High Temperature Superconductors. What can we learn from the study of the doped Mott insulator within plaquette Cellular DMFT. Gabriel Kotliar Center for Materials Theory Rutgers University CPhT Ecole Polytechnique Palaiseau, and SPhT CEA Saclay, France Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U. Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke) $$Support : NSF DMR. Blaise Pascal Chair Fondation de l’Ecole Normale. Geneve February 10 th 2006

Outline Strongly Correlated Electrons. Basic Dynamical Mean Field Ideas and Cluster Extensions. High Temperature Superconductivity and Proximity to the Mott Transition. Early Ideas. Slave Boson Implementation. CDMFT results for the 2x2 plaquette. a) Normal State Photoemission. [Civelli et. al. PRL (2005) Stanescu and Kotliar cond-mat] b) Superconducting State Tunnelling Density of States. [ Kancharla et.al. Capone et.al] c) Optical Conductivity near optimal doping and near Tc [K. Haule and G. Kotliar]

Correlated Electron Materials Are not well described by either the itinerant or the localized framework. Do not fit in the “Standard Model” Solid State Physics. Reference System: QP. [Fermi Liquid Theory and Kohn Sham DFT+GW ] Compounds with partially filled f and d shells. Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, colossal magneto-resistance, huge volume collapses…………….. Need new starting point for their description. Non perturbative problem. DMFT New reference frame for thinking about correlated materials and computing their physical properties.

Breakdown of the Standard Model Large Metallic Resistivities (Takagi)

Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Van der Marel (2005) Takagi (2003 ) Neff depends on T

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

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

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)

For reviews of cluster methods see: Georges et.al. RMP (1996) Maier et.al RMP (2005), Kotliar et.al cond-mat to appear in RMP (2006) Kyung et.al cond-mat  Weiss Field Alternative (T. Stanescu and G. K. ) periodize the cumulants rather than the self energies. Parametrizes the physics in terms of a few functions.

U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69, (2004) ]

Effective Action point of view. Identify observable, A. Construct a free energy functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.). DMFT Local Spectral Function. (R. Chitra and G.K (2000) (2001). H=H 0 + H 1.  [a,J 0 ]=F 0 [J 0 ]–a J 0 _ +  hxc [a] Functional of two variables, a,J 0. H 0 + A J 0 Reference system to think about H. J 0 [a] Is the functional of a with the property 0 =a 0 computed with H 0 + A J 0 Many choices for H 0 and for A Extremize a to get  [J 0 ]= ext a  [a,J 0 ]

Finite T, DMFT and the Energy Landscape of Correlated Materials T

Pressure Driven Mott transition How does the electron go from the localized to the itinerant limit ?

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

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

Ising critical endpoint: prediction Kotliar Lange Rozenberg Phys. Rev. Lett. 84, 5180 (2000) Observed! In V 2 O 3 P. Limelette et.al. Science 302, 89 (2003)

. Three peak structure, predicted Georges and Kotliar (1992) Transfer of spectral weight near the Mott transtion. Predicted Zhang Rozenberg and GK (1993). ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, (2003).

Conclusions. Three peak structure, quasiparticles and Hubbard bands. Non local transfer of spectral weight. Large metallic resistivities. The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. Coherent and incoherence crossover. Real and momentum space. Theory and experiments begin to agree on the broad picture.

Some References Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, (1996). Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2006). Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Cuprate superconductors and the Hubbard Model. PW Anderson Schematic Phase Diagram (Hole Doped Case)

Methodological Remarks Leave out inhomogeneous states and ignore disorder. What can we understand about the evolution of the electronic structure from a minimal model of a doped Mott insulator, using Dynamical Mean Field Theory ? Approach the problem directly from finite temperatures,not from zero temperature. Address issues of finite frequency –temperature crossovers. As we increase the temperature DMFT becomes more and more accurate. DMFT provides a reference frame capable of describing coherent and incoherent regimes within the same scheme.

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. coherence order parameter.  singlet formation order parameters.Baskaran Zhou Anderson, Ruckenstein et.al (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, T BE 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, (2001)

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:  [T]=x-Ta x 2, Exp:  [T]= x-T a. Mean field is too uniform on the Fermi surface, in contradiction with ARPES. No quantitative computations in the regime where there is a coherent-incoherent crossover,compare well with experiments. [e.g. Ioffe Kotliar 1989] CDMFT may solve some of these problems.!!

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.

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

. 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. 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. 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 the self energy to finite T and higher frequency. CDMFT study of cuprates

Can we continue the superconducting state towards the Mott insulating state ?

Competition of AF and SC AF AF+SC SC or AF SC 

Competition of AF and SC M. Capone M. Civelli and GK (2006)

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.

Superconducting State t’=0 Does the Hubbard model superconduct ? Is there a superconducting dome ? Does the superconductivity scale with J ? Is it BCS like ?

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

Order Parameter and Superconducting Gap do not always scale! ED study in the SC state Capone Civelli Parcollet and GK (2006)

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

Evolution of DOS with doping U=12t. Capone et.al. : Superconductivity is driven by transfer of spectral weight, slave boson b 2 !

Superconductivity is destroyed by transfer of spectral weight. M. Capone et. al. Similar to slave bosons d wave RVB.

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.

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

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

Dependence on periodization scheme.

Comparison of 2 and 4 sites

Spectral shapes. Large Doping Stanescu and GK cond-matt

Small Doping. T. Stanescu and GK cond-matt

Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K cond-matt

Lines of Zeros and Spectral Shapes. Stanescu and GK cond-matt

Connection between superconducting and normal state. Transfer of spectral weight in optics. Elucidate how the spin superexchange energy and the kinetic energy of holes changes upon entering the superconducting state! Origin of the powerlaws discovered in the groups of N. Bontemps and D. VarDerMarel. K. Haule and GK development of an ED+DCA+NCA approach to the problem. New tool for addressing the neighborhood of the dome.

Optical Conductivity near optimal doping. [ Theory DCA ED+NCA study, K. Haule and GK ]

Kristjan Haule: there is an avoided quantum critical point near optimal doping.

What is happening near optimal doping ?? Avoided Quantum Criticality [K. Haule and GK]

Optical conductivity t-J. K. Haule

Behavior of the optical mass and the plasma frequency.

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

DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena. Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes] Formulate the problem in terms of local quantities (which we can usually compute better). Allows to perform quantitative studies and predictions. Focus on the discrepancies between experiments and mean field predictions.[Substantiates and improves over early slave boson studies of the phenomena] Generate useful language and concepts. Follow mean field states as a function of parameters. K dependence gets strong as we approach the Mott transition. Psedogap. Fermi surfaces and lines of zeros of Tsvelik (quasi-one dimensional systems ) T. Stanescu and GK (proximity to a Mott transition in 2 d). Conclusions

Conclusions Superconductivity Reproduced the basic general features of the early slave boson treatment. The approach naturally introduces a strong anisotropy and particle hole asymmetry in the problem. It reveals the frequency dependence of the self energy which is growing with doping [ J and U ] Establishes the clear differences between superconductivity above and below the Mott transition.

Conclusions Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! Truncation of the Fermi surface as a STRONG COUPLING instability (compare weak coupling RG e.g. Honerkamp, Metzner, Rice ) General phenomena, but the location of the cold regions depends on parameters. Fundamental difference between electron and hole doped cuprates.

oQualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near ( ,  /2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. oElectron doped case, has in the underlying normal state quasiparticles leave in the (  0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (  /2,  /2).

Approaching the Mott transition: from high T CDMFT Picture Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! It can be seen starting from high temperatures. D wave gapping of the single particle spectra as the Mott transition is approached. Real and Imaginary part of the self energies grow approaching half filling. Unlike weak coupling! Scenario was first encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, (2004).

High Temperature Superconductors. What can we learn from the study of the doped Mott insulator within plaquette Cellular DMFT ? We can learn a lot, but there is still a lot of work to be done until we reach the same level of understanding that we have of the single site DMFT solution. This work is definitely in progress. a) Either that we can account semiquantitatively for the large body of experimental data once we study more realistic models of the material. Or b) 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. It is still too early to tell, but some evidence in favor of a) was presented in this seminar. Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U. Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke) $$Support : NSF DMR. Blaise Pascal Chair Fondation de l’Ecole Normale.

What is the origin of the asymmetry ? Comparison with normal state near Tc. K. Haule Early slave boson work, predicted the asymmetry, and some features of the spectra. Notice that the superconducting gap is smaller than pseudogap!!

Magnetic Susceptibility

Outline Theoretical Point of View, and Methodological Developments. : Local vs Global observables. Reference Frames. Functionals. Adiabatic Continuity. The basic RVB pictures. CDMFT as a numerical method, or as a boundary condition.Tests. The superconducting state. The underdoped region. The optimally doped region. Materials Design. Chemical Trends. Space of Materials.

Connection with large N studies.

oDynamical Mean Field Theory and a cluster extension, CDMFT: G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) oCluser Dynamical Mean Field Theories: Causality and Classical Limit. G. Biroli O. Parcollet G.Kotliar Phys. Rev. B Cluster Dynamical Mean Field Theories a Strong Coupling Perspective. T. Stanescu and G. Kotliar ( 2005) References

Evolution of the normal state: Questions. Origin of electron hole asymmetry in electron and doped cuprates. Detection of lines of zeros and the Luttinger theorem.

ED and QMC

U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69, (2004) ]

Electron Hole Asymmetry Puzzle

What about the electron doped semiconductors ?

Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k electron doped P. Armitage et.al Civelli et.al Momentum space differentiation a we approach the Mott transition is a generic phenomena. Location of cold and hot regions depend on parameters.

Approaching the Mott transition: CDMFT Picture Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! D wave gapping of the single particle spectra as the Mott transition is approached. Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, (2004).

High Temperature Superconductors. What can we learn from the study of the doped Mott insulator within plaquette Cellular DMFT ? We can learn a lot, but there is still a lot of work to be done until we reach the same level of understanding that we have of the single site DMFT solution. Either that we can account semiquantitatively for the large body of experimental data once we study more realistic models of the material. 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. It is still too early to tell. Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U. Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke) $$Support : NSF DMR. Blaise Pascal Chair Fondation de l’Ecole Normale.

Conclusion

OPTICS

CDMFT and NCS as truncations of the Baym Kadanoff functional

Ex: Baym Kadanoff functional,a= G, H 0 = free electrons. Viewing it as a functional of J 0, Self Energy functional(Potthoff)

Weiss Field Functional

Example: single site DMFT semicircular density of states. GKotliar EPJB (1999) Extremize Potthoff’s self energy functional. It is hard to find saddles using conjugate gradients. Extremize the Weiss field functional.Analytic for saddle point equations are available Minimize some distance

Approaching the Mott transition: CDMFT Picture Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! D wave gapping of the single particle spectra as the Mott transition is approached. Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, (2004).

Dynamical RVB brings in strong anistropy in the underdoped regime.

What about the electron doped semiconductors ?

Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k electron doped P. Armitage et.al Civelli et.al Momentum space differentiation a we approach the Mott transition is a generic phenomena. Location of cold and hot regions depend on parameters.

oQualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near ( ,  /2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. oElectron doped case, has in the underlying normal state quasiparticles leave in the (  0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (  /2,  /2).

 Can we connect the superconducting state with the “underlying “normal” state “ ?  Yes, within our resolution in the hole doped case.  No in the electron doped case.  What does the underlying “normal state “ look like ?  Unusual distribution of spectra (Fermi arcs) in the normal state.

To test if the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

Introduce much larger frustration: t’=.9t U=16t n=

Approaching the Mott transition: Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, but the location of the cold regions depends on parameters. With the present resolution, t’ =.9 and.3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

Fermi Surface Shape Renormalization ( t eff ) ij =t ij + Re(  ij 

Fermi Surface Shape Renormalization Photoemission measured the low energy renormalized Fermi surface. If the high energy (bare ) parameters are doping independent, then the low energy hopping parameters are doping dependent. Another failure of the rigid band picture. Electron doped case, the Fermi surface renormalizes TOWARDS nesting, the hole doped case the Fermi surface renormalizes AWAY from nesting. Enhanced magnetism in the electron doped side.

Understanding the location of the hot and cold regions. Interplay of lifetime and fermi surface.

Superconductivity as the cure for a “sick” normal state.