Download Correlated Electrons: A Dynamical Mean Field

Extensions of Single Site DMFT and its
Applications to Correlated Materials
On the road towards understanding superconductivity in
strongly correlated materials
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Workshop on Quantum Materials
Heron Island Resort
New Queensland Australia
1-4 June 2005
Mott Transition in the Actinide Series
Lashley et.al.
Mott transition in open (right) and
closed (left) shell systems.
Superconductivity is
an
unavoidable
S
S
gT
Log[2J+1]
consequence to the
approach to the Mott
transition
with a
Uc
singlet closed shell
U
state.
g ~1/(Uc-U)
Tc
???
J=0
U
•Cuprate superconductors
and the Hubbard Model .
PW Anderson 1987 .
Connect superconductivity
to an RVB Mott insulator.
Science 235, 1196 (1987).
Hubbard , t-J model .
•Baskaran Zhou and
Anderson (1987). slave
boson approach, S-wave
Pairing. Connection to an
insulator with a Fermi
surface.
.
RVB phase diagram of the Cuprate
Superconductors and Superexchange.
•
The approach to the Mott
insulator renormalizes the
kinetic energy . Kinetic energy
renormalizes to zero.
• Attraction in the d wave
channel of order J Not
renormalized. 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) k, D singlet
formation order parameters
Problems with the approach.
• Neel order. How to continue a Neel insulating state ?
• Stability of the pseudogap state at finite temperature.
[Ubbens and Lee]
• Missing incoherent spectra . [ fluctuations of slave
bosons ]
• Temperature dependence of the penetration depth [Wen
and Lee , Ioffe and Millis ] . Theory:r[T]=x-Ta x2 , Exp:
r[T]= x-T a.
• Mean field is too uniform on the Fermi surface, in
contradiction with ARPES.
The development of DMFT solves many of these problems.!!
Also, one would like to be able to evaluate from the theory itself when the
approximation is reliable!!
Cluster Extensions of Single Site DMFT
latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )   2 ( )(cos kx.cos ky )  .......
H  H cluster  H cluster exterior  H exterior
H
H cluster  H cluster exterior  H exterior
Simpler "medium" Hamiltonian
Medium of free
electrons :
impurity
model.Solve for
the medium
usingSelf
Consistency.
Extraction of
lattice quantities.
G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001)
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.
Finite T Mott transtion in CDMFT
Parcollet Biroli and GK PRL, 92, 226402. (2004))
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  Ek 2
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.
Evolution of the k resolved Spectral
Function at zero frequency. (QMC
study Parcollet Biroli and GK PRL, 92, 226402. (2004))
A( )
 0, k )vs k
U/D=2
U/D=2.25
Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W
Physical Interpretation
• Momentum space differentiation. The
Fermi liquid –Bad Metal, and the Bad
Insulator - Mott Insulator regime are
realized in two different regions of
momentum space.
• Cluster of impurities can have different
characteristic temperatures. Coherence
along the diagonal incoherence along x
and y directions.
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987

 (t
ij
i , j  ,
  ij )(c c j  c c )  U  nini
†
i
†
j i
i
CDMFT study of cuprates
.
• 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 (Katsnelson and Lichtenstein, M. Jarrell, M
Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett
85, 1524 (2000) ) used QMC as an impurity solver and DCA as
cluster scheme.
• We use exact diag ( Krauth Caffarel 1995 with effective
temperature 32/t=124/D ) as a solver and Cellular DMFT as the
mean field scheme.
Superconducting State t’=0
• Does the Hubbard model superconduct ?
• Is there a superconducting dome ?
• Does the superconductivity scale with J ?
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) .
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 .
Competition of AF and SC
or
SC
AF
SC
AF
AF+SC


D wave Superconductivity and
Antiferromagnetism t’=0 M. Capone
V. Kancharla (see also VCPT Senechal and
Tremblay ).
Antiferromagnetic (left) and d wave superconductor (right) Order Parameters
Competition of AF and SC
U /t << 8
U f 8t
or
SC
AF
AF
SC
AF+SC


•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. Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  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.
:
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16 t
hole doped
K.M. Shen et.al. 2004
2X2 CDMFT
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) .
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 (p/2,p/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 (p, 0) region, there
is no direct road to the superconducting state (or
at least the road is tortuous) since the latter has
QP at (p/2, p/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.
Mott transition into a low entropy insulator. Is superconuctivity
realized in Am ?
“Soft”
Mott Transition?
“Hard”
Density functional based electronic structure calculations:
 Non magnetic LDA/GGA predicts volume 50% off.
 Magnetic GGA corrects most of error in volume but gives m~6B
(Soderlind et.al., PRB 2000).
 Experimentally, Am has non magnetic f6 ground state with
J=0 (7F0)
Am under pressure: J.C. GriveauJ. Rebizant G.
Lander G. Kotliar PRL (2005)
• Mott transition into a low entropy insulator.
Is it realized in Am ?
• Yes! But there are additional suprises,
which are specific to Am, such as the
second maximum in Tc vs pressure.
Additional system specific properties.
Conclusions
• Correlated Electron materials, as a second
frontier in materials science research, the “in
between “ regime between itinerant and
localizedis very interesting.
• Getting the general picture, and the material
specific details are both important..
• Mott transition : open shell (finite T Mott
endpoint in V2O3, NiSeS, K-organics, Pu )
closed shell case (Am,
cuprates…….)connection to superconductivity.
• The challenge of material design using
correlated materials.
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.
• Generate useful language and concepts. Follow mean
field states as a function of parameters.
• Controlled approach!
Conjecture, Mott transition with
Zcold finite ? Continuity with the
insulator at one point in the zone.
Conjecture, Mott transition with
Zcold finite ? Continuity with the
insulator at one point in the zone.
Conjecture, Mott transition with
Zcold finite ? Continuity with the
insulator at one point in the zone.
Is the formation of the hot and cold
regions is the result of the proximity
to Antiferromagnetism ?
Study various values of t’/t, U=16.
Introduce much larger frustration:
t’=.9t U=16t
n=.69 .92 .96
 Is the momentum space differentiation a result
of proximity to an ordered state , e.g.
antiferromagnetism?
 Fermi Surface Breakup or Momentum space
differentiation takes place irrespectively of the
value of t’. The gross features are the result of
the proximity to a Mott insulating state
irrespective of whether there is magnetic long
range order.
How is the Mott insulator
approached from the
superconducting state ?
Work in collaboration with M. Capone
Evolution of the low energy tunneling density of
state with doping. Decrease of spectral weight
as the insulator is approached. Low energy
particle hole symmetry.
Alternative view
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.
• Further understanding of phenomena of
momentum space differentiation.
• Analyze the results in terms of a few
(three!) self energy functions.
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.
LDA+DMFT spectra. Notice the
rapid occupation of the f7/2 band,
(5f)7
Photoemission Spectrum from 7F0 Americium
LDA+DMFT Density of States
S. Savrasov et. al.
Multiplet Effects
F(0)=4.5 eV
F(2)=8.0 eV
F(4)=5.4 eV
F(6)=4.0 eV
Experimental Photoemission Spectrum
(after J. Naegele et.al, PRL 1984)
J. C. Griveau et. al. (2004)
H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am
under pressure Griveau et. al.
Superconductivity due to
valence fluctuations ?
Cluster DMFT for organics ?
CDMFT for organics ?
Evidence for unconventional interaction underlying in
two-dimensional correlated electrons
F. Kagawa,1 K. Miyagawa,1, 2 & K. Kanoda1, 2
Conclusions and Outlook
• Motivation: Mott transition in Americium and
Plutonium. In both cases theory (DMFT) and
experiment suggest gradual subtle changes.
• DMFT: Physical connection between spectra and
structure. Studied the Mott transition open
and closed shell cases. .
• DMFT: method under construction, but it already
gives quantitative results and qualitative insights.
Interactions between theory and experiments.
• Pu: simple picture of alpha delta and epsilon.
Interplay of lattice and electronic structure near
the Mott transition.
• Am: Rich physics, mixed valence under
pressure ? Superconductivity near the Mott
transition.
Actinides and The Mott Phenomena
Evolution of the electronic structure between the atomic
limit and the band limit in an open shell situation.
The “”in between regime” is ubiquitous central theme in
strongly correlated systems.
Actinides allow us to probe this physics in ELEMENTS.
Mott transition across the actinide series [ B.
Johansson Phil Mag. 30,469 (1974)] . Revisit the
problem using a new insights and new techniques
from the solution of the Mott transition problem within
DMFT in a model Hamiltonian.
Use the ideas and concepts that resulted from this
development to give physical qualitative insights into
real materials.
Turn the technology developed to solve simple models
into a practical quantitative electronic structure
Collaborators 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 (2005).
• Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)
Understanding the result in terms of
cluster self energies (eigenvalues)
A
(0,p )
 B ~ (p ,p )
A
(0,0)
Systematic Evolution
Dynamical Mean-Field Theory

 (t
ij
i , j  ,
b
  ij )(ci† c j  c †j ci )  U  ni ni 
i
b
b
¶
†
c
(
t
)[
ò ò os ¶ t + m- D (t - t ')]cos (t ') +U ò no­ no¯
0 0
0
D ( w)
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
å
k
1
1
[D (iwn ) - tk + m]
G (iwn )[D ]
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
Mean-Field Classical vs Quantum
Classical case
-
å
J ij Si S j - h å Si
i, j
i
H MF = - heff So
heff
å
j

 (t
ij
i , j  ,
b
  ij )(ci† c j  c †j ci )  U  ni ni 
i
b
b
¶
†
c
(
t
)[
ò ò os ¶ t + m- D (t - t ')]cos (t ') +U ò no­ no¯
0 0
0
G
D ( w)
m0 = áS0 ñHMF ( heff )
heff =
Quantum case
J ij m j + h
- 1
0
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
å
k
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
1
1
[D (iwn ) - tk + m]
G (iwn )[D ]
DMFT as an approximation to
the Baym Kadanoff functional
[G, ]  TrLn[i  tij   ij   ]  Tr[G ]  [G ]
[G ]  Sum 2PI graphs with G lines andU vertices
DMFT  [Gii, ii, Gij  0, ij  0, i  j ] 
 TrLn[i  tij   ij  ii ]  iTr[iiGii ]  i atom[Gii ]
CDMFT vs single site DMFT and other
cluster methods.
Cellular DMFT
1
4
2
3
Site Cell. Cellular DMFT. C-DMFT.
G.. Kotliar,S. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice notations.
•Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler
et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein
periodized scheme, Nested Cluster Schemes , causality
issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Estimates of upper bound for Tc
exact diag. M. Capone. U=16t, t’=0, (
t~.35 ev, Tc ~140 K~.005W)
DMFT : What is the dominant atomic configuration
,what is the fate of the atomic moment ?
• Snapshots of the f electron :Dominant
configuration:(5f)5
• Naïve view Lz=-3,-2,-1,0,1, ML=-5 B,
,S=5/2 Ms=5 B . Mtot=0
• More realistic calculations,
(GGA+U),itineracy, crystal fields 7 8,

ML=-3.9
Mtot=1.1. S. Y. Savrasov and G.
Kotliar, Phys. Rev. Lett., 84, 3670 (2000)
• This moment is quenched or screened by
spd electrons, and other f electrons. (e.g.
alpha Ce).
 Contrast Am:(5f)6

Anomalous Resistivity
PRL 91,061401 (2003)
The delta –epsilon transition
• The high temperature phase, (epsilon) is body
centered cubic, and has a smaller volume than
the (fcc) delta phase.
• What drives this phase transition?
• LDA+DMFT functional computes total energies
opens the way to the computation of phonon
frequencies in correlated materials (S. Savrasov
and G. Kotliar 2002). Combine linear response
and DMFT.
Epsilon Plutonium.
Phonon entropy drives the
epsilon delta phase transition
• Epsilon is slightly more delocalized than delta,
has SMALLER volume and lies at HIGHER
energy than delta at T=0. But it has a much
larger phonon entropy than delta.
• At the phase transition the volume shrinks but
the phonon entropy increases.
• Estimates of the phase transition following
Drumont and G. Ackland et. al. PRB.65, 184104
(2002); (and neglecting electronic entropy). TC
~ 600 K.
Total Energy as a function of volume for Pu
W
(ev) vs iw (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
Expt. Wong et. al.
Alpha and delta Pu
ARPES measurements on NiS2-xSex
.Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)
One Particle Local Spectral Function and
Angle Integrated Photoemission
e
•
Probability of removing an
electron and transfering
energy =Ei-Ef,
f() A() M2
• Probability of absorbing an
electron and transfering
energy =Ei-Ef,
(1-f()) A() M2
• Theory. Compute one
particle greens function and
use spectral function.
n
n
e
QP in V2O3 was recently found
Mo et.al
k organics
• ET = BEDT-TTF=Bisethylene dithio
tetrathiafulvalene
 K (ET)2 X
Increasing pressure ----- increasing t’ ----------X0
X1
X2
X3
• (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br
Cu(NCS)2
• Spin liquid Mott transition
Vanadium Oxide Transport
under pressure. Limelette
etal
Failure of the Standard
Model: Anomalous Spectral Weight Transfer
Optical Conductivity o of FeSi for
T=20,40, 200 and 250 K from Schlesinger
et.al (1993)


0
 ( )d
Neff depends on T
RESTRICTED SUM RULES
H hamiltonian, J electric current , P polarization


0
 ( )d 
Below energy
p
  


0
( )d

iV
p
p ne2
, J ,P
  P, J H
eff
iV
eff
eff
m

  Peff , J eff  
ApreciableT dependence
found.  n  

k
2
k
k
k
2
M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).
DMFT Impurity cavity construction


i , j  ,
b
D0-
1
i
  Vij ni n j
i , j 
b
òò
0
(tij   ij )(ci† c j  c †j ci )  U  ni  ni 
co†s ( t )Go( t , t ')cos ( t ') + no­ no¯U d( t , t ')
+ Do(t , t ')n0n0
0
é
(iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
Vk - P (iwn ) ú
ú
û
+ P (iwn )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
P (iwn )[G0 ] = D0- 1 (iwn ) + [án0 (iwn )n0(iwn )ñS () ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
1
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
+ S (iwn )
Site Cell. Cellular DMFT. C-DMFT.
G.. Kotliar,S. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice notations.
•Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler
et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein
periodized scheme, Nested Cluster Schemes , causality
issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Mean-Field Classical vs Quantum
Quantum case

 (t
ij
i , j  ,
b
  ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
b
¶
†
c
(
t
)[
ò ò os ¶ t + m- D (t - t ')]cos (t ') +U ò no­ no¯
0 0
0
D ( w)
1
[iwn + m- S (iwn )] = D (iwn ) G (iwn )[D ]
G (iwn )[D ] =
å
k
1
[[iwn + m- S (iwn )] - tk ]
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
å
k
1
[D (iwn ) -
1
- tk ]
G (iwn )[D ]
Realistic DMFT loop
iw ® iwOk
é H LL
ê
êH HL
ë
tk ® H LMTO (k ) ­ E
H LH ù
ú= H LMTO
H HH ú
û
iG0- 1 = iwnO + e - D
é0
0 ù
ú
D=ê
ê0 D HH ú
ë
û
é0
0 ù
ú
S=ê
ê0 S HH ú
ë
û
S HH (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 )
é
G0- 1 (iwn ) = êê
êë
å
k
ù -1
1
ú + S HH (iwn )
iwnOk - H LMTO (k ) - E - S (iwn ) ú
ú
ûHH
Other cluster extensions (DCA Jarrell Krishnamurthy, M
Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and
Lichtenstein periodized scheme. Causality issues O. Parcollet,
G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Success story : Density Functional Linear Response
Tremendous progress in ab initio modelling of lattice dynamics
& electron-phonon interactions has been achieved
(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)
Limit of large lattice coordination
1
d
tij ~
d   ij nearest neighbors
1
 c c j  ~
d
†
i
  tij  ci† c j  ~ d
j ,
Uni  ni  ~O(1)
1
d
1
~ O (1)
d
Metzner Vollhardt, 89
1
G ( k , i ) 
i   k  (i )
Muller-Hartmann 89
Mean-Field
Quantum Case
H 

i , j  ,
H=Ho
(tij   ij )(ci† c j  c†j ci )  U  ni ni 
i
+Hm
+Hm0
H m    t 'l ,l ' A Al '
H 0  c c  Un0n0
†
0 0
l ,l ',
†
l
H m0   t '0 l (c Al  A c )
l ,
†
0
†
l 0
Determine the parameters of the mediu t’ so as to get translation
invariance on the average. A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
DMFT as an approximation to
the Baym Kadanoff functional
[G, ]  TrLn[i  tij   ij   ]  Tr[G ]  [G ]
[G ]  Sum 2PI graphs with G lines andU vertices
DMFT  [Gii, ii, Gij  0, ij  0, i  j ] 
 TrLn[i  tij   ij  ii ]  iTr[iiGii ]  i atom[Gii ]
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)
LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c k a = H LMTO ( k )
Impurity
Solver
G
0
Edc
G
U

S.C.C.
DMFT
r (r) = T
å
iw
G( r, r, iw)e
iw0+
nHH = T
å
iw
+
GHH (r , r , iw)eiw 0
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.
Mean-Field Classical vs Quantum
Quantum case

 (t
ij
i , j  ,
b
  ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
b
¶
†
c
(
t
)[
ò ò os ¶ t + m- D (t - t ')]cos (t ') +U ò no­ no¯
0 0
0
D ( w)
1
[iwn + m- S (iwn )] = D (iwn ) G (iwn )[D ]
G (iwn )[D ] =
å
k
1
[[iwn + m- S (iwn )] - tk ]
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn )[D ] =
A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)
å
k
1
[D (iwn ) -
1
- tk ]
G (iwn )[D ]
DMFT and the Invar Model
A. Lawson et. al. LA UR 04-6008 (LANL)
A. C. Lawson et. al. LA UR 046008
F(T,V)=Fphonons+F2level
 =125 K
D
g =.5
D= 1400 K
Invar model A. C. Lawson et. al. LA
UR 04-6008
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987 . Connect superconductivity to an RVB
Mott insulator. Science 235, 1196 (1987)


i , j  ,
(tij   ij )(ci† c j  c †j ci )  U  ni ni 
i
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.
• Baskaran Zhou and Anderson Slave boson
approach.
<b> coherence order
parameter.
 k, D singlet formation order parameters.
Site Cell. Cellular DMFT. C-DMFT.
G.. Kotliar,S. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice notations.
•Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler
et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein
periodized scheme, Nested Cluster Schemes , causality
issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987


i , j  ,
(tij   ij )(ci† c j  c †j ci )  U  ni ni 
i
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987


i , j  ,
(tij   ij )(ci† c j  c †j ci )  U  ni ni 
i
Momentum Space
Differentiation the high
temperature story
T/W=1/88
Hole doped case t’=-.3t, U=16 t
n=.71 .93 .97
Color scale x= .37 .15 .13
CDMFT one electron spectra
n=.96 t’/t=.-.3 U=16 t
• i
Experiments. Armitage et. al. PRL (2001).
Momentum dependence of the low-energy
Photoemission spectra of NCCO
K.M . Shen et. al. Science (2005).
For a review Damascelli et. al. RMP (2003)
Evolution of the real part of the self
energies.
RVB states
• G. Baskaran Z. Shou and P.W Anderson Solid State
Comm 63, 973 (1987). RVB state with Fermi surface ( 2
d, line of zeros ).
• G. Kotliar Phys. Rev. B37 ,3664 (1998). I Affleck and B.
Marston. Phys.Rev. B 37, 3774 (1998). RVB State with
four point zeros in 2d. Two states are related by Su(2)
symmetry I Affleck Z.Zhou, T. Hsu P.W. Anderson PRB
38,745 (1998).
o G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988). Doping
selects the d –wave superconductor as the most favorable
RVB state away from half filling.
o Parallel development of RVG ideas with variational wave
functions. C. Gross R. Joynt and T.M.Rice PRB 36, 381
(1987) F. C. Zhang C. Gros T M Rice and H Shiba
Supercond. Scie Tech. 1, 36 (1988).
Comparison with Experiments in Cuprates:
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
hole doped
K.M. Shen et.al. 2004
2X2 CDMFT
electron doped
P. Armitage et.al. 2001
Civelli et.al. 2004