Superconductors: a Dynamical

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Optical Conductivity of Cuprates

Superconductors: a Dynamical
RVB perspective

Work with K. Haule (Rutgers)

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



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

A.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
eff
eff eff
d P J
iV

 

 
  
 

,,
eff eff eff
H J P


2
0
( ),
ne
d 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 electriccurrent P polarizatio
n
Below energy


2
2
k
k
k
n
k






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 s
d 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 H

Kinetic 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
 
0
b

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 Field


Powerful 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.
k, 

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 eff
d P J
iV

 

 
  
 

,,
eff eff eff
H J P


2
0
( ),
ne
d 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 electriccurrent P polarizatio
n
Below energy


2
2
k
k
k
n
k





Optical Spectral Weight Can be
Used to infer the mechanism of
superconductivity.


RESTRICTED SUM RULES

0
( ),
eff eff
d P J
iV

 

 
  
 

,,
eff eff eff
H J P


2
0
( ),
ne
d 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 electriccurrent P polarizatio
n
Below energy


2
2
k
k
k
n
k





RESTRICTED SUM RULES

0
( ),
eff eff
d P J
iV

 

 
  
 

,,
eff eff eff
H J P


2
0
( ),
ne
d 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 electriccurrent P polarizatio
n
Below energy


2
2
k
k
k
n
k





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 Field


Alternative (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.