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The Mott Transition and the
Challenge of Strongly Correlated
Electron Systems.
G. Kotliar
Physics Department and Center for
Materials Theory
Rutgers
PIPT Showcase Conference UBC Vancouver May 12th 2005
The Standard Model of Solid
State Physics.
• Itinerant limit. Band Theory. Wave picture
of the electron in momentum space. .
Pauli susceptibility.
• Localized model. Real space picture of
electrons bound to atoms. Curie
susceptibility at high temperatures, spinorbital ordering at low temperatures.
Correlated Electron Materials
• Are not well described by either the itinerant or
the localized framework .
• Compounds with partially filled f and d shells.
Need new starting point for their description.
Non perturbative problem. New reference frame
for computing their physical properties.
• Have consistently produce spectacular “big”
effects thru the years. High temperature
superconductivity, colossal magneto-resistance,
huge volume collapses……………..
Large Metallic Resistivities
 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), Vander Marel
(2005) Takagi (2003 ) Neff
depends on T
Breakdown of the standard model
of solids.
• Qualitative Issues : i.e. Large metallic
resistivities. Breakdown of the rigid band picture.
Anomalous transfer of spectral weight in
photoemission and optics……………
• The quantitative tools of the standard model fail
(i.e. density functional + GW )
Two paths for calculation of
electronic structure of
strongly correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
DMFT ideas can be used in both cases.
MODEL HAMILTONIAN AND OBSERVABLES

 (t
ij
i , j  ,
  ij )(c c j  c c )  U  ni ni
Parameters:
†
i
†
j i
i
U/t , T, carrier concentration, frustration :
1
A( k ,  )   Im[G( k ,  )]   Im[
]
   k  ( , k )
A( )   A( k ,  )
Local Spectral Function
k
A( )    ( k   )
Limiting case itinerant electrons
k
Limiting case localized electrons
Hubbard bands
A( )   ( B   )   ( A   )
U   A  B
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
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)
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 ]
Mott transition in V2O3 under pressure
or chemical substitution on V-site. How does the electron go
from localized to itinerant.
The Mott transition and Universality
Same behavior at high
tempeartures, completely
different at low T
COHERENCE INCOHERENCE CROSSOVER
T/W
Phase diagram of a Hubbard model with partial frustration at integer
filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .
Transfer of optical spectral weight
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P
Metcalf Phys. Rev. Lett. 75, 105 (1995)
Transfer of optical spectral weight
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P
Metcalf Phys. Rev. Lett. 75, 105 (1995)
Anomalous transfer of optical spectral
weight, NiSeS. [Miyasaka and Takagi
2000]
Anomalous Resistivity and Mott
transition Ni Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator. M.
Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and
P Metcalf Phys. Rev. Lett. 75, 105 (1995)
Single-site DMFT and expts
Conclusions: lessons from the
application of DMFT to toy model.
• Three peak structure, quasiparticles and
Hubbard bands.
• The Mott transition is driven by transfer of
spectral weight from low to high energy as we
approach the localized phase.
• The method can describe coherent and
incoherent phenomena and their crossover.
Access to non perturbative regime . Real and
momentum space.
• Theory and experiments begin to agree on a
broad picture.
Realistic Descriptions of Materials
and a First Principles Approach to
Strongly Correlated Electron
Systems.
• Incorporate realistic band structure and orbital
degeneracy.
• Incorporate the coupling of the lattice degrees
of freedom to the electronic degrees of freedom.
• Predict properties of matter without empirical
information.
LDA+DMFT V. Anisimov, A. Poteryaev, M.
Korotin, A. Anokhin and G. Kotliar, J. Phys.
Cond. Mat. 35, 7359 (1997).
• Realistic band structure and orbital degeneracy.
Describes the excitation spectra of many strongly
correlated solids. .
Spectral Density Functionals. Chitra and Kotliar PRB 2001
Savrasov et. al. Nature (2001) Savrasov and Kotliar PRB (2005)
•Determine the self energy , the density and the structure of the
solid by extremizing a functional of these quantities.
Coupling of electronic degrees of freedom to structural degrees
of freedom.
Mott Transition in the Actinide Series
Pu phases: A. Lawson Los Alamos Science 26,
(2000)
LDA underestimates the volume of fcc Pu by 30%.
Within LDA fcc Pu has a negative shear modulus.
LSDA predicts  Pu to be magnetic with a 5 ub moment.
Experimentally it is not.
Treating f electrons as core overestimates the volume by 30 %
Total Energy as a function of volume for PU
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Double well structure and 
Pu
Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low
Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil.
Mag. B 82, 1837 ]
Natural consequence of the conclusions on the model Hamiltonian level. We
had two solutions at the same U, one metallic and one insulating. Relaxing the
volume expands the insulator and contract the metal.
Phonon freq (THz) vs q in delta Pu X.
Dai et. al. Science vol 300, 953, 2003
Inelastic X Ray. Phonon energy
10 mev, photon energy 10 Kev.
E = Ei - Ef
Q =ki - kf
DMFT Phonons in fcc -Pu
C11 (GPa)
C44 (GPa)
C12 (GPa)
C'(GPa)
Theory
34.56
33.03
26.81
3.88
Experiment
36.28
33.59
26.73
4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
J. Tobin et. al. PHYSICAL REVIEW B 68,
155109 ,2003
First Principles DMFT Studies of Pu
• Pu strongly correlated element, at the
brink of a Mott instability, which could not
be described within the standard model.
• Quantitative computations : total energy,
photoemission spectra and phonon
dispersions of delta Pu.
• Qualitative Insights and quantitative
studies. Double well. Alpha and Delta
Pu.Other Pu anomalies.
Approach the Mott point from the right Am under
pressureExperimental Equation of State (after Heathman et.al, PRL 2000)
“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)
Mott transition in open (right) and
closed (left) shell systems.
Realization in Am ??
S
gT
Log[2J+1]
S
Tc
???
Uc
U
J=0
g ~1/(Uc-U)
U
Cluster Extensions of Single Site DMFTCaptures new
physics beyond single site DMFT , i.e. d wave superconductivity, and
other novel aspects of the Mott transition in two dimensional systems.
latt (k , ) 0 ( ) 
1 ( )(cos kx  cos ky )   2 ( )(cos kx.cos ky )  .......
Conclusions Future Directions
• DMFT: Method under development, but it
already gives new insights into materials…….
• Exciting development: cluster extensions. Allows
us to see to check the accuracy of the single
site DMFT corrections, and obtain new physics
at lower temperatures and closer to the Mott
transition where the single site DMFT breaks
down.
• Development of DMFT allow us to focus on
deviations of experiments from mean field
theory.
• DMFT and RG developments
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 (2005).
• Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)