Superconductivity near the Mott transition a Cluster Dynamical Mean Field Theory (CDMFT) perspective Superconductivity near the Mott transition a Cluster Dynamical Mean Field Theory (CDMFT) perspective Gabriel Kotliar Rutgers Coherence and incoherence in stronly correlated systems. July 3-7 Rome Italy Collaborators : G. Biroli M. Capone M Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla A.M.Tremblay B. Kyung D. Senechal A. Georges

References Collaborators M. Capone and GK PRB 74, 54513(2006) M. Civelli M. Capone A. Georges K. Haule O Parcollet T. Stanescu and GK cond-mat M. Civelli et. al. PRL (2005) B. Kyung S. Kancharla D. Senechal A. Ms Tremblay M. CIvellli and GK PRB ( K. Haule and GK ( preprint) Closely related work: M. Capone M. Fabrizio C. Castellani and E. Tosatti Phys. Rev. Lett 93, (2004). M. Capone M. Fabrizio C. Castellani and E. Tosatti Phys. Rev. Lett 93, (2004). Science (2002).

Cuprates Damascelli, Shen, Hussain, RMP 75, 473 (2003)

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

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

t-J Hamiltonian RVB P.W. Anderson (1987) Slave Boson Formulation: Baskaran Zhou Anderson (1987) Ruckenstein Hirschfeld and Appell (1987) b + i bi +f +  i f  i = 1 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 like a a d –wave superconductor.

Superexchange Mechanism: proximity to the Mott transition renormalizes down kinetic energy, but not the superexchange. Superexchange Mechanism: proximity to the Mott transition renormalizes down kinetic energy, but not the superexchange. Coherent Quasiparticles Re Slave Boson Mean Field Theory Phase Diagram. Formation of Singlets

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). Gauge fluctuations destablize the mean field [Ubbens and Lee] Temperature dependence of the penetration depth [Wen and Lee, Ioffe and Millis ]. Theory:  [T]=x-T x 2, Exp:  [T]= x-T Z= x. Mean field is too uniform on the Fermi surface, in contradiction with ARPES. No proper description the incoherent regime and the coherent-incoherent and the incoherent regime.

Dynamical Mean Field Theory Map lattice model into quantum impurity problem in a self consistent medium. The quantum impurity problem is used to generate local quantities, i.e. a local self energy. From local quantities one reconstruct k dependent spectral functions, susceptibilities, etc. Single site, DMFT k independent self energy (cumulant).] Cluster extensions, incorporate additional k dpendence. Follow different mean field states, AF, normal, supeconductor, etc as a function of parameters.

CLUSTER EXTENSIONS: umbiased reduction of the many body problem to a plaquette in a medium. 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 (2006). Employ different impurity solvers. ED (Civelli Capone) CTQMC (Haule)NCA (Haule)

Single site DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W Synthesis: Synthesis: Brinkman Rice Hubbard Castellani, C., C. DiCastro, D. Feinberg, and J. Ranninger, 1979, Phys. Rev. Lett. 43, 1957.

Good description of the evolution of the spectra and transport, not too close to the Mott transition, at relatively high temperatures. For example V2O3 ( Rozenberg et. al. 1996) K-organics (Limelette et.al. 2002). At lower temperatures, closer to the Mott transition, cluster description is necessary. Study at low temperatures the doping driven Mott transition.

The approach validates many crucial features of the RVB theory.

Tunnelling DOS (NCA-tJ): Gap (distance between coherence peaks) increases with decreasing doping.

Order Parameter and Superconducting Gap do not scale for large U ! ED study in the SC state Capone and GK PRB (2006) Kancharla et. al. cond-mat

CDMFT on a plaquette gives rise to a “Dynamical RVB “pictures which retains all the good features of the previous slave boson mft treatment

The quasiparticle residue, decreases with doping but the effective mass (Fermi velocity) remains finite. [M. Grilli and GK] PRL (1990) The gap in the tunneling density of states increases with decreasing doping. The ph asymmetry grows with the approach to the Mott insulator. Superconducting order parameter does not scale with the gap.

But with substantial two differences!!! which have important consequences a) nodal antinodal dichotomy b) v  decreses with decreasing doping in superconductor. [Two- gap picture]

Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK PRB (2006)

Nodal Antinodal Dichotomy [Civelli et. al. (2007)]

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

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)

Scaling of the || velocity in the superconductor with doping. M. Civelli et. al. cond-mat K. Haule and GK>

Consequences for linear term coefficient of the penetration depth.. K. Haule and GK

Experiments:two superconducing gaps, with opposite dependence on doping ? Antinodal gap increases towards the Mott insulator while v  decreases? Coherence and single-particle excitations in the high-temperature superconductors. Guy Deutscher,Nature 397, (1999) Andreev reflection. M. Opel et. al. PRB 61, 9752 (2000) Venturini, F. et al., Doping dependence of the electronic Raman spectra in Phys. Chem. Solids, 63, 2345 (2001). Raman scattering.

LeTacon et. al. Two Energy Scales and two Quasiparticle Dynamics in the Superconducting. Nature Physics 2, 537 (2006) Raman scattering.. K. Tanaka, et. al Distinct Fermi-Momentum Dependent Energy Gaps in Deeply Underdoped Bi2212. arXiv:cond- mat/ ARPES M. C. Boyer et. al. arXiv: Imaging the Two Gaps of the High-TC Superconductor Pb-Bi2Sr2CuO6+x Tunnelling. arXiv: Spectroscopic distinction between the normal state pseudogap and the superconducting gap of cuprate high T_{c} superconductors Li Yu, et. al.. C- Axis Optical Spectrsocopy. arXiv: Spectroscopic distinction between the normal state pseudogap and the superconducting gap of cuprate high T_{c} superconductors Li Yu, et. al.. C- Axis Optical Spectrsocopy.

Metodological advantages. We can follow well defined phases as a function of parameters, doping temperature. Well defined (meta) stable states, in contrast to the old slave boson MFT approach. CDMFT treats properly the incoherent state, with short ranged magnetic correlations.

AF and superconductivity: M. Capone and GK PRB 74, AFM blue dashed line with circles and dSC red solid line with squares order parameters as a function of doping for four values of the repulsion U/ t=4,8,12, and 16. The dSC order parameter is multiplied by a factor of 10 for graphical purposes.

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.

Evolution of the q integrated staggered spin susceptilibty K. Haule and GK (2006) Evolution of the q integrated staggered spin susceptilibty K. Haule and GK (2006)

Conclusions: CDMFT studies of superconductivity near a Mott insulator. Captures the essential RVB physics of the interplay of the Mott transition and superconductivity. Kinetic energy supression. Retains the good aspects of the slave boson MFT. Solves many problems of the earlier slave boson. [e.g.doping dependence of T linear term in the penetration depth ] Allows the continuation of spin liquid states as metastable states. Functional of local spectral functions. Nodal Antinodal dichotomy, emerges naturally. Work in progress. No full solution of the CDMFT eqs.and its lattice interpretation, (on the same level of single site DMFT), is available yet.

Happy Birthday Carlo!!!!

Temperature dependence of the arcs. doping=.09 (underdoped) Plaquette DMFT. K. Haule and GK Temperature dependence of the arcs. doping=.09 (underdoped) Plaquette DMFT. K. Haule and GK

Lines of Zeros and Spectral Shapes. 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

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

On the accuracy of CDMFT

U/t=4. Two Site Cellular DMFTin the 1D Hubbard model Two Site Cellular DMFT ( G.. Kotliar et.al. PRL (2001)) in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69, (2004)T. D Stanescu and GK PRB (2006)24

On the interpretation of CDMFT

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)

RVB phase diagram of the Cuprate Superconductors. Superexchange. The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. Approach the Mott insulator, Z, charge stiffness, T B E =Tcoh goes to zero. M* finite. 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)

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)

Pseudoparticle picture

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

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

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

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.

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

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

. 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:

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.

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.

Dependence on periodization scheme.

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

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

Pseudoparticle picture

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

Behavior of the optical mass and the plasma frequency.

Magnetic Susceptibility

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)

P. W. Anderson, Science 235, 1196 (1987)

RVB phase diagram of the Cuprate Superconductors. 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)

RVB Approach Anderson (1987) 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…..] Solve the plaquette mean field equations!!!! Work in progress.

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/

. 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

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 continuously 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)

Copper oxide superconducors CuO 2

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

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.

Origin of the ph asymmetry

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).

Copper oxide superconducors CuO 2

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

U t  t’ t’’ Model Hamiltonians

Photoemission spectra near the antinodal direction in a Bi2212 underdoped sample. Campuzano et.al

RVB phase diagram of the Cuprate Superconductors. Superexchange. Proximity to Mott insulator renormalizes the kinetic energy Trvb increases. Proximity to the Mott insulator reduce the charge stiffness, and QPcoherence scale. T BE goes to zero. Superconducting dome. Pseudogap with d wave symmetry. 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)

Approach Understand the physics resulting from the proximity in the context of the simplest models. Leave out disorder, electronic structure,phonons,inhomogeneous structures. Follow different “states” as a function of parameters. Second step compare free energies which will depend more on the detailed modelling Local (plaquette ) Mott physics. Leave out long wavelength collective modes. Look at experiments. Work in progress. The framework and the resulting CDMFT equations are very non trivial to solve.

Lower Temperature, AF and SC M. Capone and GK, AF AF+SC SC AF SC 

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..

Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). A(  ) 10

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

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

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)

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. ( Parcollet Biroli and GK PRL, 92, (2004)) ) Uc= , Tc/D=1/44. Tmott~.01 W U/D=2 U/D=2.25

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)

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

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