Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Gabriel Kotliar Rutgers CPHT Ecole Polytechnique Palaiseau Support :National Science Foundation. Chaire Blaise Pascal Fondation de l’Ecole Normale. MS2-High Tc Dresden July Collaborators : M. Capone M Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla A.M.Trembaly B. Kyung D. Senechal.

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

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.

RVB phase diagram of the Cuprate Superconductors. Superexchange. 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)

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.!!

Exact Baym Kadanoff functional ofwo variables. ,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: 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

. 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 8t ) Use exact diag ( Krauth Caffarel 1995 ) as a solver to reach larger U’s and smaller Temperature 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  -  (k,  )+  =  /b 2 -(  +b 2 t) (cos kx + cos ky)/b 2 + b > b(k),    (  ),    k  Extends the functional form of self energy to finite T and higher frequency. Larger clusters can be studied with VCPT CPT [Senechal and Tremblay, Arrigoni, Hanke ] CDMFT study of cuprates

Optics and RESTRICTED SUM RULES 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 Below energy

Optics and RESTRICTED SUM RULES Notice that n is only defined for T> Tc, while s exists only for T<Tc, so the use of this equation implies some sort of mean field picture to continue the normal state below Tc.

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

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

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

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) Senechal et.al PRL94 (2005)

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

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/ 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 andA.Tremblay. Cond mat M. Capone (2006).

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

Superconducting DOS    =.08    =.16 Superconductivity is destroyed by transfer of spectral weight.. Similar to slave bosons d wave RVB. M. Capone et. al

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

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..

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.

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=4t, 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 ?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, (2005) Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16 t, t’=-.3 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. K.M. Shen et.al X2 CDMFT

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

Optics and RESTRICTED SUM RULES 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 Below energy

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

Larger frustration: t’=.9t U=16t n= M. Civelli M. CaponeO. 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 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 G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, (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. coherence order parameter.  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 zerosUpon 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 spacecluster 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 Fermi surface.

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).

Classical case Quantum case A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum Easy!!! Hard!!! 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), 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),……………………………………...

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, (2004)) CDMFT results Kyung et.al. (2006)

. 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  -  (k,  )+  =  /b 2 -(  +b 2 t) (cos kx + cos ky)/b 2 + b > b(k),    (  ),    k  Better description of the incoherent state, more general functional form of the self energy to finite T and higher frequency. CDMFT : methodological comments 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=16t Left U=8t

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.

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

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

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