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Dynamical RVB: Cluster Dynamical RVB: Cluster Dynamical Mean Field Studies of Dynamical Mean Field Studies of Doped Mott Insulators. Doped Mott Insulators. Gabriel Kotliar Gabriel Kotliar Rutgers CPHT Ecole Polytechnique Palaiseau Support :National Science Foundation. Chaire Blaise Pascal Fondation de l’Ecole Normale. MS2-High Tc Dresden July 11-25 2006 MS2-High Tc Dresden July 11-25 2006 Collaborators : M . Capone M Collaborators : M . Capone M Civelli K. Haule O. Parcollet Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla T.D. Stanescu V. Kancharla A.M.Trembaly B. Kyung D. Senechal A.M.Trembaly B. Kyung D. Senechal . .

Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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Page 1: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Dynamical RVB: Cluster Dynamical Mean Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators.Field Studies of Doped Mott Insulators.

Gabriel KotliarGabriel Kotliar

Rutgers CPHT Ecole Polytechnique Palaiseau

Support :National Science Foundation. Chaire Blaise Pascal Fondation de l’Ecole Normale.

MS2-High Tc Dresden July 11-25 2006MS2-High Tc Dresden July 11-25 2006

Collaborators : M . Capone M Civelli K. Haule Collaborators : M . Capone M Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla O. Parcollet T.D. Stanescu V. Kancharla A.M.Trembaly B. Kyung D. SenechalA.M.Trembaly B. Kyung D. Senechal . .

Page 2: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

References and CollaboratorsReferences and Collaborators

• References:• M. Capone et. al. in preparation• M. Capone and G. Kotliar cond-mat 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)

Page 3: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

ApproachApproach

• 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.

Page 4: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

RVB phase diagram of the Cuprate RVB phase diagram of the Cuprate Superconductors. Superexchange.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, 217002 (2001)

Page 5: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Problems with the approach.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.!!

Page 6: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Exact Baym Kadanoff functional ofwo variables. Exact Baym Kadanoff functional ofwo variables. ,G]. Restric to the ,G]. Restric to the degrees of freedom that live on a plaquette and its supercell extension.. degrees of freedom that live on a plaquette and its supercell extension.. Maps the many body problem onto Maps the many body problem onto a self consistenta self consistent impurity model impurity model

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

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 0511334

Page 7: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

.. • 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,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • 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

Page 8: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Optics and RESTRICTED SUM RULESOptics and RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

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

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Page 9: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Optics and RESTRICTED SUM RULESOptics and RESTRICTED SUM RULES

0( ) ( ) ( ) ( )n s n sd T T T T

Notice that <T>n is only defined for T> Tc, while <T>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.

Page 10: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Hubbard model

DrudeU

t2/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

Page 11: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Phys Rev. B 72, 092504 (2005)

cluster-DMFT, cond-mat/0601478

Kinetic energy change in t-J K Haule and GK

Kinetic energy decreases

Kinetic energy increases

Kinetic energy increases

Exchange energy decreases and gives

largest contribution to condensation energy

cond-mat/0503073

Page 12: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Finite temperature view of the phase Finite temperature view of the phase diagram t-J model.diagram t-J model.

K. Haule and GK (2006) K. Haule and GK (2006)

Page 13: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Doping Driven Mott transiton at low temperature, in 2d Doping Driven Mott transiton at low temperature, in 2d ((U=16 t=1, t’=-.3U=16 t=1, t’=-.3 ) Hubbard model ) Hubbard model

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k0) vs kK.M. Shen et.al. 2004

2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)Senechal et.al Senechal et.al PRL94 (2005)PRL94 (2005)

Page 14: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Page 15: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Superconductivity in the Hubbard model Superconductivity in the Hubbard model role of the role of the Mott transitionMott transition and and influence of the super-exchangeinfluence of the super-exchange. .

( work with M. Capone et.al V. Kancharla.et.al ( work with M. Capone et.al V. Kancharla.et.al CDMFT+ED, 4+ 8 sites t’=0) . CDMFT+ED, 4+ 8 sites t’=0) .

Page 16: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Kotliar andA.Tremblay. Cond mat Kotliar andA.Tremblay. Cond mat 05082050508205 M. Capone M. Capone (2006). (2006).

Page 17: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Page 18: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Superconducting DOS Superconducting DOS

=.08

= .16

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

Page 19: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Anomalous Self EnergyAnomalous Self Energy. (from Capone et.al.) Notice the . (from Capone et.al.) Notice the remarkable increase with decreasing doping! True remarkable increase with decreasing doping! True

superconducting pairing!! U=8tsuperconducting pairing!! U=8t

Significant Difference with Migdal-Eliashberg.

Page 20: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT 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.

.

Page 21: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 22: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 23: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 24: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 25: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

OutlineOutline

• 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.

Page 26: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 27: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 28: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

How is the Mott insulatorHow is the Mott insulatorapproached from the approached from the

superconducting state ?superconducting state ?

Work in collaboration with M. Capone M Civelli O Parcollet

Page 29: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

• 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 ?

Page 30: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Superconducting State t’=0Superconducting 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!

Page 31: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

• 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.

Page 32: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

•Can we connect the Can we connect the superconducting state with the superconducting state with the “underlying “normal” state “ ? “underlying “normal” state “ ?

What does the underlying “normal” What does the underlying “normal” state look like ?state look like ?

Page 33: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Follow the “Follow the “normal state”normal state” with doping. with doping. Civelli et.al. PRL 95, Civelli et.al. PRL 95, 106402 (2005)106402 (2005)

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k U=16 t, 0) vs k U=16 t,

t’=-.3t’=-.3

( 0, )vs k A k

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

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

K.M. Shen et.al. 2004

2X2 CDMFT

Page 34: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Page 35: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Optics and RESTRICTED SUM RULESOptics and RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

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

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Page 36: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Temperature dependence of the spectral Temperature dependence of the spectral weight of CDMFT in normal state. weight of CDMFT in normal state.

Carbone, see also ortholani for CDMFT. Carbone, see also ortholani for CDMFT.

Page 37: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Larger frustration: t’=.9t U=16tLarger frustration: t’=.9t U=16tn=.69 .92 .96n=.69 .92 .96

M. Civelli M. CaponeO. Parcollet and GK M. Civelli M. CaponeO. Parcollet and GK

PRL (20050PRL (20050

Page 38: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Add equation for the difference between Add equation for the difference between the methods. 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.

Page 39: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

. Spectral weight integrated up to 1 eV of the three BSCCO . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-films. a) under-

doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼overdoped, Tc=63 K; the fulloverdoped, Tc=63 K; the full

symbols are above Tc (integration from 0+), the open symbols symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the below Tc, (integrationfrom 0, including th weight of the

superfuid).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:

Page 40: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT 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.

.

Page 41: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

IssuesIssues• 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.

Page 42: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

• 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. 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). .

Page 43: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Page 44: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

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.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Page 45: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators
Page 46: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

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

Page 47: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Classical case Quantum case

A. Georges, G. Kotliar (1992)

Mean-Field : Classical vs QuantumMean-Field : Classical vs Quantum

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ¯

¶+ D-

¶- +òò ò

( )wD

†( )( ( ) )) (

MFo n oo n n Sc i c iG i s ss ww w D=- á ñ

( )

(()

)

11

([ ]

)[ ]n

n

kn

G i

G it ki m

w

wwD

D

=- - +

å

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

effh

00 ( )MF effH hm S=á ñ

ijff jj

e mh J h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Easy!!!

0 [ ]S th heffbá ñ=

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),……………………………………...

Page 48: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

DMFT Qualitative Phase diagram of a DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillingfrustrated Hubbard model at integer filling

T/W

Georges et.al. Georges et.al. RMP (1996) RMP (1996)

Kotliar Kotliar Vollhardt Vollhardt

Physics Today Physics Today (2004)(2004)

Page 49: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Single site DMFT and kappa organics. Qualitative phase Single site DMFT and kappa organics. Qualitative phase

diagram Coherence incoherence crosoverdiagram Coherence incoherence crosover. .

Page 50: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

Finite T Mott tranisiton in CDMFT Finite T Mott tranisiton in CDMFT O. Parcollet O. Parcollet

G. Biroli and GK PRL, 92, 226402. (2004))G. Biroli and GK PRL, 92, 226402. (2004))

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

Page 51: Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators

.. • 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,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • 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

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

Further extensions by periodizing cumulants rather than self energies. Stanescu and Further extensions by periodizing cumulants rather than self energies. Stanescu and GK (2005)GK (2005)

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Early SB DMFT. 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.

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• Normal State at low temperatures.

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Dependence on periodization scheme. Dependence on periodization scheme.

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

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Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

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.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

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t’=0

Phase diagram

Temperature Depencence of Integrated spectral weight

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E Energy difference between the normal E Energy difference between the normal and superconducing state of the t-J model. and superconducing state of the t-J model.

K. Haule (2006)K. Haule (2006)

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Temperature dependence of the spectral Temperature dependence of the spectral weight of CDMFT in normal state. weight of CDMFT in normal state.

Carbone, see also Toschi et.al for CDMFT. Carbone, see also Toschi et.al for CDMFT.