Transcript Electronic Raman Scattering in cuprates Evidence of new

High Temperature Superconductors. What can we
learn from the study of the doped Mott insulator
within plaquette Cellular DMFT.
• Gabriel Kotliar
• Center for Materials Theory Rutgers University
• CPhT Ecole Polytechnique Palaiseau, and
SPhT CEA Saclay , France
Geneve February 10th 2006
Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U.
Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu,
(Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M
Tremblay, D. Senechal B. Kyung
(Sherbrooke)
$$Support : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.
Outline
•
Strongly Correlated Electrons. Basic Dynamical Mean
Field Ideas and Cluster Extensions.
• High Temperature Superconductivity and Proximity
to the Mott Transition. Early Ideas. Slave Boson
Implementation.
• CDMFT results for the 2x2 plaquette.
• a) Normal State Photoemission. [Civelli et. al. PRL
(2005) Stanescu and Kotliar cond-mat]
b) Superconducting State Tunnelling Density of States. [
Kancharla et.al. Capone et.al]
c) Optical Conductivity near optimal doping and near Tc
[K. Haule and G. Kotliar]
Correlated Electron Materials
• Are not well described by either the itinerant or the
localized framework . Do not fit in the “Standard Model”
Solid State Physics. Reference System: QP. [Fermi
Liquid Theory and Kohn Sham DFT+GW ]
• Compounds with partially filled f and d shells.
• Have consistently produce spectacular “big” effects thru
the years. High temperature superconductivity, colossal
magneto-resistance, huge volume collapses……………..
• Need new starting point for their description. Non
perturbative problem. DMFT New reference frame for
thinking about correlated materials and computing their
physical properties.
Breakdown of the Standard Model
Large Metallic Resistivities (Takagi)
 1
e2 k F (k F l )
  1Mott
h
(100 cm)1
Transfer of optical spectral weight non local in
frequency Schlesinger et. al. (1994), Van der Marel
(2005) Takagi (2003 ) Neff
depends on T
DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479
(1992). First happy marriage of atomic and band physics.
1
G(k , i ) 
i   k  (i )
Reviews: A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter
Vollhardt Physics Today 57,(2004)
Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - hå Si
i, j
Quantum case
 (t

i , j  ,
i
HMF = - heff So
b
ij
  ij )(ci† c j  c†j ci )  U  ni  ni 
i
b
b
¶
†
ò ò cos (t )[ ¶ t + m- D (t - t ')]cos (t ') + U ò no­ no¯
0 0
0
heff
D (w)
m0 = áS0 ñHMF ( heff )
heff =
å
J ij m j + h
j
Phys. Rev. B 45, 6497
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn ) =
å
k
1
[D (iwn ) -
1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
Cluster Extensions of Single Site DMFT
latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )  2 ( )(cos kx.cos ky )  .......
Many Techniques for
solving the impurity
model: QMC, (FyeHirsch), NCA,
ED(Krauth –Caffarel),
IPT, …………For a
review see Kotliar et.
Al to appear in RMP
(2006)
For reviews of cluster methods see: Georges et.al. RMP (1996) Maier et.al
RMP (2005), Kotliar et.al cond-mat 0511085. to appear in RMP (2006) Kyung
et.al cond-mat 0511085
Parametrizes the physics in
terms of a few functions .
D , Weiss Field
Alternative (T. Stanescu and
G. K. ) periodize the cumulants
rather than the self energies.
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) ]
U/t=4.
Effective Action point of view.
• Identify observable, A. Construct a free energy
functional of <A>=a, G [a] which is stationary at the
physical value of a.
• Example, density in DFT theory. (Fukuda et. al.).
• DMFT Local Spectral Function. (R. Chitra and
G.K (2000) (2001).
• H=H0+l H1. G [a,J0]=F0[J0 ]–a J0 _ + Ghxc [a]
• Functional of two variables, a ,J0.
• H0 + A J0 Reference system to think about H.
• J0 [a] Is the functional of a with the property
<A>0 =a < >0 computed with H0 + A J0
• Many choices for H0 and for A
Finite T, DMFT and the Energy
Landscape of Correlated Materials
T
Pressure Driven Mott transition
How does the electron
go from the localized to
the itinerant limit ?
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)
T/W
Phase diagram of a Hubbard model with partial frustration at
integer filling. Thinking about the Mott transition in single site
DMFT. High temperature universality
Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
Ising critical endpoint: prediction Kotliar Lange
Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
Observed! In V2O3
P. Limelette et.al. Science 302, 89 (2003)
Three peak structure, predicted Georges and Kotliar (1992) Transfer of
.spectral weight near the Mott transtion. Predicted Zhang Rozenberg and
GK (1993) . ARPES measurements on NiS2-xSex
Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et
al., Phys. Rev.Lett. 90, 186403 (2003).
Conclusions.
• Three peak structure, quasiparticles and
Hubbard bands.
• Non local transfer of spectral weight.
• Large metallic resistivities.
• The Mott transition is driven by transfer of
spectral weight from low to high energy as we
approach the localized phase.
• Coherent and incoherence crossover. Real and
momentum space.
• Theory and experiments begin to agree on the
broad picture.
Some References
• Reviews: A. Georges G. Kotliar W.
Krauth and M. Rozenberg RMP68 , 13,
(1996).
• Reviews: G. Kotliar S. Savrasov K.
Haule V. Oudovenko O. Parcollet and C.
Marianetti. Submitted to RMP (2006).
• Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987 . Schematic Phase Diagram (Hole
Doped Case)


i , j ,
(tij   ij )(ci† c j  c†j ci )  U  nini
i
Methodological Remarks
• Leave out inhomogeneous states and ignore disorder.
• What can we understand about the evolution of the
electronic structure from a minimal model of a doped
Mott insulator, using Dynamical Mean Field Theory ?
• Approach the problem directly from finite
temperatures,not from zero temperature. Address
issues of finite frequency –temperature crossovers. As
we increase the temperature DMFT becomes more and
more accurate.
• DMFT provides a reference frame capable of describing
coherent and incoherent regimes within the same
scheme.
RVB physics and 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. t-J limit.
• Slave boson approach.
<b>
coherence order parameter. k, D singlet formation
order parameters.Baskaran Zhou Anderson ,
Ruckenstein et.al (1987) .
Other states flux phase or s+id ( G. Kotliar (1988) Affleck and
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
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, 217002 (2001)
Problems with the approach.
• Neel order. How to continue a Neel insulating state ?
Need to treat properly finite T.
• Temperature dependence of the penetration depth [Wen
and Lee , Ioffe and Millis ] . Theory:[T]=x-Ta x2 , Exp:
[T]= x-T a.
• Mean field is too uniform on the Fermi surface, in
contradiction with ARPES.
• No quantitative computations in the regime where there
is a coherent-incoherent crossover,compare well with
experiments. [e.g. Ioffe Kotliar 1989]
CDMFT may solve some of these problems.!!
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.
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)
T/W
Phase diagram of a Hubbard model with partial frustration at
integer filling. Thinking about the Mott transition in single site
DMFT. High temperature universality
•
•
•
•
•

•
•
CDMFT
study
of
cuprates
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.
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 .
DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS
-(k,)+= /b2 -(D+b2 t) (cos kx + cos ky)/b2 +l
b--------> b(k), D ----- D(), l  l (k )
Extends the functional form of the self energy to finite T and
higher frequency.
• Can we continue the superconducting
state towards the Mott insulating state ?
Competition of AF and SC
or
SC
AF
SC
AF
AF+SC


Competition of AF and SC M.
Capone M. Civelli and GK (2006)
• 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.
Superconducting State t’=0
• Does the Hubbard model superconduct ?
• Is there a superconducting dome ?
• Does the superconductivity scale with J ?
• Is it BCS like ?
Superconductivity in the Hubbard model role of
the Mott transition and influence of the superexchange. ( work with M. Capone V. Kancharla.
CDMFT+ED, 4+ 8 sites t’=0) .
Order Parameter and Superconducting Gap do not
always scale! ED study in the SC state Capone Civelli
Parcollet and GK (2006)
How is the Mott insulator
approached from the
superconducting state ?
Work in collaboration with M. Capone.
Evolution of DOS with doping U=12t. Capone et.al. :
Superconductivity is driven by transfer of spectral weight ,
slave boson b2 !
Superconductivity is destroyed by
transfer of spectral weight. M. Capone
et. al. Similar to slave bosons d wave
RVB.
• 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 ?
Superconducting 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!
• 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.
Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U=8t
Significant Difference with Migdal-Eliashberg.
•Can we connect the
superconducting state with the
“underlying “normal” state “ ?
What does the underlying
“normal” state look like ?
Follow the “normal state” with doping. Civelli et.al. PRL 95,
106402 (2005)
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16
t, t’=-.3
K.M. Shen et.al. 2004
A(  0, k )vs k
Ek=t(k)+Re(k ,   0)  
 k = Im(k ,   0)
k
A(k ,   0)  2
 k  Ek 2
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.
2X2 CDMFT
Dependence on periodization
scheme.
Comparison of 2 and 4 sites
Spectral shapes. Large Doping
Stanescu and GK cond-matt
0508302
Small Doping. T. Stanescu and GK
cond-matt 0508302
Interpretation in terms of lines of zeros and lines of poles of G T.D.
Stanescu and G.K cond-matt 0508302
Lines of Zeros and Spectral Shapes.
Stanescu and GK cond-matt 0508302
Connection between
superconducting and normal state.
• Transfer of spectral weight in optics. Elucidate how the
spin superexchange energy and the kinetic energy of
holes changes upon entering the superconducting state!
• Origin of the powerlaws discovered in the groups of N.
Bontemps and D. VarDerMarel.
• K. Haule and GK development of an ED+DCA+NCA
approach to the problem. New tool for addressing the
neighborhood
of the dome.
Optical Conductivity near optimal
doping. [ Theory DCA ED+NCA study, K. Haule and GK]
Kristjan Haule: there is an avoided
quantum critical point near optimal
doping.
What is happening near optimal
doping ?? Avoided Quantum
Criticality [K. Haule and GK]
Optical conductivity t-J . K. Haule
Behavior of the
optical mass and the
plasma frequency.
RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy


iV


0
Heff , J eff , Peff
 2 k
 nk 2
k
k
  P, J  
 ( )d 

iV
 ne2
m
  Peff , J eff  
Low energy sum rule can
have T and doping
dependence . For nearest
neighbor it gives the
kinetic energy.
Treatement needs refinement
• The kinetic energy of the Hubbard model
contains both the kinetic energy of the
holes, and the superexchange energy of
the spins.
• Physically they are very different.
• Experimentally only measures the kinetic
energy of the holes.
Conclusions
• DMFT is a useful mean field tool to study correlated
electrons. Provide a zeroth order picture of a physical
phenomena.
• Provide a link between a simple system (“mean field
reference frame”) and the physical system of interest.
[Sites, Links, and Plaquettes]
• Formulate the problem in terms of local quantities (which
we can usually compute better).
• Allows to perform quantitative studies and predictions .
Focus on the discrepancies between experiments and
mean field predictions.[Substantiates and improves over
early slave boson studies of the phenomena]
• Generate useful language and concepts. Follow mean
field states as a function of parameters.
• K dependence gets strong as we approach the Mott
transition. Psedogap. Fermi surfaces and lines of zeros
of Tsvelik (quasi-one dimensional systems ) T.
Stanescu and GK (proximity to a Mott transition in 2 d).
Conclusions Superconductivity
• Reproduced the basic general features of the
early slave boson treatment.
• The approach naturally introduces a strong
anisotropy and particle hole asymmetry in the
problem.
• It reveals the frequency dependence of the self
energy which is growing with doping [ J and U ]
• Establishes the clear differences between
superconductivity above and below the Mott
transition.
Conclusions
• Qualitative effect, momentum space
differentiation. Formation of hot –cold regions is
an unavoidable consequence of the approach to
the Mott insulating state!
• Truncation of the Fermi surface as a STRONG
COUPLING instability (compare weak coupling
RG e.g. Honerkamp, Metzner, Rice )
• General phenomena, but the location of the cold
regions depends on parameters.
• Fundamental difference between electron and
hole doped cuprates.
o Qualitative Difference between the hole doped and
the electron doped phase diagram is due to the
underlying normal state.” In the hole doped, it has
nodal quasiparticles near (/2,/2) which are
ready “to become the superconducting
quasiparticles”. Therefore the superconducing
state can evolve continuously to the normal state.
The superconductivity can appear at very small
doping.
o Electron doped case, has in the underlying normal
state quasiparticles leave in the (, 0) region, there
is no direct road to the superconducting state (or
at least the road is tortuous) since the latter has
QP at (/2, /2).
Approaching the Mott transition:
from high T CDMFT Picture
• Fermi Surface Breakup. Qualitative effect,
momentum space differentiation. Formation of
hot –cold regions is an unavoidable
consequence of the approach to the Mott
insulating state! It can be seen starting from high
temperatures.
• D wave gapping of the single particle spectra as
the Mott transition is approached. Real and
Imaginary part of the self energies grow
approaching half filling. Unlike weak coupling!
• Scenario was first encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
High Temperature Superconductors. What can we
learn from the study of the doped Mott insulator
within plaquette Cellular DMFT ?
• We can learn a lot, but there is still a lot of work to be done until we
reach the same level of understanding that we have of the single
site DMFT solution. This work is definitely in progress.
• a) Either that we can account semiquantitatively for the large body
of experimental data once we study more realistic models of the
material.
• Or b) 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.
• It is still too early to tell, but some evidence in favor of a) was
presented in this seminar.
Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U.
Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu,
(Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M
Tremblay, D. Senechal B. Kyung
(Sherbrooke)
$$Support : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.
What is the origin of the asymmetry
? Comparison with normal state
near Tc. K. Haule
Early slave boson work, predicted the asymmetry, and some features of the
spectra.
Notice that the superconducting gap is smaller than pseudogap!!
Magnetic Susceptibility
Outline
•Theoretical Point of View, and Methodological Developments. :
•Local vs Global observables.
•Reference Frames. Functionals. Adiabatic Continuity.
•The basic RVB pictures.
•CDMFT as a numerical method, or as a boundary condition.Tests.
•The superconducting state.
•The underdoped region.
•The optimally doped region.
•Materials Design. Chemical Trends. Space of Materials.
Connection with large N studies.
References
o Dynamical Mean Field Theory and a cluster
extension, CDMFT: G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401
(2001)
o Cluser Dynamical Mean Field Theories: Causality and
Classical Limit.
G. Biroli O. Parcollet G.Kotliar Phys. Rev. B 69 205908
• Cluster Dynamical Mean Field Theories a Strong
Coupling Perspective. T. Stanescu and G. Kotliar
( 2005)
Evolution of the normal state:
Questions.
• Origin of electron hole asymmetry in
electron and doped cuprates.
• Detection of lines of zeros and the
Luttinger theorem.
ED and QMC
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) ]
U/t=4.
Electron Hole Asymmetry Puzzle
What about the electron doped semiconductors ?
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
electron doped
Momentum space differentiation
a we approach the Mott
transition is a generic
phenomena.
Location of cold and hot regions
depend on parameters.
P. Armitage et.al. 2001
Civelli et.al. 2004
Approaching the Mott transition:
CDMFT Picture
• Qualitative effect, momentum space
differentiation. Formation of hot –cold regions is
an unavoidable consequence of the approach to
the Mott insulating state!
• D wave gapping of the single particle spectra as
the Mott transition is approached.
• Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
High Temperature Superconductors. What can we
learn from the study of the doped Mott insulator
within plaquette Cellular DMFT ?
• We can learn a lot, but there is still a lot of work to be done until we
reach the same level of understanding that we have of the single
site DMFT solution.
•
Either that we can account semiquantitatively for the large body
of experimental data once we study more realistic models of the
material.
• 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.
• It is still too early to tell.
Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U.
Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu,
(Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M
Tremblay, D. Senechal B. Kyung
(Sherbrooke)
$$Support : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.
Conclusion
OPTICS
CDMFT and NCS as truncations
of the Baym Kadanoff functional
1
G[G, ]  TrLn[G0  ]  Tr[G]  [G]
[G]  Sum 2PI graphs with G lines andU vertices
GCDMFT  G[Gij , ij , ij  C , Gij  0, ij  0, ij  C ]
Gncs  G[Gij, ij,| i  j | r, Gij  0, ij  0,| i  j | r ]
Ex: Baym Kadanoff functional,a=
G, H0 = free electrons.
1
1
1
GBK[G]  TrLn[G ]  Tr[(G0  G )G]  [G]
[G]  Sum 2PI graphs with G lines andU vertices
1
G[G, ]  TrLn[G0  ]  Tr[G]  [G]
Viewing it as a functional of J0, Self Energy functional(Potthoff)
Gself []  TrLn[G01  ]  extG (Tr[G]  [G])
Weiss Field Functional
Example: single site DMFT semicircular density of
states. GKotliar EPJB (1999)
D(i ) 2
F [D]  T 
 Fimp [D]
2
t
 Lloc [ f † , f ] 
f† ( i ) D ( i ) f ( i )
†
 ,
  Fimp  Log[  df dfe
]
Extremize Potthoff’s self energy functional. It
is hard to find saddles using conjugate
gradients.
Extremize the Weiss field functional.Analytic
for saddle point equations are available
Minimize some distance
Approaching the Mott transition:
CDMFT Picture
• Fermi Surface Breakup. Qualitative effect,
momentum space differentiation. Formation of
hot –cold regions is an unavoidable
consequence of the approach to the Mott
insulating state!
• D wave gapping of the single particle spectra as
the Mott transition is approached.
• Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
Dynamical RVB brings in strong
anistropy in the underdoped
regime.
What about the electron doped semiconductors ?
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
electron doped
Momentum space differentiation
a we approach the Mott
transition is a generic
phenomena.
Location of cold and hot regions
depend on parameters.
P. Armitage et.al. 2001
Civelli et.al. 2004
o Qualitative Difference between the hole doped and
the electron doped phase diagram is due to the
underlying normal state.” In the hole doped, it has
nodal quasiparticles near (/2,/2) which are
ready “to become the superconducting
quasiparticles”. Therefore the superconducing
state can evolve continuously to the normal state.
The superconductivity can appear at very small
doping.
o Electron doped case, has in the underlying normal
state quasiparticles leave in the (, 0) region, there
is no direct road to the superconducting state (or
at least the road is tortuous) since the latter has
QP at (/2, /2).
 Can we connect the superconducting state with the
“underlying “normal” state “ ?
 Yes, within our resolution in the hole doped case.
 No in the electron doped case.
 What does the underlying “normal state “ look like ?
 Unusual distribution of spectra (Fermi arcs) in the normal
state.
To test if the formation of the hot and
cold regions is the result of the
proximity to Antiferromagnetism, we
studied various values of t’/t, U=16.
Introduce much larger frustration:
t’=.9t U=16t
n=.69 .92 .96
Approaching the Mott transition:
• Qualitative effect, momentum space
differentiation. Formation of hot –cold regions is
an unavoidable consequence of the approach to
the Mott insulating state!
• General phenomena, but the location of the
cold regions depends on parameters.
• With the present resolution, t’ =.9 and .3 are
similar. However it is perfectly possible that at
lower energies further refinements and
differentiation will result from the proximity to
different ordered states.
Fermi Surface Shape
Renormalization ( teff)ij=tij+ Re(ij(0))
Fermi Surface Shape
Renormalization
• Photoemission measured the low energy
renormalized Fermi surface.
• If the high energy (bare ) parameters are doping
independent, then the low energy hopping
parameters are doping dependent. Another
failure of the rigid band picture.
• Electron doped case, the Fermi surface
renormalizes TOWARDS nesting, the hole
doped case the Fermi surface renormalizes
AWAY from nesting. Enhanced magnetism in
the electron doped side.
Understanding the location of the
hot and cold regions. Interplay of
lifetime and fermi surface.
Superconductivity as the cure for a
“sick” normal state.