kutepov Haule Savrasov and Kotliar PRB (2010). Mass renormalization = 3 without satellites Exp: W.Z. Hu et.al., PRL 101, (2008)

LDA+DMFT Magnetic moment .95 muB Expt 1 muB
EXPT: Hu, W. Z. et al. Phys. Rev. Lett. 101, (2008). EXPT: Nakajima, M. et al. Phys. Rev. B 81, (2010) Theory Yin et. al. (2010) L

Origin of the anisotropy is electronic
Optical features sharpen in the polarized spectra. Experimental predictions. Measurements underway ( not easy !)

Spin polarization of the frequency dependent self energy (real part)
Spin polarization of the frequency dependent self energy (real part). Frequency dependent exchange splitting. Large at high energies.

Orbital polarization of the frequency dependent hybridization Weiss field. Lives only at very low energies.

Magnetic Stripe Phase of the FeAs materials: new insights from LDA+DMFT Z. Yin K. Haule and GK [ in preparation] Focus on changes of Neff(L, T) at various energy scales L, in going to the magnetic state. a) At low energies conductivity goes up. Rapid coherence crossover from an incoherent normal state compensates for a loss of carriers. Gain kinetic energy at very low energies! For intermediate L, loss in carriers (kinetic energy )

Mass enhancement, plasma frequency Optical conductivity
U=5eV, J=0.7eV Interband peak ~ 0.6eV PRB 82, (2010) Plasma frequency: LDA ~ 2.6eV DMFT ~ 1.6eV Exp ~ 1.6eV Drude weight Mass enhancement of Fe-5d bands m*/mLDA~3 Exp: W.Z. Hu et.al., PRL 101, (2008) . Theory : Fourth generation of LDA+DMFT methods and codes. Kutepov Haule Savrasov and Kotliar (2010). Mass renormalization without satellites !

Yin Haule and GK (2010) . How well do we fare in the magnetic phase ?
Structure of Magnetic ordering correctly predicted by LSDA. LDA+DMFT is also OK in this respect. . Experiments: 0.87 µB Ba122 LSDA: µB at experimental crystal geometry LSDA: 0.87 µB , relaxed structure, As 0.1 angstrom away from experiments, bad DOS and wrong optics. LSDA+U: 0.87 µB with U=-0.85 eV, slightly improved DOS and optics, but not in good agreement with experiments. LDA+DMFT: 0.87 µB with U=5.0 and J=0.7 eV experimental crystal structure, excellent ARPES, optics, anisotropy

Go back to basics: U’s for DMFT. ( Kutepov et. al
Go back to basics: U’s for DMFT. ( Kutepov et. al. building on the PT in G and W by R. Chitra and G. K) Rigorous Definition of the Hubbard U and the Weiss Field Delta in a Solid. Kutepov et.al (2010) . Define a projector. Use the same projector in calculating the U’s that you will use in your DMFT calculation Approximate Wloc and Piloc using Self Consistent GW. Kutepov et. al. 2010 Eliminate the hybridization to the semicore states included in GW but not in LDA +DMFT by rescaling

DMFT Concepts and Tools in electronic structure
Local Self Energies and Correlated Bands Local Spectral Function Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Valence Histograms. Describes the history of the “atom” in the solid, multiplets! Functionals of density and spectra give total energies

Correlation phase diagram and ordered moment of Hunds metals. Yin et al.

DMFT Concepts Local Self Energies and Correlated Bands Local Spectral Function Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Valence Histograms. Describes the history of the “atom” in the solid, multiplets! Functionals of density and spectra give total energies

Electronic Structure Meets DMFT
LDA+DMFT Functional formulations, life without U Further extensions, clusters, GW+DMFT … V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtentsein and Katsnelson. PRB 57,6884 (1998). Almbladh et.al.(1999), Chitra and Kotliar (2000) (2001). Savrasov Kotliar and Abrahams (2001) Large Number of Groups and Many Compounds have been studied.

Conceptual Underpinning Diagrams: PT in W and G.
Introduce projector Gloc Wloc : Chitra and Kotliar Phys. Rev. B 62, (2000) and Phys. Rev.B (2001).

DMFT meets electronic structure. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Spectra=- Im G(k,w) DMFT Bands in a frequency dependent potential DMFT atom in a medium described Determine energy density and S self consistently from extremizing a functional . Savrasov and Kotliar PRB 69, , (2001) Full self consistent implementation 9

Standard Model of Solid State Physics
In many materials ( Cu, Au, …)electrons in solids behave as waves, quasiparticles [Sommerfeld Bloch] . Simple conceptual picture of excitations [Landau] Powerful computational tools Density functional theory , Kohn Sham . First order perturbation theory in the screened Coulomb interactions [GW ] [Hedin] 2

Henry Enhenreich in Electronic Structure for Materials Science, Science (1987)
DMFT in 2010, is also in an early stage of development. DMFT concepts and algorithms serve as a very primitive compass to guide us in journeys thru the vast space of possible materials. 2

Materials processing historicall speaking was a craft
Materials processing historicall speaking was a craft. It became an experimental science at a later stage. Then theory started to play a supporting role with the advent of density functinoal theory. DMFT in 2010, is still in an early stage of development. DMFT concepts and algorithms significantly extended the theoretical capablities. It serves as a very primitive compass to guide us in journeys thru the vast space of possible materials. Combination of theory and experiment opens an exciting range of possiblities. 29

Hubbard Model DMFT Collective field describing the localization delocalization phenomena Can be sublattice dependent, spin dependent, etc.. … 8 DMFT self consistency condition A. Georges and G. Kotliar PRB 45, 6479 (1992).

Thanks!! for your attention!
$upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, materials world network. J. Shim C Marianetti M Rozenberg A. Kutepov K. Haule Z. Yin C. Weber S. Savrasov

Optical conductivity in LDA+DMFT Shim, K Haule G Kotliar Science (2007)‏
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 22

P. W. Anderson, Science 235, 1196 (1987) High Tc , strongly correlated
Doped quasi- 2d- spin ½ Mott insulator. Hubbard model High temperature superconductivity as a result of introducing holes in an RVB backround. Superexchange. Anomalous properties result from doping a Mott insulating state. Caveat: controversial, no accepted way to evaluate strenght of correlations, no complete solution of even the (plaquette ) mean field theory of these models. 22

Slave Boson MFT: Difficulties.
Review : Lee Nagaosa and Wen RMP (2006). Mean field is too uniform on the Fermi surface. Poor description of the incoherent finite temperature regime. Mean field is static. Cluster: DMFT in its plaquette version may solve some of these problems!! (see also papers by Rome-Trieste collaboration Castellani Capone Tossati ) Reviews: T. Maier, M. Jarrell, T Pruschke, and M.H. Hettler. RMP 77, 1027 (2005) G. Kotliar et. al. RMP 78, (2006)

Trvb, onset of spin pairing.
RVB phase diagram of the Cuprate Superconductors. Correct prediction d wave symmetry of SC order parameter (and generic cuprate phase diagram) many years before the experiment. Tc controlled by J. Trvb, onset of spin pairing. < b>, TBE , coherence temperature, formation of QP.. Superconducting dome. Pseudogap evolves into SC Problems: a) poor description of the incoherent part b) MFT too uniform c) other states i.e. AF. d) Restricted form of the electron self energy. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi , A. Paramenkanti PRL 87, (2001)

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