Transcript Strongly Correlated Electron Systems: a DMFT Perspective Colloquium UBC September (2004)

Strongly Correlated
Electron Systems: a DMFT
Perspective
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
Colloquium UBC September (2004)
Outline
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Introduction to the strong correlation
problem.
Essentials of DMFT
The Mott transition problem: some insights
from studies of models.
Towards an electronic structure method:
applications to materials: Ce, Pu
Outlook
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The electron in a solid: wave picture

Momentum Space (Sommerfeld)
k2
k 
2m
e 2 k F (k F l )
h
Maximum metallic
resistivity 200 mohm cm
Standard model of solids
Periodic potential, waves form
bands , k in Brillouin zone
Landau: Interactions renormalize away
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Standard Model of Solids
 ~
const
CV ~ T
RH ~ const
RIGID BAND PICTURE. Optical response, transitions
between bands.
Quantitative tools: DFT, LDA, GGA, total energies,good
starting point for spectra, GW,and transport
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The electron in a solid: particle picture.

NiO, MnO, …Array of atoms is insulating if a>>aB. Mott:
correlations localize the electron
e_
e_
e_
e_
•Superexchange

•Think in real space , solid collection of atoms
•High T : local moments, Low T spin-orbital order
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1
T
Mott : Correlations localize the electron
Low densities, electron behaves as a particle,use
atomic physics, work in real space.
•One particle excitations: Hubbard Atoms: sharp excitation
lines corresponding to adding or removing electrons. In
solids they broaden by their incoherent motion, Hubbard
bands (eg. bandsNiO, CoO MnO….)
•H H H+ H H H
motion of H+ forms the lower
Hubbard band
•H H H H- H H
motion of H_ forms the upper
Hubbard band
• Quantitative calculations of Hubbard bands and
exchange constants, LDA+ U, Hartree Fock. Atomic
RUTGERS
Physics.
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Localization vs Delocalization
Strong Correlation Problem
• A large number of compounds with electrons in partially filled
shells, are not close to the well understood limits (localized
or itinerant). Non perturbative problem.
•These systems display anomalous behavior
(departure from the standard model of solids).
•Neither LDA –GW or LDA+U or Hartree Fock work
well.
•Dynamical Mean Field Theory: Simplest approach
to electronic structure, which interpolates correctly
between atoms and bands. Treats QP bands and
Hubbard bands. New reference point, to replace the
Kohn Sham system.
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DFT+GW program has been less
succesful in correlated situations.
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Strong interactions localize the particles. Atoms
with open shells are not easily connected to band
theory.
The spectrum in this case, contain Hubbard bands
which are NOT simply perturbatively connected to
the Kohn Sham orbitals.
Need an alternative reference point for doing
perturbation theory! Situation is worse “in between
the atomic and the localized limit”
DMFT!
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Correlated Materials do “big” things
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Mott transition.Huge resistivity changes
V2O3.
Copper Oxides. .(La2-x Bax) CuO4 High
Temperature Superconductivity.150 K in the
Ca2Ba2Cu3HgO8 .
Uranium and Cerium Based Compounds.
Heavy Fermion Systems,CeCu6,m*/m=1000
(La1-xSrx)MnO3 Colossal Magnetoresistance.
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Strongly Correlated Materials.
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Large thermoelectric response in CeFe4 P12
(H. Sato et al. cond-mat 0010017). Ando
et.al. NaCo2-xCuxO4 Phys. Rev. B 60,
10580 (1999).
Large and ultrafast optical nonlinearities
Sr2CuO3 (T Ogasawara et.a Phys. Rev.
Lett. 85, 2204 (2000) )
Huge volume collapses, Ce, Pu……
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Breakdown of standard model
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LDA+GW program fails badly.
Large metallic resistivities exceeding the
Mott limit. [Anderson, Emery and Kivelson]
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Breakdown of the rigid band
picture. Need new ways to think
about the excitations.
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Anomalous transfer of spectral weight in
photoemission and optics. [G. Sawatzki]
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Strongly correlated systems are usually
treated with model Hamiltonians
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In practice other methods (eg
constrained LDA are used)
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Strongly correlated systems are usually
treated with model Hamiltonians
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They are hard to derive and hard to
solve.
In practice other methods (eg.
constrained LDA are used)
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Outline
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Introduction to the strong correlation problem
and to the Mott transition.
DMFT ideas
Applications to the Mott transition problem:
some insights from studies of models.
Towards an electronic structure method:
applications to materials: Pu……….
Outlook
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Mean-Field : Classical vs Quantum
Classical case
-
å
J ij Si S j - h å Si
i, j
Quantum case
 (t

i , j  ,
i
H MF = - heff So
b
ij
 m 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 ñH MF ( heff )
heff =
å
J ij m j + h
G = ­ áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn ) =
å
k
j
Phys. Rev. B 45, 6497
1
[D (iwn ) -
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1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
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Insert transparency from nijmeigen
About infinite dimensions, and about
Greens functions.
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DMFT: Effective Action point of view.
Identify observable, A. Construct an exact functional of
<A>=a, G [a] which is stationary at the physical value of a.
 Example, density in DFT theory. (Fukuda et. al.)
 When a is local, it gives an exact mapping onto a local
problem, defines a Weiss field.
 The method is useful when practical and accurate
approximations to the exact functional exist. Example:
LDA, GGA, in DFT.
 DMFT, build functionals of the LOCAL spectral function.
[Density of states for adding or removing and electron]
 Exact functionals exist. We also have good
approximations!
 Extension to an ab initio method.

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LDA+DMFT References
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V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and
G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).
A Lichtenstein and M. Katsenelson Phys. Rev. B 57,
6884 (1988).
S. Savrasov and G.Kotliar and Abrahams funcional
formulation for full self consistent Nature {\bf 410},
793(2001).
Reviews: Held et.al. , Psi-k Newsletter \#{\bf 56}
(April 2003), p. 65 Lichtenstein Katsnelson and and
Kotliar cond-mat/0211076:
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How good is the LOCAL
approximation?
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C-DMFT: test in one dimension.
(Bolech, Kancharla GK cond-mat 2002)
Gap vs U, Exact solution
Lieb and Wu,
Ovshinikov
Nc=2 CDMFT
vs Nc=1
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N vs mu in one dimension.
Compare 2+8 vs exact Bethe Anzats, [M.
Capone and M.Civelli]
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Outline
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
Introduction to the strong correlation
problem.
Essentials of DMFT
Applications to the Mott transition problem:
some insights from studies of models.
Towards an electronic structure method:
applications to materials
Outlook
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The Mott transition
Electronically driven MIT.
 Forces to face directly the localization
delocalization problem.
 Relevant to many systems, eg V2O3
 Techniques applicable to a very broad
range or problems.

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Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Resistivity.
Limelette et. al.
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How good is the local
approximation ?
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Single site DMFT study of the Mott transition, based
on a study of the Hubbard model on frustrated
lattices made several interesting qualitative
predictions.
New experiments and reexamination of old ones
give credence to that the local picture is quite
good.
DMFT is a new reference frame to approach
strongly correlated phenomena, and describes
naturally , NON RIGID BAND picture, highly
resistive states, etc….
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Insight
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Phase diagram in the T, U plane of a
frustrated ((the magnetic order is
supressed)) correlated system at integer
filling.
At high temperatures, the phase diagram is
generic, insensitive to microscopic details.
At low temperatures, details matters.
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Schematic DMFT phase diagram one band
Hubbard model (half filling, semicircular
DOS, partial frustration) Rozenberg et.al PRL
(1995)
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Mott transition in layered organic conductors
al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
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S Lefebvre et
Insight, in the strongly correlated
region the one particle density of
states has a three peak structure
low energy quasiparticle peak
plus Hubbard bands.
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DMFT has bridged the gap
between band theory and
atomic physics.

Delocalized picture, it
should resemble the
density of states,
(perhaps with some
additional shifts and
satellites).
Localized picture. Two
peaks at the ionization
and affinity energy of the
atom.

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One electron spectra near the
Mott transition, three peak
structure.
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ARPES measurements on NiS2-xSex
.Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57,
3829 (1998)
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QP in V2O3 was recently
found Mo et.al
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Insights from DMFT
The Mott transition is driven
by transfer of spectral weight
from low to high energy as we
approach the localized phase
Control parameters: doping,
temperature,pressure…
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Evolution of the Spectral Function
with Temperature
Anomalous transfer of spectral weight connected to the
proximity to the Ising Mott endpoint (Kotliar Lange nd
Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
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ARPES measurements on NiS2-xSex
.Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57,
3829 (1998)
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Anomalous metallic resistivities
In the “ in between region “ anomalous
resistivities are the rule rather than the
exception.

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Failure of the Standard Model:
Miyasaka and
NiSe2-xSx
Takagi (2000)
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Anomalous Resistivity and Mott
transition (Rozenberg et. Al. ) Ni Se2-x
Sx
Insights from DMFT: think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
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More recent work, organics,
Limelette et. al.(PRL 2003)
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Title:
Anomalous Resistivities when wave
picture does not apply. Doped
Hubbard model
Qualitative single site DMFT
predictions: Optics
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Spectra of the strongly correlated metallic
regime contains both quasiparticle-like and
Hubbard band-like features.
Mott transition is drive by transfer of spectral
weight. Consequences for optics.
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Anomalous transfer of spectral
weight in v2O3
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Anomalous transfer of optical
spectral weight V2O3
:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf
Phys. Rev. Lett. 75, 105 (1995)
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Anomalous transfer of optical spectral
weight, NiSeS. [Miyasaka and Takagi
2000]
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Anomalous transfer of
spectral weight heavy
fermions Rozenberg Kajueter Kotliar
(1996)
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Anomalous transfer of optical weight
[A. Damascelli D. Van der Marel ]
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Anomalous Spectral Weight Transfer:
Optics
H hamiltonian, J electric current , P polarization


 ne 2
0  ( )d  iV   P, J   m
Below energy


0

H eff , J eff , Peff

ApreciableT dependence
 ( )d 
  Peff , J eff  found.
iV
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL
72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).
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DMFT and the strong correlation
anomalies: crossover from momentum
space to real space picture
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Metals with resistivities which exceed the
Mott Ioffe Reggel limit.
Three peak structure of DOS
Transfer of spectral weight which is non
local in frequency.
Dramatic failure of DFT based
approximations in predicting physical
properties.
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Outline
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Introduction to the strong correlation problem.
Essentials of DMFT
Applications to the Mott transition problem: some
insights from studies of models.
Towards an electronic structure method:
applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3,
NiO,MnO,CrO2,K3C60,2d and quasi-1d organics,
magnetic semiconductors,SrRuO4,V2O3………….
Outlook
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Generalized phase diagram
T
Relax Structure,
bands,
orbitals
U/W
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Pu in the periodic table
actinides
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Electronic Physics of Pu
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DFT studies.
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Underestimates the volume by 35 %
Predicts Pu to be magnetic.
Largest quantitative failure of DFT-LDA-GA
Fail to predict a stable delta phase.
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Phonon Spectra
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Electrons are the glue that hold the atoms
together. Vibration spectra (phonons) probe
the electronic structure.
Phonon spectra reveals instablities, via soft
modes.
Phonon spectrum of Pu had not been
measured until recently.
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Phonon freq (THz) vs q in delta Pu X.
Dai et. al. Science vol 300, 953, 2003
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Inelastic X ray scattering. Wong et.
al. Science 301, 1078 (2003).
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Pu: DMFT total energy vs Volume
(Savrasov Kotliar and Abrahams 2001)
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Alpha and delta Pu
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Outline

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Introduction to the strong correlation
problem.
Essentials of DMFT
The Mott transition problem: some insights
from studies of models.
Towards an electronic structure method:
applications to materials: Pu
Outlook
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What do we want from
materials theory?
New concepts , qualitative ideas
 Understanding, explanation of existent
experiments, and predictions of new ones.
 Quantitative capabilities with predictive
power.

Notoriously difficult to achieve in strongly
correlated materials. DMFT is delivering on
both fronts.
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Outlook
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Local approach to strongly correlated
electrons.
Many extensions, make the approach
suitable for getting insights and quantitative
results in correlated materials.
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Conclusion
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
The character of the localization
delocalization in simple( Hubbard) models
within DMFT is now fully understood, nice
qualitative insights.
This has lead to extensions to more realistic
models, and a beginning of a first principles
approach to the electronic structure of
correlated materials.
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Outlook
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Systematic improvements, short range
correlations, cluster methods, improved
mean fields.
Improved interfaces with electronic
structure.
Exploration of complex strongly correlated
materials. Correlation effects on surfaces,
large molecules, systems out of equilibrium,
illumination, finite currents, aeging.
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Acknowledgements: Development of DMFT
Collaborators: V. Anisimov,G. Biroli, R. Chitra, V.
Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H.
Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein,
G. Moeller, Y. Motome, O. Parcollet , G. Palsson, M.
Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang,
X.Y. Zhang
Support: NSF DMR 0096462
Support: Instrumentation. NSF DMR-0116068
Work on Fe and Ni: ONR4-2650
Work on Pu: DOE DE-FG02-99ER45761 and
LANL subcontract No. 03737-001-02
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High Performance
Computing
http://beowulf.rutgers.edu
High Performance Computing Project
Department of Physics and Astronomy
National Science Foundation
- NSF0116068: Acquisition of a
Network Cluster of Advanced
Workstations for First Principles Electronic Structure
Calculations of
Complex Materials
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TOP 500 (ICL-UT)
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TOP 500
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Shear anisotropy fcc Pu (GPa)

C’=(C11-C12)/2

C44= 33.59


= 4.78
C44/C’ ~ 8 Largest shear anisotropy in any
element!
LDA Calculations (Bouchet et. al.) C’= -48
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Dai et. al.
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Epsilon Plutonium.
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Anomalous transfer of
spectral weight heavy
fermions
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Anomalous transfer of
spectral weight
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Anomalous transfer of
spectral weigth heavy
fermions
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V2O3 resistivity
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Example: DMFT for lattice model (e.g.
single band Hubbard).Muller Hartman 89,
Chitra and Kotliar 99.
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Observable: Local Greens function Gii ().
Exact functional G [Gii () .
DMFT Approximation to the functional.
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Spectral Density Functional : effective
action construction (Chitra and GK).
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Introduce local orbitals, aR(r-R)orbitals, and local
GF
G(R,R)(i ) =  dr ' dr  (r ) *G(r , r ')(i ) a (r ')
R
R
The exact free energy can be expressed as a
functional of the local Greens function and of the
density by introducing sources for r(r) and G and
performing a Legendre transformation,
Gr(r),G(R,R)(i)]
Approximate functional using DMFT insights.
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