Electronic Structure of Strongly Correlated Materials : a DMFT Perspective
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Electronic Structure of Strongly Correlated Materials : a DMFT
Perspective
Gabriel Kotliar
Physics Department andCenter for Materials Theory
Rutgers University
Supported by the NSF -DMR
THE STATE UNIVERSITY OF NEW JERSEY
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Outline
Introduction to the electronic structure of correlated electrons
Dynamical Mean Field Theory Delocalization - Localization
Transition in frustrated systems:universality at high temperatures
A case study of system specific properties: Pu (S. Savrasov, supported by DOE, Basic Energy Sciences)
Outlook
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Standard Model of Solids
High densities, electron as a wave, band theory, k-space
One particle excitations: quasi-particle bands
Au, Cu, Si……
How to think about an electron in a solid ?
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Standard Model
Typical Mott values of the resistivity 200 Ohm-cm
Residual instabilites SDW, CDW, SC
Odd # electrons -> metal
Even # electrons -> insulator
Theoretical foundation: Sommerfeld, Bloch and Landau
Computational tools DFT in LDA
Transport Properties, Boltzman equation , low temperature dependence of transport coefficients
2 ( )Mott
F Fe k k lh
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Mott : correlations localize the electron
Low densities, electron as a particle, atomic physics, real space
One particle excitations: Hubbard bands
NiO, CoO MnO….
Magnetic and Orbital Ordering at low T
Quantitative calculations of Hubbard bands and exchange constants, LDA+ U
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Localization vs Delocalization
•A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant).•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U works well•Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands
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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455
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Failure of the Standard Model: NiSe2-xSx
Miyasaka and Takagi (2000)
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Failure of the StandardModel: Anomalous Spectral Weight Transfer
Optical Conductivity o of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993)
0( )d Neff
0( )d
Neff depends on T
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Strong Correlation Problem Large number of f and d
electrons based compounds Hamiltonian is known. Identify
the relevant degrees of freedom at a given scale.
Treat the itinerant and localized aspect of the electron
The Mott transition, head on confrontation with this issue
Dynamical Mean Field Theory simplest approach interpolating between that bands and atoms
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Hubbard model
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
U/t
Doping d or chemical potential
Frustration (t’/t)
T temperature
Mott transition as a function of doping, pressure temperature etc.
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Two routes to the Mott transition: Bandwidth-Control (“U/D”) and Filling-Control (doping)Imada et.al RMP (1999)
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Limit of large lattice coordination
1~ d ij nearest neighborsijt d
† 1~i jc cd
†
,
1 1~ ~ (1)ij i jj
t c c d Od d
~O(1)i iUn n
Metzner Vollhardt, 89
1( , )( )k
G k ii i
Muller-Hartmann 89
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Mean-Field : Classical vs Quantum
Classical case Quantum case
Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)
†
0 0 0
( )[ ( ')] ( ')o o o oc c n nb b b
s st m t t tt ¯¶ + - D - +¶òò ò
( )wD†
( )( ) ( )MFL o n o n HG c i c iw w D=- á ñ
1( ) 1( )( )[ ]
[ ]n
kn k
n
G ii
G i
ww e
w
=D - -D
å
,ij i j i
i j i
J S S h S- -å åMF eff oH h S=-effh
0 0 ( )MF effH hm S=á ñ
eff ij jj
h J m h= +å
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
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Solving the DMFT equations
G 0 G
Impu rit ySolv er
S.C .C.
•Wide variety of computational tools (QMC, NRG,ED….)
•Analytical Methods
G0 G
Im puritySo lver
S .C .C .
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DMFT: Methods of Solution
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Reviews of DMFT
Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)
A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
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DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al. ).
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DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)
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Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT
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Schematic DMFT phase diagram Hubbard model (partial frustration) Rozenberg et.al. PRL (1995)
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Kuwamoto Honig and AppellPRB (1980)
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A time-honored example: Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Phase Diag: Ni Se2-x Sx
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Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl 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 Mott endpoint (Kotliar Lange and Rozenberg 2000)
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Ising character of the transfer of spectral weight
Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint
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. ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690
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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
Spectral Evolution at T=0 half filling full frustration
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Parallel development: Fujimori et.al
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Anomalous Resistivities and Mott phenomena (Rozenberg et. al 1995)
Resistivity exceeds the Mott limit
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Anomalous Resistivity and Mott transition Ni Se2-x Sx
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Miyasaka and takagi
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Insights from DMFT Mott transition as a bifurcation
of an effective action
Important role of the incoherent part of the spectral function at finite temperature
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Landau Functional
† †,
2
2
[ , ] ( ) ( ) ( )†
† † † †
0
†
Mettalic Order Para
( )[ ] [ ]
mete
[ ]
[ , ] [ [ ] ]
( )( )
r: ( )
( ) 2 ( )[ ]( )
loc
LG imp
L f f f i i f i
imp
loc f
imp
iF T Ft
F Log df dfe
dL f f f e f Uf f f f dd
F iT f i f i TG ii
i
2
2
Spin Model An
[ ] [[ ]2 ]
alogy:
2LG
t
hF h Log ch hJ
G. Kotliar EPJB (1999)
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Realistic Calculationsof the Electronic Structure of Correlated materials Combinining DMFT with state of
the art electronic structure methods to construct a first principles framework to describe complex materials
Hubbard bands and QP bands
The puzzle of elemental plutonium (S. Savrasov)
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Delocalization Localization Transition across the actinide series
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Problems with LDA
o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
o Many studies (Freeman, Koelling 1972, ….Beottger et.al 1998, Wills et.al. 1999) give
o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than Is 35% lower than experimentexperiment
o This is the largest discrepancy ever known in DFT based calculations.
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Pu: Complex Phase Diagram (J. Smith LANL)
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Anomalous ResistivityJ. Smith LANL
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Pu Specific Heat
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Pu: DMFT total energy vs Volume (Savrasov 00)
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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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S. Savrasov: DMFT Lab
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Strongly Correlated Electrons
Competing Interaction
Low T, Several Phases Close in Energy
Complex Phase Diagrams
Extreme Sensitivity to Changes in External Parameters
Need for Realistic Treatments
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Outlook Strongly correlated electron
exhibit unusual characteristics, complex systems.
Two recent examples: largeThermoelectric response in CeFe4
P12 (H. Sato et al. cond-mat 0010017).
Large Ultrafast Optical Nonlinearities Sr2CuO3 (T Ogasawara et.al cond-mat 000286)
Theory will play an important role in optimizing their physical properties.
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Outlook
The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR
Strategy: advancing our understanding scale by scale
Generalized cluster methods to capture longer range magnetic correlations
New structures in k space?
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