Dynamical Mean Field Approach to strongly Correlated Electrons

Dynamical Mean Field Approach to strongly Correlated Electrons Gabriel Kotliar Rutgers University Theoretical and Experimental Magnetism Meeting 3 -4 August Cosener house , Abingdon, Oxfordhisre UK Support : National Science Foundation. Department of Energy (BES).

Outline • Motivation. Introduction to DMFT ideas. • Application to the late actinides. • Application to Cuprate Supeconductors. Collaborators M. Civelli K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis ) A. M. Tremblay B. Kyung V. Kancharla (Sherbrook) M. Capone (Rome) O Parcollet(Saclay).

The Mott transition across in actinides

Cuprate Superconductors: doping the Mott insulator.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. Extremize a functional of the local spectra. Local self energy. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP 68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57, (2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti (to appear in RMP).

Mott transition in one band model. Review Georges et. al. RMP 96 T/W Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U, and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993). .

DMFT + electronic structure method Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites problem, in a medium of non interacting electrons obeying a self-consistency condition. (A. Georges et al. , RMP 68, 13 (1996)). DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated bands (s, p): use LDA or GW For correlated bands (f or d): with DMFT add all local diagrams. Gives total energy and spectra Technical Implementation is Involved. Different Impurity Solvers. [ED-NCA- Expansions in t and U , etc ] Different forms of Self consistency conditions for the bath in the clusters case. Different levels of complexity in the description of the electronic structure, simple models to all electron calculations. Review: G. Kotliar, S. Savrasov K. Haule, V. Oudovenko O Parcollet, C. Marianetti. Review of Modern Physics 2006.

Mean Field Approach Follow different “states” as a function of parameters. • Second step compare free energies. • Work in progress. Solving the DMFT equations are non trivial. T Configurational cordinate, doping, T, U, structure

Photoemission and Localization Trends in Actinides alpa->delta volume collapse transition F 0=4, F 2=6. 1 F 0=4. 5, F 2=7. 15 Curie-Weiss F 0=4. 5, F 2=8. 11 Curium has large magnetic moment and orders antif Pu does is non magnetic. Tc

The “DMFTvalence” in the late actinides

Minimum in melting curve and divergence of the compressibility at the Mott endpoint

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (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)

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar (2005)

Theoretical Approach [P. WAnderson, 1987] • Connection of the cuprate anomalies to the proximity to a doped Mott insulator without magnetic long range order. [Spin Liquid] • Study low energy one band models, Hubbard and t-J. Needed. a good mean field theory of the problem. RVB physics requires a plaquette as a reference frame.

CDMFT. study of cuprates • A functional of the cluster Greens function. Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. • Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. al T. Maier et. al. (2000). ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8 t ) • Use exact diag ( Krauth Caffarel 1995 ) and vertex corrected NCA as a solvers to study larger U’s and CDMFT as the mean field scheme.

RVB phase diagram of the Cuprate Superconductors. Superexchange. Flux-S+i. D spin liquid. [Affleck and Marston , G Kotliar] 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)

Superconductivity in the Hubbard model role of the Mott transition and influence of the super-exchange. ( work with M. Capone et. al V. Kancharla. et. al CDMFT+ED, 4+ 8 sites t’=0).

cond-mat/0508205 Anomalous superconductivity in doped Mott insulator: Order Parameter and Superconducting Gap. They scale together for small U, but not for large U. S. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G. Kotliar and. A. Tremblay. Cond mat 0508205 M. Capone (2006).

Superconducting DOS d =. 06 d =. 08 d= . 1 d =. 16 Superconductivity is destroyed by transfer of spectral weight. . Similar to slave bosons d wave RVB. M. Capone et. al

Doping Driven Mott transiton at low temperature, in 2 d (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 Senechal et. al PRL 94 (2005) Nodal Region 2 X 2 CDMFT Civelli et. al. PRL 95 (2005)

Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302

Optics and RESTRICTED SUM RULES Below energy Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state

Optics and RESTRICTED SUM RULES <T>n is defined for T> Tc, while <T>s exists only for T<Tc. <T>n is a strong function of temperature in the normal state. Carbone et. al (2006).

Hubbard versus t-J model Kinetic energy in Hubbard model: • Moving of holes • Excitations between Hubbard bands Hubbard model U Drude t 2/U t Experiments Excitations into upper Hubbard band Kinetic energy in t-J model • Only moving of holes Drude J-t intraband interband transitions t-J model no-U ~1 e. V

Kinetic energy change in t-J K Haule and GK Kinetic energy increases cluster-DMFT, cond-mat/0601478 Kinetic energy decreases Kinetic energy increases cond-mat/0503073 Phys Rev. B 72, 092504 (2005) Exchange energy decreases and gives largest contribution to condensation energy

Haule and Kotliar (2006) Coarsed grained or “local “ susceptibility around (pp) Scalapino White PRB 58, 8222 (1988)

Conclusion • DMFT versatile tool for advancing our understanding, and predicting properties of strongly correlated materials. • Theoretical spectroscopy in the making. Substantial work is needed to refine the tool. • Great opportunity for experimental-theoretical interactions. • Refine the questions and our understanding by focusing on differences between the DMFT results and the experiments.

Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case A. Georges, G. Kotliar (1992)

Anomalous Self Energy. (from Capone et. al. ) Notice the remarkable increase with decreasing doping! True superconducting pairing!! U=8 t Significant Difference with Migdal-Eliashberg.

<l. s> in the late actinides [DMFT results: K. Haule and J. Shim ]

a-U

XAS and EELS J. Tobin et. al. PRB 72, 085109 (2005)

Double well structure and d Pu Qualitative explanation of negative thermal expansion[Lawson, A. C. , Roberts J. A. , Martinez, B. , and Richardson, J. W. , Jr. Phil. Mag. B, 82, 1837, (2002). G. Kotliar J. Low Temp. Physvol. 126, 1009 27. (2002)] F(T, V)=Fphonons+ Finvar Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

“Invar model “ for Pu-Ga. Lawson et. al. Phil. Mag. (2006) Data fits if the excited state has zero stiffness.

References and Collaborators • • • References: M. Capone et. al. in preparation M. Capone and G. Kotliar cond-mat/0603227 Kristjan Haule, Gabriel Kotliar cond-mat/0605149 M. Capone and G. K cond-mat/0603227 Kristjan Haule, Gabriel Kotliar cond-mat/0601478 • Tudor D. Stanescu and Gabriel Kotliar cond-mat/0508302 • S. S. Kancharla, M. Civelli, M. Capone, B. Kyung, D. Senechal, G. Kotliar, A. -M. S. Tremblay cond-mat/0508205 • M. Civelli M. Capone S. S. Kancharla O. Parcollet and G. Kotliar Phys. Rev. Lett. 95, 106402 (2005)

Mott Phenomeman and High Temperature Superconductivity Began Study of 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 understand some qualitative aspects found in the experiment. In which case the next step LDA+CDMFT or GW+CDMFT could be then be used make realistic modelling of the various spectroscopies. • B) Or we do not, in which case other degrees of freedom, or inhomogeneities or long wavelength non Gaussian modes are essential as many authors have surmised. • Too early to tell, talk presented some evidence for A. .

Correlations Magnetism and Structure across the actinide series : a Dynamical Mean Field Theory Perspective G. Kotliar Physics Department and Center for Materials Theory Rutgers University. . Collaborators K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis ) A. M. Tremblay B. Kyung (Sherbrook) M. Capone (Rome) O Parcollet(Saclay). Support: DOE- BES DOE-NNSA. Expts. : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J. Singleton J. Smith J Thompson J. Tobin Plutonium Futures Asilomar July 9 -13 (2006).

M. Capone and GK cond-mat 0511334. Competition fo superconductivity and antiferromagnetism.

Temperature dependence of the spectral weight of CDMFT in normal state. Carbone, see also ortholani for CDMFT.

Finite temperature view of the phase diagram t-J model. K. Haule and GK (2006)

Outline • Introduction. Mott physics and high temperature superconductivity. Early Ideas: slave boson mean field theory. Successes and Difficulties. • Dynamical Mean Field Theory approach and its cluster extensions. • Results for optical conductivity. • Anomalous superconductivity and normal state. • Future directions.

UPS of alpha-U GGA - He I (hv=21. 21 e. V), He II (hv=40. 81 e. V) - f-electron features is enhanced in He II spectra. Opeil et al. PRB(2006)

-LDA+DMFT reproduces peaks near -1 e. V, 0. 3 e. V, and EF -The peak near -3 e. V corresponds to U 6 d states. n_f=2. 94

Cluster Extensions of Single Site DMFT Many Techniques for solving the impurity model: QMC, (Fye. Hirsch), NCA, ED(Krauth –Caffarel), IPT, …………For a review see Kotliar et. Al to appear in RMP (2006)

n_5/2=2. 41 n_7/2=0. 53

How is the Mott insulator approached from the superconducting state ? Work in collaboration with M. Capone M Civelli O Parcollet

• In BCS theory the order parameter is tied to the superconducting gap. This is seen at U=4 t, but not at large U. • How is superconductivity destroyed as one approaches half filling ?

Superconducting State t’=0 • • Does it superconduct ? Yes. Unless there is a competing phase. Is there a superconducting dome ? Yes. Provided U /W is above the Mott transition. Does the superconductivity scale with J ? Yes. Provided U /W is above the Mott transition. Is superconductivity BCS like? Yes for small U/W. No for large U, it is RVB like!

• The superconductivity scales with J, as in the RVB approach. Qualitative difference between large and small U. The superconductivity goes to zero at half filling ONLY above the Mott transition.

• Can we connect the superconducting state with the “underlying “normal” state “ ? What does the underlying “normal” state look like ?

Follow the “normal state” with doping. Civelli et. al. PRL 95, 106402 (2005) Spectral Function A(k, ω→ 0)= -1/π G(k, ω → 0) vs k U=16 t, t’=-. 3 K. M. Shen et. al. 2004 If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface. 2 X 2 CDMFT

Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302

Optics and RESTRICTED SUM RULES Below energy Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state

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

Add equation for the difference between the methods. • Can compute kinetic energy from both the integral of sigma and the expectation value of the kinetic energy. • Treats normal and superconducting state on the same footing. •

. Spectral weight integrated up to 1 e. V of the three BSCCO films. a) underdoped, 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 Began Study of 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 understand some qualitative aspects found in the experiment. In which case LDA+CDMFT or GW+CDMFT could be then be used to account semiquantitatively for the large body of experimental data by studying 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. .

Issues • What aspects of the unusual properties of the cuprates follow from the fact that they are doped Mott insulators using a DMFT which treats exactly and in an umbiased way all the degrees of freedom within a plaquette ? • Solution of the model at a given energy scale, Physics at a given energy • Recent Conceptual Advance: DMFT (in its single site a cluster versions) allow us to address these problems. • A) Follow various metastable states as a function of doping. • B) Focus on the physics on a given scale at at time. What is the right reference frame for high Tc.

• 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). Uniform Solutions. S-wave superconductors. Uniform RVB states. Other RVB states with d wave symmetry. Flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988). Spectrum of excitation have point zeros. Upon doping they become a d –wave superconductor. (Kotliar and Liu 1988). .

The simplest model of high Tc’s t-J, PW Anderson Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations) BK Functional, Exact cluster in k space cluster in real space

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

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)G. Kotliar and D. Vollhardt Physics Today (2004).

Mean-Field : Classical vs Quantum Classical case Easy!!! Quantum case Hard!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) 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) , …………………. . . • Pruschke et. al Adv. Phys. (1995) • Georges et. al RMP A. (1996) Georges, G. Kotliar (1992)

DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W Georges et. al. RMP (1996) Kotliar Vollhardt Physics Today (2004)

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

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

• • • CDMFT : methodological comments Functional of the cluster Greens function. Allows the. investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. Can study different states on the same footing allowing for the full frequency dependence of all the degrees of freedom contained in the plaquette. DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS w-S(k, w)+m= w/b 2 -(D+b 2 t) (cos kx + cos ky)/b 2 +l b----> b(k), D ----- D(w), l ----- l (k ) Better description of the incoherent state, more general functional form of the self energy to finite T and higher frequency. Further extensions by periodizing cumulants rather than self energies. Stanescu and GK (2005)

Early SB DMFT. • There are two regimes, one overdoped one underdoped. • Tc has a dome-like shape. • High Tc superconductivity is driven by superexchange. • Normal state at low doping has a pseudogap a low doping with a d wave symmetry.

• Normal State at low temperatures.

Dependence on periodization scheme.

Energetics and phase separation. Right U=16 t Left U=8 t

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

Temperature Depencence of Integrated spectral weight Phase diagram t’=0

E Energy difference between the normal and superconducing state of the t-J model. K. Haule (2006)

Conclusion • DMFT studies of electrons and lattice displacements. • Valence changes and transfers of spectral weight. [ Consistent picture of Pu-Am-Cm]. • Alpha and delta Pu, screened (5 f)^5 configuration. Differ in the degree of screening. Different views [ Pu non magnetic (5 f)^6, Pu magnetic ] • Magnetism and defects. • Important role of phonon entropy in phase transformations.

LS vs jj coupling in Am and Cm

Temperature dependence of the spectral weight of CDMFT in normal state. Carbone, see also Toschi et. al for CDMFT.

UPS of alpha-U GGA - He I (hv=21. 21 e. V), He II (hv=40. 81 e. V) - f-electron features is enhanced in He II spectra. Opeil et al. PRB(2006)

-LDA+DMFT reproduces peaks near -1 e. V, 0. 3 e. V, and EF -The peak near -3 e. V corresponds to U 6 d states. n_f=2. 94

<l. s> in the late actinides [DMFT results: K. Haule and J. Shim ]

a-U

Why is Epsilon Pu (which is smaller than delta Pu) stabilized at higher temperatures ? ? Compute phonons in bcc structure.

Phonon entropy drives the epsilon delta phase transition • Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. • At the phase transition the volume shrinks but the phonon entropy increases. • Estimates of the phase transition following Drumont and G. Ackland et. al. PRB. 65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Double well structure and d Pu Qualitative explanation of negative thermal expansion[Lawson, A. C. , Roberts J. A. , Martinez, B. , and Richardson, J. W. , Jr. Phil. Mag. B, 82, 1837, (2002). G. Kotliar J. Low Temp. Physvol. 126, 1009 27. (2002)] F(T, V)=Fphonons+ Finvar Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

“Invar model “ for Pu-Ga. Lawson et. al. Phil. Mag. (2006) Data fits if the excited state has zero stiffness.

Approach • Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. [ Leave out disorder, electronic structure, phonons …] • Follow different “states” as a function of parameters. [Second step compare free energies which will depend more on the detailed modelling…. . ] • Work in progress. The framework and the resulting equations are very non trivial to solve.

Approach the Mott point from the right Am under pressure. Experimental Equation of State (after Heathman et. al, PRL 2000) “Soft” Mott Transition? “Hard” Density functional based electronic structure calculations: q Non magnetic LDA/GGA predicts volume 50% off. q Magnetic GGA corrects most of error in volume but gives m~6 m. B (Soderlind et. al. , PRB 2000). q Experimentally, Am has non magnetic f 6 ground state with J=0 (7 F 0)

Am equation of state. LDA+DMFT. New acceleration technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar cond-mat. 0507552 (2005)

Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884, (1998 ), Svane cond-mat 0508311] and Sunca. [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is determined by multiplet splittings.

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (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)

Phonon entropy drives the epsilon delta phase transition • Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. • At the phase transition the volume shrinks but the phonon entropy increases. • Estimates of the phase transition following Drumont and G. Ackland et. al. PRB. 65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Double well structure and d Pu Qualitative explanation of negative thermal expansion[Lawson, A. C. , Roberts J. A. , Martinez, B. , and Richardson, J. W. , Jr. Phil. Mag. B, 82, 1837, (2002). G. Kotliar J. Low Temp. Physvol. 126, 1009 27. (2002)] F(T, V)=Fphonons+ Finvar Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

“Invar model “ for Pu-Ga. Lawson et. al. Phil. Mag. (2006) Data fits if the excited state has zero stiffness.

a-U

Why is Epsilon Pu (which is smaller than delta Pu) stabilized at higher temperatures ? ? Compute phonons in bcc structure.

What can we learn from “small” Cluster-DMFT? Phase diagram t’=0

The simplest model of high Tc’s t-J, PW Anderson Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations) BK Functional, Exact cluster in k space cluster in real space

CDMFT. study of cuprates • • • AFunctional of the cluster Greens function. Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. T. Maier et. al. (2000). ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8 t ) Use exact diag ( Krauth Caffarel 1995 ) and vertex corrected NCA as a solvers to study larger U’s and CDMFT as the mean field scheme. Recently (K. Haule and GK ) the region near the superconducting –normal state transition temperature near optimal doping was studied using NCA + DCA-CDMFT. DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS w-S(k, w)+m= w/b 2 -(D+b 2 t) (cos kx + cos ky)/b 2 +l b----> b(k), D ----- D(w), l ----- l (k ) Extends the functional form of self energy to finite T and higher frequency. Larger clusters can be studied with VCPT [Senechal and Tremblay, Arrigoni, Hanke ]

Exact Baym Kadanoff functional ofwo variables. G[S, G]. Restric to the degrees of freedom that live on a plaquette and its supercell extension. . Maps the many body problem onto a self consistent impurity model 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. Tremblay Kyung Senechal cond-matt 0511334

Problems with the approach. • Stability of the MFT. Ex. Neel order. Slave boson MFT with Neel order predicts AF AND SC. [Inui et. al. 1988] Giamarchi and L’huillier (1987). • Mean field is too uniform on the Fermi surface, in contradiction with ARPES. [Penetration depth, Wen and Lee ][Raman spectra, sacutto’s talk, Photoemission ] • Description of the incoherent finite temperature regime. Development of DMFT in its plaquette version may solve some of these problems. !!