‘Botany’ of correlated fermion...
Transcript of ‘Botany’ of correlated fermion...
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
‘Botany’ of correlated fermion physics
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Example: 3He
Phase diagram
Fermions: short ranged interaction
Low temperatures: quantum liquid phase, unstable to localized (solid) phase and to other phases with interesting quantum orders (superfluid)
ltl.tkk.fi
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Normal liquidD. Greywall PRB 27 2747 (1983)
D. Greywall Fig. 6
Low T: specific heat linear in T but coefficient enhanced. Higher T; saturation
C
T! m!kF
P ! 0 (vol=36.8)m! = 2.7mbare
P ! 29bar (vol=26.2)m! = 5.4mbare
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Other quantitiescharge susceptibility !
spin susceptibility !spin = !01+F a
0
Vollhardt RMP 56 p. 99 (1984) from Greywall data
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Values of renormalization
While fermi liquid theory tells you what kind of renormalizations to expect, the values of the mass enhancement, landau parameters etc are
beyond the scope of FLT
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
At higher T specific heat ‘flattens off’
=> ‘free spin’ like behavior T>0.4K + positional entropy of classical liquidQualitative idea: particle motion
quenches spin entropy. Interaction-induced ‘blocking’ of particle motion=>quenching scale low, C/T high
S=R ln(2)/particle
This is beyond the scope of FLT
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Transitions to other states
While some idea of which states are favored and a rough order of magnitude of the transitions can be found from fermi liquid theory, a detailed understanding
is beyond the scope of FLT
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Digression: consequences of long ranged order(not discussed here)
http://www.aip.org/history/acap/biographies/bio.jsp?wilsonk
http://www.statemaster.com/encyclopedia/Image:Andersonphoto.jpg
Spontaneously broken symmetry (superconductivity, antiferromagnetism, ....) => order parameter (e.g. gap magnitude and phase; neel vector amplitude and direction). Many properties associated with the order parameter and its fluctuations are ‘universal’ (depending only on a few properties of host material) and can be studied from a relatively general field-theoretic point of view. Fermi liquid theory itself can be formulated in this language
P. W. Anderson, Basic Notions in Condensed Matter PhysicsD. Forster, Hydrodynamics, Broken Symmetry and Correlation Functions
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Questions beyond the scope of fermi liquid theory
•Higher T behavior•Instability to other phases•Value of m* and other fermi liquid parameters
Modern (correlated fermion) ‘materials theory’:
2 themes:
--New (esp. non-classical) orders+fluctuations--beyond fermi liquid theory in above sense
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Summary•Fermi liquid theory: successful description of low T properties of strongly coupled fermion system
•Simple physics: fermions+short range repulsion (basically ‘hard core’ contact interaction) => nontrivial behavior
•Electron Green function (esp coherent (‘quasiparticle’) part is a good thing to look at
These insights are the basis of our understanding of the physics of ‘correlated electron’ materials
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Basic Solid State Physics:Fermi liquid approach: start from non-interacting electrons then add interaction-driven renormalizationsNew feature: periodic potential
H = !!
i
"2i
2m+ Vlattice(ri)
=>Energy bands
LaTiO3
http://dmft.rutgers.edu/LDA/lmto/lmto_run.htm
all bands full or empty: insulatorsome partly filled bands: metal
If number of electrons/cell not even:insulator impossible (in band theory)
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
‘Band’ classification doesnt always work
LaTiO3
http://dmft.rutgers.edu/LDA/lmto/lmto_run.htm
LaTiO3: •# electrons/unit cell is odd•material is insulating!•YTiO3 even more insulating
Okimoto et al. PRB51 9581 (1995)
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
‘Mott’ insulating behavior
www.picsearch.com http://www.engr.utexas.edu/news/6544-goodenough-royal-society
Imada et al. Rev. Mod. Phys., 70, 1039,1998
•Many materials (transition metal oxides, low-d organic materials, C-60 variants...) are insulating when band theory says they should be metallic•Common feature: narrow bands, spatially confined orbitals, separated from other bands
LaTiO3
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
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LaTiO3 Lattice constant 3.91A Important orbital Ti 3d
0.000 1.955 3.9100.000
1.955
3.910
Ti0.5 (integrated in z direction ±1Å)
x (Å)
y (Å
)
-0.02000
0.02000
0.06000
0.1000
0.1400
0.1800
Here: Ti 3d charge density, integrated over z and plotted using false color (for LTO-STO superlattice).
S. Okamoto, N, Spaldin and AJM PRL 97 056802 (2006)
Spatial extent of charge in important orbitals is small relative to lattice constant
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Implications of nearly localized wave function
•Bandwidth (‘bare mass’ or kinetic energy) arises from overlap of orbitals: => small and controllable by changes in crystal structure•e2/r=>expensive to put 2 electrons into same orbital=>strong local repulsion is dominant interaction
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
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=>Model Hamiltonian: ‘Hubbard model’
H = !!
ij
ti!jc†i!cj! + U
!
i
ni"ni#
One (spin degenerate) orbital per lattice site. hopping t short ranged.
Important parameters: --relative interaction strength U/t--electron density n
?Where do parameters come from?
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Qualitative view of physics of Hubbard Model
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
Pauli principle: no motion possible=>insulator
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
Add a “hole”:
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
Add a “hole”:
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
Add a “hole”: motion possible
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Conventional (band) Insulator:physics of filled shells
Add a “hole”: motion possible
in k-space: filled and empty bands, with gap
0.0 0.1 0.2 0.3 0.4 0.5
!1.0
!0.5
0.0
0.5
1.0
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Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
strong repulsion: no charge motion possible=>insulator
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
strong repulsion: no motion possible=>insulator
BUT: virtual hopping =>‘superexchange’:
magnetic scale J~t2/U
1 2 3
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
add a hole
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
add a hole
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
add a hole: charge motion possible
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Unconventional (‘Mott’) Insulator:physics of “Coulomb blockade”
add a hole: charge motion possible. Properties scale with hole density
=>
Deep theoretical questions:
•Metal-insulator transition as vary U/t and n •role of long and short ranged order•nature of hole motion in background of correlated spins•consequences for other physical properties
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Data: LaTiO3 reveal divergent effective mass (specific heat) associated with
approach to insulating phase
Fig 104 Imada et al Rev Mod Phys
Consistent with idea: entropy comes from spins, which are correlated by hole motion
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
But there is more to life than the 1 orbital Hubbard model
Orbital degeneracy:
in free space: d-level is 5 fold degeneratein solid state, O(3) rotation symmetry broken
=>degeneracy lifted (mainly by `ligand field’ i.e. hybridization to neighboring atoms)
Cubic ligand field: 5->2+3
eg: 3z2 ! r2, x2 ! y2
t2g: xy, xz, yz
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digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial
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digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
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digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
eg anti- bonding
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
eg anti- bonding
eg bond- ing
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
eg anti- bonding
eg bond- ing
t2g anti- bonding
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
eg anti- bonding
eg bond- ing
t2g anti- bonding
t2g bonding
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
digression: in transition metal oxides hybridization of transition metal (Tm)
to oxygen is crucial Tm O Tm OO O Tm O Tm O
J. Rondinelli ANL prvt comm
LaNiO3
eg anti- bonding
eg bond- ing
t2g anti- bonding
t2g bonding
Rule of thumb: eg has wider bands, bigger bonding/antibonding split than t2g
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Back to orbital degeneracy
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Transition metal ion in free space(For simplicity neglect spin orbit coupling here)
Fundamental perspective: atom as fully entangled multielectron system. Eigenstates A characterized by quantum numbers
Electron number N
Orbital Angular momentum L, Lz
Spin Angular momentum S, Sz
=> Energy EA(N,S,L)
J. C. Slater: Quantum Theory of Atomic Structure
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Needed: a representation in terms of single-particle orbitals (for intuition,
and to discuss band formation)Idea (Slater):
choose basis (e.g. d-symmetry atomic orbitals+...)devise Hamiltonian whose eigenstates are EA(N,S,L)
(for relevant range of energies)!2,m,! = Y2,m(", #)fd(r)Example:
H =!
(m!)i=1...4
U!1!2!3!4m1m2m3m4
d†m1!1
d†m2!2
dm3!3dm4!4
Slater, J. C. (1960). Quantum Theory of Atomic Structure
Choose Us to fit EA(N,S,L)--or compute them
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Parametrizing the interaction: Coulomb integrals
U!1!2!3!4m1m2m3m4
=
e2
!d3rd3r! !
†m1!1
(r)!m3!3(r)!†m2!2
(r!)!m4!4(r!)|"r ! "r!|
Spherical symmetry+spin rotation: express all U in terms of interactions between multipoles of charge density
F0 = e2
!d3rd3r! !(r)!(r!)
|̃r! r̃!|charge-charge (monopole) interaction
F2, F4, ..: interaction between higher multipolesSpin-orbit--also need ‘G’ parameters
Crucial parameter:
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
d-orbitalsweak spin-orbit coupling
•Need only F0, F2, F4
•*In free space*: definite relation between F2, F4 => Only 2 parameters: Conventional notation: U, J
Conventional (Slater-Kanamori) representation
85 interaction terms, determined by 2 parameters
H = U!
a
na!na" + (U ! 2J)!
a>b,!=!,"na!nb!
+(U ! 3J)!
a#=b!
na!nb!̄ ! J!
a#=b
c†a!c†a"cb!cb" + c†a!c
†b"cb!ca"
Lower symmetry: more parameters, more complicated form--but generically: charging energy +smaller terms
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Orbital degeneracy, interactions make a difference
Werner, Gull, AJM, Phys. Rev. B 79, 115119 (2009)
3 orbital model Insulating phase inside lobes
J=0
Critical U for 1 orbital model
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Effect of J
Werner, Gull, AJM, Phys. Rev. B 79, 115119 (2009)
3 orbital model Insulating phase inside lobes
J=0 J=U/6
Phase boundaries move sustantially if J>0\Note also effect of lifting orbital degeneracy slightly
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
The case of LaTiO3
Argument: LaTiO3 is insulating only because crystal structure lifts orbital degeneracy
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Second example: High Tc (Cu-O2) superconductors
Data: Phase diagram
A.Zimmers Ph.D. thesis
Uchida, S., T. Ido, H. Takagi, T. Arima, Y. Tokura, and S.Tajima, 1991, Phys. Rev. B 43, 7942
Electron counting: 1 el/unit cell at 0 doping
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
High Tc CuO2 superconductors are not quite Mott Insulators
O: very electronegative: p-shell almost full Cu: ‘late’ transition metal oxide. d-shell almost full.Ti: ‘early transition metal oxide: d-shell almost empty
Electronic configuration:La2CuO4: Cu d9 O2p6
LaTiO3: Ti d1 O2p6
Hole dope: in late TMO ‘holes go on the O
Cu d9 O2p5
Ti d0 O2p6
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Mott vs Charge Transfer Insulators
Zaanen, Sawatzky, Allen, Phys. Rev. Lett. 55, 418 (1985)
2dn=>dn-1dn+1 not the only process Fragment of CuO2 plane
Cuprates: 2d9=>d8d10 8eV2d9=>d9Op5d10 2eV
Cu---O---Cu
Cu---O---Cu
Cu---O---Cu
EnergiesCu d10
Cu d9
Cu d8
µO p
!U
! > U
Charge transfer insulator
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Zaanen-Sawatzky-Allen Classification
Mott-Hubbard
Charge Transfer
Zaanen, Sawatzky, Allen, Phys. Rev. Lett. 55, 418 (1985)
Questions:•Where do you place different materials in this scheme•More generally, what is the role of covalency with other orbitals•what difference does it make?
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Charge transfer insulators:important of other (non-d) orbitals
More extreme example: heavy fermion compounds
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Charge transfer insulators:important of other (non-d) orbitals
More extreme example: heavy fermion compounds
Fig 2 G R Stewart RMP
Low T: constant specific heat coefficient; magnitude =>m ~ 1000me
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Charge transfer insulators=>importance of other (non-d) orbitals
More extreme example: heavy fermion compounds
Fig 2 G R Stewart RMP 56 755 1984
Low T: constant specific heat coefficient; magnitude =>m ~ 1000me
Large specific heat is due to degrees of freedom that can superconduct
Fig 3
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
What is going on?
CeCu2Si2Cu, Si (also Ce): wide spd bands => metallic state
Ce f: small orbital, weakly hybridized. Large U. Holds ~1 electron.
High T: ‘free spin’ like behavior
Low T: ‘heavy fermi liquid’
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Anderson Lattice Model
H =!
im!
Emf f†
imfim +12U
"N i
F (N if ! 1)
#+ .
+!
ikam!
eik·RiVkm!f†mcka! + H.c. +
!
ka!
!akc†ka!cka!
Lattice version of ‘Kondo resonance’
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Summary:‘botany’ of correlated materials
•Want to understand•Physics at scales above fermi liquid•Nature of instabilities to other phases•Metal-insulator transitions
•Wish to relate novel behavior to crystal structure
•Need to deal with strong local interactions involving multiple quantum numbes
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Department of PhysicsColumbia UniversityCopyright A. J. Millis 2010
Issues
•Derivation of model parameters (Hubbard of Anderson lattice) from underlying chemistry
•Solution of models
More ambitious goal: fully ‘ab initio’ theory