1. Gabriel Kotliar Rutgers
Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators.
Gabriel Kotliar
Rutgers
Statistical Mechanics Meeting Rutgers University
December
Collaborators : G. Biroli M . Capone M Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla A.M.Tremblay B. Kyung D. Senechal.

2. Outline Anderson’s RVB framework.
(Cluster ) Dynamical Mean Field Theory
Optical conductivity and kinetic energy changes in the superconducting state.(K. Haule and GK)
Other results
Outlook

3. P. W. Anderson, Science 235, 1196 (1987)
Doped quasi- 2d- spin ½ Mott insulator
t-J model , Hubbard model
High temperature superconductivity as a result of introducing holes in an RVB backround.
Anomalous properties result from doping a Mott insulating state.

4. t-J Hamiltonian
Slave Boson Formulation: Baskaran Zhou Anderson (1987) Ruckenstein Hirschfeld and Appell (1987)
b+i bi +f+si fsi = 1
What is this model I talked about earlier waying we more or less agree on.
Recall that the materials are planar, as ween on the top left, and a single band is necessary as seen on the bottom left.
This means, we can consider one electron per unit cell (CuO2 unit) with a hopping amplitude that can be fitted from the band structure calculation and a short range (screened) interaction when two electrons are on the same site.
Even with such a simple model, exact diagonalizations are difficult.
Even with just 16 site, it takes 4 Giga bits just to store the 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 like a a d –wave superconductor.

5. 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 , TBE 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)

6. Cuprates: phase diagram
Unusual
Metal
More
conventional
metal (FL?)
“Pseudo-gap”
Antiferromagnetic
Mott
Insulator
superconductor
Underdoped
Overdoped
Optimal doping

7. 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, Photoemission ]
Description of the incoherent finite temperature regime.
Development of DMFT in its plaquette version may solve some of these problems.!!

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

9. A(w)
Animate, and expand.
A. Georges, G. Kotliar (1992) 11

10. CLUSTER EXTENSIONS: umbiased reduction of the many body problem to a plaquette in a medium. Natural framework to formulate RVB. Can continue, metastable states.
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) Tremblay Kyung Senechal cond-matt

11. Two Site Cellular DMFT (G. Kotliar et. al
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)
U/t=4.
Edit. LISA. 24

12. Order Parameter and Superconducting Gap do not scale for large U
Order Parameter and Superconducting Gap do not scale for large U ! ED study in the SC state Capone et.al. (2006) Sarma et. al.

13. 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

14. Optics and RESTRICTED SUM RULES
<T>n is only defined for T> Tc, while <T>s exists only for T<Tc
Experiment: use of this equation implies extrapolation.
Theory : use of this equation implies of mean field picture to continue the normal state below Tc.

15. Excitations into upper
Optical Conductivity
Hubbard model U
Experiments
intraband
interband
transitions
Excitations into upper
Hubbard band
Kinetic energy in t-J model
Only moving of holes
Drude
t-J model
J-t
no-U
~1eV

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

17. H.J.A. Molegraaf et al., Science 295, 2239 (2002).
. 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:

18. Where is the change of exchange energy? K. Haule (2006)

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

20. Doping Driven Mott transiton at low temperature, in 2d (U=16 t=1, t’=-
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. 2004
Antinodal Region
2X2 CDMFT
Senechal et.al PRL94 (2005)
Nodal Region
Civelli et.al. PRL 95 (2005)

21. Conclusions: Dynamical RVB
Captures the essential RVB physics. Slave boson method.
Solves many problems of the earlier RVB.
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 eqs. is available yet.

22. Happy Birthday Elihu and Phil !

24. Pseudoparticle picture

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

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

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

28. 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) .

29. 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).

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

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

32. Anomalous Self Energy. (from 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.

33. 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. .

39. 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.

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

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

44. H.J.A. Molegraaf et al., Science 295, 2239 (2002).
. 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:

45. P. W. Anderson. Connection between high Tc and Mott physics
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 zerosUpon doping they become a d –wave superconductor. (Kotliar and Liu 1988). .

46. 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

47. 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.

49. Georges et.al. RMP (1996) Kotliar Vollhardt Physics Today (2004)
DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling
T/W
Georges et.al. RMP (1996) Kotliar Vollhardt Physics Today (2004)
Different way of thinking was generated by the study of the Mott transition at integer filling.
Universality and system specificity. . Bridge atomic physic and band physics. Crossovers with changing degrees of freedom.

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

51. Dependence on periodization scheme.

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

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

55. Pseudoparticle picture

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

60. Behavior of the optical mass and the plasma frequency.

61. Magnetic Susceptibility

62. References and Collaborators
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)

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

64. 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)

65. 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.

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

72. CDMFT study of cuprates.
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 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
w-S(k,w)+m= w/b2 -(D+b2 t) (cos kx + cos ky)/b2 +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 CPT [Senechal and Tremblay, Arrigoni, Hanke ]

73. 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 , TBE 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)