CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France Strongly Correlated Electron Materials: A Dynamical Mean Field Theory Perspective Gabriel Kotliar and Center for Materials Theory & CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France Seminaire de la federation PHYSTAT-SUD 11 Mai 2006 Support : Chaire Blaise Pascal Fondation de l’Ecole Normale.

Collaborators A. Georges (Ecole Polytechnique) O. Parcollet (CEA-Saclay) G. Biroli (CEA-Saclay) M Civelli (ILL-Grenoble) M. Capone ( Rome ) T. Stanescu (U. Illinois) K. Haule (Rutgers) B. Kyung, A. M. Tremblay (Sherbrook)

Correlated Electron Materials Not well described the “Standard Model” Solid State Physics based on Band Theory. Reference System: QP. [Fermi Liquid Theory and Kohn Sham DFT+GW ] Partially filled f and d shells, some organic materials. Spectacular “big” effects. High temperature superconductivity, colossal magneto-resistance, huge volume collapses…………….. Anomalous physical properties, large resistivities. Breakdown of the rigid band picture. Non trivial evolution of spectral functions.

Kappa Organics Phase diagram of (X=Cu[N(CN)2]Cl) F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005) Phase diagram of (X=Cu[N(CN)2]Cl) S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

Cuprate Experimental Phase diagram Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Copper oxide superconducors CuO2

Kappa organics H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998) Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003) t’/t ~ 0.6 - 1.1

Model Hamiltonians t’ t’’ m U t What is this model I talked about earlier waying we more or less agree on. Recall that the materials are planar, as ween on the top left, and a single band is necessary as seen on the bottom left. This means, we can consider one electron per unit cell (CuO2 unit) with a hopping amplitude that can be fitted from the band structure calculation and a short range (screened) interaction when two electrons are on the same site. Even with such a simple model, exact diagonalizations are difficult. Even with just 16 site, it takes 4 Giga bits just to store the states.

Perspective U/t d t’/t Doping Driven Mott Transition Pressure Driven Mott transtion d t’/t

Open Problems Controversial Issues What is the mechanism for high temperature superconductivity. Why is realized in the copper oxides? What are the essential low energy degrees of freedom to describe the physics of these materials at a given energy scale? Proper reference frame for understanding the correlated solid.

RVB physics and Cuprate Superconductors P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987) Connection between the anomalous normal state of a doped Mott insulator and high Tc. t-J limit. Slave boson approach. <b> coherence order parameter. k, D singlet formation order parameters.Baskaran Zhou Anderson , (1987)Ruckenstein Hirshfeld and Appell (1987) . Other states flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) have point zeors.

RVB phase diagram of the Cuprate Superconductors. Superexchange. The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero. Superconducting dome. Pseudogap evolves continously into the superconducting state. 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, 217002 (2001)

PWA: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. 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.

Problems with the approach. Neel order. How to continue a Neel insulating state ? Need to treat properly finite T. Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] . Theory:r[T]=x-Ta x2 , Exp: r[T]= x-T a. Doping and polarization dependence of the Raman scattering intensity. [LeTacon et.al. 2006] Mean field is too uniform on the Fermi surface, in contradiction with ARPES. Difficulties in describing quantitavely the incoherent regime. Develoment of cluster DMFT may solve some of these problems.!!

Approach Leave out inhomogeneous states and ignore disorder. Study minimal model of a doped Mott insulator to understand the electronic structure in that regime. Approach the problem directly from finite temperatures,not from zero temperature. Address issues of finite frequency –temperature crossovers. Compare with experiments. Reconsider. CDMFT has made this program feasible. Reference frame describes coherent and incoherent regimes on the same footing. The framework and the resulting equations are very non trivial to solve and to interpret. Very exciting time.

Dynamical Mean Field Theory. Cavity Construction. A. Georges and G Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Reviews: A. Georges W. Krauth G.Kotliar and M. Rozenberg RMP (1996)

Limit of large lattice coordination Metzner Vollhardt, 89 Neglect k dependence of irreducible quantities such as self energy Muller-Hartmann 89

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) Animate, and expand. Pruschke et. al Adv. Phys. (1995) Georges et. al RMP (1996) IPT: Georges Kotliar (1992). . QMC: M. Jarrell, (1992), NCA 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) ,……………………………………... A. Georges, G. Kotliar (1992)

CDMFT: removes limitations of single site DMFT No k dependence of the self energy. No d-wave superconductivity. No Peierls dimerization. No (R)valence bonds. Various cluster approaches, DCA momentum spcace. Cellular DMFT G. Kotliar et.al. PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, 205108 (2004) T. D. Stanescu and G. Kotliar cond-mat/0508302 Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP in Press.

DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W Synthesis: Brinkman Rice Hubbard Castellani et.al. Kotliar Ruckenstein Fujimori 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.

Single site DMFT and kappa organics Single site DMFT and kappa organics. Qualitative phase diagram Coherence incoherence crosover.

CDMFT results Kyung et.al. (2006) Finite T Mott tranisiton in CDMFT O. Parcollet G. Biroli and GK PRL, 92, 226402. (2004)) CDMFT results Kyung et.al. (2006)

Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

Evolution of the k resolved Spectral Function at zero frequency Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) U/D=2.25 U/D=2 Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

Doping Driven Mott transiton at low temperature, in 2d (U=16 t=1, t’=- 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. 2004 Antinodal Region 2X2 CDMFT Nodal Region Civelli et.al. PRL 95 (2005)

Larger frustration: t’=.9t U=16t n=.69 .92 .96 M. Civelli M. CaponeO. Parcollet and GK PRL (20050

Spectral shapes. Large Doping Stanescu and GK cond-matt 0508302

Small Doping. T. Stanescu and GK cond-matt 0508302

Photoemission spectra near the antinodal direction in a Bi2212 underdoped sample. Campuzano et.al EDC along different parts of the zone, from Zhou et.al.

Lower Temperature, AF and SC M. Capone and GK, Kancharla et. al. AF+SC d d

Can we continue the superconducting state towards the Mott insulating state ? For U > ~ 8t YES. For U ~ < 8t NO, magnetism really gets in the way.

Connection between superconducting and normal state properties. K Connection between superconducting and normal state properties. K. Haule and G. Kotliar cond-mat (2006) Elucidate how the spin superexchange energy and the kinetic energy of holes changes upon entering the superconducting state! Mechanism of superconductivity. Temperature dependence of optical spectral weigths.

RESTRICTED SUM RULES Below energy Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy.

Treatement needs refinement The kinetic energy of the Hubbard model contains both the kinetic energy of the holes, and the superexchange energy of the spins. Physically they are very different. Experimentally only measures the kinetic energy of the holes.

Dca vs CDMFT critical temperatures for the t-J model. K. Haule (2006) Dca vs CDMFT critical temperatures for the t-J model.

Energy Balance between the normal and superconducing state.

H.J.A. Molegraaf et al., Science 295, 2239 (2002). . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under- doped, Tc=70 K; b) ∼ optimally doped, Tc=80 K; c) overdoped, Tc=63 K; the full symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the superfuid). H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) . Recent review:

Mott Phenomeman and High Temperature Superconductivity Studied minimal model of a doped Mott insulator within plaquette Cellular DMFT ? Rich Structure of the normal state and the interplay of the ordered phases. Work needed to reach the same level of understanding of the single site DMFT solution. A) Either that we will account semiquantitatively for the large body of experimental data once we study more realistic models of the material. B) Or we do not, in which case other degrees of freedom, or inhomgeneities or long wavelength non Gaussian modes are essential as many authors have surmised. Too early to tell, talk presented some evidence for A. .

Collaborators O. Parcollet (CEA-Saclay) G. Biroli (CEA-Saclay) M Civelli (ILL-Grenoble) M. Capone ( Rome ) T. Stanescu (U. Illinois) K. Haule (Rutgers) B. Kyung, A. M. Tremblay (Sherbrook)