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Theory of correlated fermionic condensed matter
1. Correlated electrons made simplea. What are electronic correlations and where do they show up?
Supported by Deutsche Forschungsgemeinschaft through SFB 484
XIV. Training Course in the Physics of Strongly Correlated SystemsSalerno, October 5, 2009
Dieter Vollhardt
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• "Correlations"
• Electronic correlations in the periodic table
• Fermi liquid theory
• Electronic correlations in solids: Examples
• How to detect electronic correlations: e.g., photoemission spectroscopy
• Model approaches to correlated electon systems:Hubbard model
Outline:
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Correlation [lat.]: con + relatio ("with relation")
Grammar: either ... or
Mathematics, natural sciences:
AB A B
( ) ( ') ( ) ( ') r r r r
e.g., densities:
Beyond (standard) mean-field theory [Weiss/Hartree-Fock,...]
correlation causality
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Short-range spatial correlations in everyday life
Time average insufficient
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(Sempe)
Correlationsvs.
long-range order
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Electronic Correlationsin the Periodic Table
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Partially filled d-orbitals
Partially filled f-orbitals
Narrow d,f-orbitals strong electronic correlations
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Insula-tor
Solid NeNaCl
Localizedelectrons
Atomiclevels n i
ExampleRepresen-tation
Energy levels
PropertyElectronic Bands in Solids
Tran-sition +rareearthmetals/oxides(Ni, V2O3, Ce)
Narrowbands
Corre-latedmetal
n n i k
Na, AlExtendedwavesBroad
bandsSimple metal n k
overlap of wave functions: matrix element t
band overlap band width t W k W
Small W: Strong electronic correlations
1 aW
1 lattice spacing: vaverage time spent on atom:
a
k k k Consequences?
Estimate strength of correlations:
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Transition metals: Spin, charge, orbital order; electron-lattice coupling,Mott-Hubbard metal-insulator transitions, high Tc, …
Transition metal oxides: direct view of d-electrons
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Rare earth elements: Heavy fermion-, Kondo lattice-, RKKY-behavior, unconventional superconductivity, non-Fermi liquid behavior, volume anomalies
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Actinides: Heavy fermion behavior, unconventional superconductivity, volume anomalies, strong spin-orbit coupling
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Electrons vs. Quasiparticles,Fermi liquid theory
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1
2Spin = Fermion
Fermi-Dirac statistics
Fermi body/surface
Pauli exclusion principleof many fermions
Electrons
No such thing for bosons!
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Fermi gas: Ground state
kx
ky
kz
Fermi sea
Fermi surface
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kx
ky
kz
Fermi gas: Excited states (T>0)
Switch on interaction adiabatically (d=3)
Exact k-states ("particles"): infinite life time
Particle
Hole
Fermi sea
Fermi surface
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Landau Fermi liquid
kx
ky
kz
Well-defined k-states ("quasiparticles") with - finite life time - effective mass- effective interaction
(Quasi-) Particle
(Quasi-) Hole
Fermi sea
Fermi surface
Landau (1956/58) 1-1 correspondence between k-states
= elementary excitation
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Electronic Correlations in Solids: Examples
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Simple metals
00*lim V
T
cT
m m
Potassium
Consequence of elementary excitations
(quasiparticles)
Consequence of elementary excitations
(quasiparticles)
T2 (K2)
C/T
(mJ/
mol
e K2 )
CeCu2Si2, UBe13:very heavy quasiparticles
"Heavy Fermions"
* 1000 m m
0lim ,*V
T
cT
mm
v*F
F mk
Steglich et al. (1979)
Stewart et al.(1983)
C/T
(mJ/
mol
e K2 )
T2 (K2)
1.
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2.
Ce
Magnetic impurity in a metallic host:The Kondo effect
Explanation of the three peak structure?
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Detection of electronic correlations in solids byPhotoemission spectroscopy
Excursion:
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Angular Resolved PES = ARPES
Measures occupied states of electronic spectral function
1. Photoemission Spectroscopy (PES)
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PESPES
Ideal spectral function of a material
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Ideal spectral function of a material
PESPES
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Occupied states(ideal)
PESPES
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Occupied states(measured)
PESPES
Only two peaks visible
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2. Inverse Photoemission Spectroscopy (IPES)
Measures unoccupied states of electronic spectral function
X-ray Absorption Spectroscopy (XAS)
Information also available by:
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Ideal spectral function of a material
IPES/XASIPES/XAS
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Unoccupied states(ideal)
IPES/XASIPES/XAS
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Unoccupied states(measured)
IPES/XASIPES/XAS
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Photoemission spectra of Ni: -6 eV satellite
Guillot,..., Falicov (1977)
Not reproducible byDensity Functional Theory/Local Density Approximation
3.
Explanation of the -6 eV satellite?
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Photoemission spectra of (Sr,Ca)VO3
Osaka – Augsburg – Ekaterinburg collaboration: Sekiyama et al., 2004
SrVO3CaVO3
Reason for shift of spectral weight?
4.
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5.
Sawatzky, Allen (1984)
Origin of gap(antiferromagnetism)?
Photoemission spectra of NiO
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Rice, McWhan (1970); McWhan, Menth, Remeika,
Brinkman, Rice (1973)
Metal-insulator transition in V2O3
6.
•PI PM: 1. order transitionwithout lattice symmetry change
•Anomalous slope of P(T)
heating
Pomeranchuk effect in 3He
Microscopicexplanation?
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•large resistivity changes•huge volume changes•high Tc superconductivity•strong thermoelectric response
•gigantic non-linear optical effects•colossal magnetoresistance
Correlated electron materials
Fascinating topics for fundamental research
Large susceptibilities
Technological applications:• sensors, switches• magnetic storage• refrigerators• functional materials, ...
with
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Model approachesto correlated electrons
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t U
Gutzwiller, 1963Hubbard, 1963Kanamori, 1963
Hubbard model
Microscopic theoryof ferromagnetism?
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,nUH n n
k i ik
ki
†
, ,
c c U n nt
i j i ii j i
t U
Gutzwiller, 1963Hubbard, 1963Kanamori, 1963
Hubbard model
time
Local Hubbard physics:
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n n n n i i i i
Hartree-(Fock) mean-field theorygenerally insufficient
Correlation phenomena:Metal-insulator transitionFerromagnetisms,...
t U
Gutzwiller, 1963Hubbard, 1963Kanamori, 1963
Hubbard model
,nUH n n
k i ik
ki
†
, ,
c c U n nt
i j i ii j i
Local Hubbard physics:
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Beyond models: How to include material-specific details?
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Reliable, comprehensive approximation scheme
forcorrelated electron
models and materialsfor
arbitrary input parameters
Dynamical Mean-Field Theory