Optical Conductivity of CupratesSuperconductors: a Dynamical

RVB perspectiveWork with K. Haule (Rutgers)

Collaborators : G. Biroli M . Capone M Civelli A. PeraliCollaborators : G. Biroli M . Capone M Civelli A. Perali

O. Parcollet T.D. Stanescu K. Haule C. Bolech V. Kancharla O. Parcollet T.D. Stanescu K. Haule C. Bolech V. Kancharla

A.M.Tremblay B. Kyung D. Senechal M Sindel S. Savrasov A GeorgesA.M.Tremblay B. Kyung D. Senechal M Sindel S. Savrasov A Georges

K. Haule, G. Kotliar, Europhys Lett. 77, 27007 (2007).

“Optics sumrule” Conference Roma 03-07-2007

• Restricted Optical Sum Rules

• What are they ?

• What are they good for ?

Optics and RESTRICTED SUM RULES

0( ) , H

effeff effd P JiV

, ,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

J. Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 105 (1995). L. Baldassarre Poster P1 this conference.

Hubbard model single site DMFT. [ W(T) is T dependent near Mott trans.

• The temperature dependence in W(T) is a measure of the residual coupling between the low energy degrees of freedom and the rest .

• It is particularly strong in the vicinity of a Mott transition.

• Doping driven Mott transition influences the physics of cuprates!

Optics and RESTRICTED SUM RULES can be used to infer the mechanism of

superconductivity

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

<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.

Hirsch, Science 295, 2226 (2002).

J. E. Hirsch, Science, 295, 5563 (2226)

BCS: upon pairing potential energy of electrons decreases, kinetic energy increases(cooper pairs propagate slower)Condensation energy is the difference

non-BCS: kinetic energy decreases upon pairing(pairs propagate faster in superconductor)

• The kinetic energy of the Hubbard model contains both the kinetic energy of carriers in a spin backround , and the superexchange energy of the spins.

• Physically they are very different.

• Experimentally only measures the kinetic energy of the holes.

Low energy HKinetic energy of projected fermions

Superexchange

Hubbard model

U

t-J model

J-t

Drude

no-U

Experiments

intraband interband transitions

~1eV

Excitations into upper Hubbard band

Kinetic energy in t-J model•Only moving of holes

Optical Conductivity

Dynamical RVBPoint of view

• Study simple [“unrealistic”] models of the doped Mott insulator (RVB)

• Capture local physics. Reference frame is a plaquette in a medium.

• Recent advances thru the use of Cluster DMFT• Incorporate at a later stage, other elements, long

wavelenght collective modes, inhomogenieties, disorder.

Superexchange Mechanism

•

Coherent Quasiparticles

Re

0

0b

Slave Boson Mean Field Theory Phase Diagram.

Formation of Singlets

TBC onset of QP coherence

TRVB onset of single pairing

Crossover from BCS at large doping to correlated superconductor at low doping

Impurity Model-----Lattice Model

Weiss FieldPowerful cluster solvers, NCA, OCA, CTQMC, ED….

E Energy difference between the normal and superconducing state of the

t-J model. K. Haule GK

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, (integrate from 0, including the

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 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) .

CDMFT optics t-J model

CDMFT optics

• Optical weight increases as temperature decreases.The magnitude is approximately given by single site DMFT [as first computed by Toschi et.al, PRL (2005). ].

• Substantial new physics is brought by the cluster effects. Existence of d wave superconductivity and pseudogap. Avoided criticality, power laws.

• Crossover from pseudogap to fermi liquid as a function of doping.

• Notice that in spite of the opening of a pseudogap. The spectral weight does not decrease with decreasing temperature for reasonable cuttoffs.!!!

Cuttoff and temperature dependence of integrated optial

spectral weight

Single site DMFT vs CDMFT changes in optical weight in the

normal state

At which frequency do we recover all the spectral weight ?

• At very high frequencies. Of the order of 3t. (t, .3-.45 ev)

• It is due to the anomalous greens

function. Not visible in photoemission.

Optical Mass at low doping

Optical mass and plasma frequency

Padilla et.al.

Conclusion

• Optical anomalies, do NOT rule out the proximity to a Mott transition as a basis for a

theoretical approach to describe the cuprates. a) temperature and doping dependence of the optical

spectral weight. • CDMFT on a plaquette, is a substantial improvement

over the earlier slave boson approach, to describe the optics, and many other key experiments. [ My talk on Wendesday].

• Further work to improve: a) our understanding of the plaquette CDMFT equations, b) to make the models more realistics c) to make CDMFT more flexible and d) to incorporate vertex corrections are warranted e) refine the connection with spin liquids [J. C Domenge and GK]

Power laws in optics. A. El Azrak,et.al. PR B 49, 9846 (1994).D. van der Marel, Nature 425, 271 (2003).

Optical Weight of the lower Hubbard band

Stephan and Horsch Int. Jour Mod Phys B6, 141 (1992)

Eskes Oles Meinders and Stephan PRB 50 (1994) 17980

Optical weight of the upper Hubbard band

Avoided Quantum Criticality

• Intermediate physics phenomena.

• No analytic understanding of the dimension 2/3.

RVB phase diagram of the 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.

• Slave boson approach. <b> coherence order parameter. singlet formation order parameters.

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.

Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

Finite T, DMFT and the Energy Landscape of Correlated Materials

T

Conclusion

• More quantitative comparison with experiments On the theory side. Investigate effects of t’

t’’ and more realistic electronic structure. Effects of vertex corrections,

periodization. More extreme underdoping and

overdoping. Better impurity solvers.

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 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, 217002 (2001)

Hubbard vs t-J

Drude Transition from uper to lower Hubbard band at U

Incoherent part of the spectra

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.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Optical Spectral Weight Can be Used to infer the mechanism of

superconductivity.

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.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

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.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

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 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, 217002 (2001)

For reviews of cluster methods see: Georges et.al. RMP (1996)

Maier et.al RMP (2005), Kotliar et.al RMP (2006)

Weiss FieldAlternative (T. Stanescu and

G. K. ) periodize the cumulants rather than the self energies.

Parametrizes the physics in terms of a few functions .

Impurity solver, NCA, ED, CTQMC

Superexchange mechanism?

• Near the Mott transition the optical weight has a surprising large T dependence.

M. J. Rozenberg et al., Phys. Rev. Lett. 75, 105 (1995).

• This phenomena of buildup of spectral weight with reducing temperature was found in cuprates, and was well accounted

by single site DMFT. Toschi et. al. Phys. Rev. Lett. 95, 097002 (2005)

• At very low doping, one can separate two components. [Coherent and Incoherent]

• At large they merge into one “Drude-like” broad frequency range.

• Expected temperature dependence in overdoped region. [Narrowing of Drude peak]. Anomalous temperature dependence

at low doping.