Band structure of strongly correlated
materials from the Dynamical Mean
Field perspective
K Haule
Rutgers University
Collaborators : J.H. Shim & Gabriel Kotliar, S. Savrasov
Bonn, 2008
Outline
Dynamical Mean Field Theory in combination with band
structure

LDA+DMFT results for 115 materials (CeIrIn5)





Local Ce 4f - spectra and comparison to AIPES)
Momentum resolved spectra and comparison to ARPES
Optical conductivity
Two hybridization gaps and its connection to optics
Fermi surface in DMFT
Actinides

Absence of magnetism in Pu and magnetic ordering in
Cm explained by DMFT

Valence of correlates solids, example of Pu
References:
•J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007).
•J.H. Shim, KH, and G. Kotliar, Nature 446, 513 (2007).
Standard theory of solids
Band Theory: electrons as waves: Rigid band picture: En(k) versus k
Landau Fermi Liquid Theory applicable
Very powerful quantitative tools: LDA,LSDA,GW
Predictions:
•total energies,
•stability of crystal phases
•optical transitions
M. Van Schilfgarde
Strong correlation –
Standard theory fails



Fermi Liquid Theory does NOT work . Need new concepts
to replace rigid bands picture!
Breakdown of the wave picture. Need to incorporate a real
space perspective (Mott).
Non perturbative problem.
Universality of the Mott transition
Crossover: bad insulator to bad metal
Critical point
First order MIT
V2O3
1B HB model
(DMFT):
Ni2-xSex
k organics
1B HB model
(plaquette):
Basic questions to address

How to computed spectroscopic quantities (single particle s
pectra, optical conductivity phonon dispersion…) from first
principles?

How to relate various experiments into a unifying picture.

New concepts, new techniques….. DMFT maybe simplest ap
proach to meet this challenge
DMFT + electronic structure method
Basic idea of DMFT+electronic structure method (LDA or GW):
For less correlated orbitals (s,p): use LDA or GW
For correlated orbitals (f or d): add all local diagrams by solving QIM
(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).
D
atom
solid
Hund’s rule, SO coupling, CFS
LDA+DMFT
(G. Kotliar et.al., RMP 2006).
observable of interest is the "local“ Green's functions (spectral function)
Exact functional of the local Green’s function exists, its form unknown!
Currently Feasible approximations: LDA+DMFT:
LDA functional
ALL local diagrams
Variation gives st. eq.:
Generalized Q. impurity problem!
DMFT + electronic structure method
Dyson equation
correlated orbitals
hybridization
other “light” orbitals
obtained by DFT
Ce(4f) obtained by “impurity solution”
Includes the collective excitations of the system
Self-energy is local in localized basis,
in eigenbasis it is momentum dependent!
all bands are affected:
have lifetime
fractional weight
An exact impurity solver,
continuous time QMC - expansion in terms of hybridization
K.H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005).
General impurity problem
Diagrammatic expansion in terms of hybridization D
+Metropolis sampling over the diagrams
•Exact method: samples all diagrams!
•Allows correct treatment of multiplets
Analytic impurity solvers (summing certain types of diagrams),
expansion in terms of hybridization
K.H. Phys. Rev. B 64, 155111 (2001)
Fully dressed atomic propagators
hybridization
D
•Allows correct treatment of multiplets
•Very precise at high and intermediate
frequencies and high to intermediate
temperatures
Complementary to CTQMC
(imaginary axis -> low energy)
DMFT
“Bands” are not a good concept in DMFT!
Frequency dependent complex object
instead of “bands”
lifetime effects
quasiparticle “band” does not carry weight 1
Spectral function is a good concept
DMFT is not a single impurity calculation
Auxiliary impurity problem:
Weiss field
temperature dependent:
High-temperature D given mostly by LDA
low T: Impurity hybridization affected by the
emerging coherence of the lattice
(collective phenomena)
high T
DMFT SCC:
low T
Feedback effect on D makes the crossover from
incoherent to coherent state very slow!
Phase diagram of CeIn3 and 115’s
CeIn3
CeXIn5
4
AFM
2
*
layering
T (K)
3
1
SC
SC
?
SC
0
Co
CeCoIn5
N.D. Mathur et al., Nature (1998)
Tc[K]
Rh
CeCoIn5
CeRhIn5
CeIrIn5
PuCoG5
Na
0.2K
2.3K
2.1K
0.4K
18.3K
n/a
~50K
~50K
~50K
~370K
300
400
750
100
1000
0.5
Ir
CeRhIn5 X CeIrIn5
CeIn3
Tcrossover
Cv/T[mJ/molK^2]
0.5
1
0.5
Co
CeCoIn5
Tcrossover α Tc
Crystal structure of 115’s
Tetragonal crystal structure
Ir IrIn2 layer
In
Ce CeIn3 layer
IrIn2 layer
4 in plane In neighbors
Ce
In
8 out of plane in neighbors
In
Coherence crossover in experiment
ALM in DMFT
Schweitzer&
Czycholl,1991
Crossover scale ~50K
•High temperature
Ce-4f local moments
out of plane
in-plan
e
•Low temperature –
Itinerant heavy bands
Issues for the system specific study
•How does the crossover from localized moments
to itinerant q.p. happen?
?
•Where in momentum spac
e q.p. appear?
A(w)
•How does the spectral
weight redistribute?
w
k
•What is the momentum
dispersion of q.p.?
•How does the hybridization gap look like in momentum spa
ce?
Temperature dependence of the local Ce-4f spectra
•At 300K, only Hubbard bands
•At low T, very narrow q.p. peak
(width ~3meV)
•SO coupling splits q.p.: +-0.28eV
•Redistribution of weight up to very high
frequency
SO
Broken symmetry (neglecting strong correlations)
can give Hubbard bands, but not both Hubbard band
And quasiparticles!
J. H. Shim, KH, and G. Kotliar
(e
Science 318, 1618 (2007).
Buildup of coherence
Very slow crossover!
coherent spectral weight
Buildup of coherence in single impurity case
coherence pea
k
T
TK
Slow crossover pointed out by
S. Nakatsuji, D. Pines, and Z. Fisk
Phys. Rev. Lett. 92, 016401 (2004)
scattering rate
T*
Crossover around 50K
Consistency with the phenomenological
approach of NPF
Fraction of itinerant heavy fluid
Remarkable agreement with Y. Yang & D. Pines
cond-mat/0711.0789!
Anomalous Hall coefficient
m* of the heavy fluid
Angle integrated photoemission vs DMFT
Very good agreement, but hard to see resonance
in experiment:
resonance very asymmetric in Ce
ARPES is surface sensitive at 122eV
ARPES
Fujimori, 2006 (T=10K)
Angle integrated photoemission vs DMFT
Lower Hubbard band
Nice agreement for the
• Hubbard band position
•SO split qp peak
Hard to see narrow resonance
in ARPES since very little weight
of q.p. is below Ef
ARPES
Fujimori, 2006
Momentum resolved Ce-4f spectra Af(w,k)
Hybridization gap Fingerprint of spd’s due to hybridization
q.p. band
SO
T=10K
scattering rate~100meV
T=300K
Not much weight
Quasiparticle bands
LDA bands
LDA bands DMFT qp bands
DMFT qp bands
three bands, Zj=5/2~1/200
Momentum resolved total spectra
Most of weight transferred into
the UHB
LDA+DMFT at 10K
A(w,k)
ARPES, HE I, 15K
LDA f-bands [-0.5eV, 0.8eV] almost
disappear, only In-p bands remain
Very heavy qp at Ef,
hard to see in total spectra
Below -0.5eV: almost rigid downshift
Unlike in LDA+U, no new band at -2.5eV
Fujimori, 2003
Large lifetime of HBs -> similar to LDA(f-core)
rather than LDA or LDA+U
Optical conductivity
F.P. Mena & D.Van der Marel, 2005
Typical heavy fermion at low T:
no visible Drude peak
w
no sharp
hybridization gap
k
first mid-IR peak
at 250 cm-1
Narrow Drude peak (narrow q.p. band)
Hybridization gap
second mid IR peak
at 600 cm-1
CeCoIn5
Interband transitions across
hybridization gap -> mid IR peak
E.J. Singley & D.N Basov, 2002
Optical conductivity in LDA+DMFT
•At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV)
•At 10K:
•very narrow Drude peak
•First MI peak at 0.03eV~250cm-1
•Second MI peak at 0.07eV~600cm-1
Multiple hybridization gaps
eV
10K
non-f spectra
300K
In
Ce
In
•Larger gap due to hybridization with out of plane
In
•Smaller gap due to hybridization with in-plane In
Fermi surfaces of CeM In5
within LDA
Localized 4f:
LaRhIn5, CeRhIn5
Shishido et al. (2002)
Itinerant 4f :
CeCoIn5, CeIrIn5
Haga et al. (2001)
de Haas-van Alphen experiments
LDA (with f’s in valence) is reasonable for CeIrIn5
Experiment
LDA
Haga et al. (2001)
Fermi surface changes under
pressure in CeRhIn5
localized
itinerant
Shishido, (2005)




Fermi surface reconstruction at 2.34GPa
Sudden jump of dHva frequencies
Fermi surface is very similar on both sides,
slight increase of electron FS frequencies
Reconstruction happens at the point of
maximal Tc
We can not yet address FS change
with pressure 
We can study FS change
with Temperature -
At high T, Ce-4f electrons are excluded from the F
At low T, they are included in the FS
Electron fermi surfaces at (z=0)
Slight decrease of th
e electron FS with T
LDA
M
X
M
X
G
X
M
X
M
a2
LDA+DMFT (10 K)
a2
LDA+DMFT (400 K)
Electron fermi surfaces at (z=p)
No a in DMFT!
No a in Experiment!
LDA
A
R
A
R
Z
R
A
R
A
LDA+DMFT (10 K)
a3
a3
a
Slight decrease of th
e electron FS with T
LDA+DMFT (400 K)
Electron fermi surfaces at (z=0)
Slight decrease of th
e electron FS with T
LDA+DMFT (10 K)
LDA
M
X
M
X
G
X
M
X
M
b1
b1
b2
c
b2
LDA+DMFT (400 K)
Electron fermi surfaces at (z=p)
No c in DMFT!
Slight decrease of th
No c in Experiment! e electron FS with T
LDA+DMFT (10 K)
LDA
A
R
A
R
Z
R
A
R
A
b2
c
b2
LDA+DMFT (400 K)
Hole fermi surfaces at z=0
Big change-> from small hole like
to large electron like
LDA+DMFT (10 K)
LDA
M
X
e1
M
g
X
G
X
M
X
M
LDA+DMFT (400 K)
h
g
h
Localization – delocalization transition
in Lanthanides and Actinides
Delocalized
Localized
Electrical resistivity &
specific heat
Heavy ferm.
in an element
Itinerant
closed shell Am
J. C. Lashley et al. PRB 72 054416 (2005)
NO Magnetic moments in Pu!
Pauli-like from melting to lowest T
No curie Weiss up
to 600K
Curium versus Plutonium
nf=6 -> J=0
closed shell
(j-j: 6 e- in 5/2 shell)
(LS: L=3,S=3,J=0)
One hole in the f shell
No magnetic moments,
large mass
Large specific heat,
Many phases, small or large volume
One more electron in the f shell
Magnetic moments! (Curie-Weiss
law at high T,
Orders antiferromagnetically at
low T)
Small effective mass (small
specific heat coefficient)
Large volume
Standard theory of solids:
DFT:
All Cm, Am, Pu are magnetic in LSDA/GGA
LDA: Pu(m~5mB), Am (m~6mB) Cm (m~4mB)
Exp: Pu (m=0),
Am (m=0)
Cm (m~7.9mB)
Non magnetic LDA/GGA predicts volume up to 30% off.
In atomic limit, Am non-magnetic, but Pu magnetic with spin ~5mB
Many proposals to explain why Pu is non magnetic:
 Mixed level model (O. Eriksson, A.V. Balatsky, and J.M. Wills) (5f)4 conf. +1itt.
 LDA+U, LDA+U+FLEX (Shick, Anisimov, Purovskii) (5f)6 conf.
Cannot account for anomalous transport and thermodynamics


Can LDA+DMFT account for anomalous properties of actinides?
Can it predict which material is magnetic and which is not?
Starting from magnetic solution,
Curium develops antiferromagnetic long range order below Tc
above Tc has large moment (~7.9mB close to LS coupling)
Plutonium dynamically restores symmetry -> becomes paramagnetic
DOS (states/eV)
DOS (states/eV)
4
-Plutonium
3
Total DOS
f DOS
2
1
0
4 -6
-4
Curium
-2
0
Total DOS
f, J=5/2,jz<0
f, J=7/2,jz<0
3
2
2
f, J=5/2,jz>0
f, J=7/2,jz>0
4
6
4
6
1
0
-6
-4
-2
0
ENERGY (eV)
2
J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).
Multiplet structure crucial for correct Tk in Pu (~800K)
and reasonable Tc in Cm (~100K)
Without F2,F4,F6: Curium comes out paramagnetic heavy fermion
Plutonium weakly correlated metal
DOS (states/eV)
DOS (states/eV)
4
-Plutonium
3
Total DOS
f DOS
2
1
0
4 -6
-4
Curium
-2
0
Total DOS
f, J=5/2,jz<0
f, J=7/2,jz<0
3
2
2
f, J=5/2,jz>0
f, J=7/2,jz>0
4
6
4
6
1
0
-6
-4
-2
0
ENERGY (eV)
2
Magnetization of Cm:
Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell
(Probability for atomic configurations)
f electron
fluctuates
between these
atomic states on
the time scale
t~h/Tk
(femtoseconds)
Pu partly f5 partly f6
J=6,g =1
J=5/2, g =0
J=7/2,g =0
J=9/2,g =0
J=0,g =0
J=1,g =0
J=2,g =0
J=3,g =0
J=4,g =0
J=5,g =0
J=4,g =0
0.3
J=5,g =0
Nf =6
Nf =5
Nf =4
J
-Plutonium
J=3,g =1
J=2,g =1
J=1,g =0
J=2,g =0
Probability
0.6
•5 electrons 80%
Nf =6
•6 electrons 20%
J=6,g =0
J=5,g =0
J=4,g =0
J=3,g =0
J=2,g =0
0.3
J=7/2,g =0
Curium
0.6
Probabilities:
Nf =8
Nf =7
J=6,g =0
J=5,g =0
J=4,g =0
J=3,g =0
J=2,g =0
J=1,g =0
J=0,g =0
Probability
0.0
0.9
•4 electrons <1%
0.0
-6
-4
-2
0
ENERGY (eV)
2
4
6
One dominant atomic state – ground state of the atom
J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).
Fingerprint of atomic multiplets
- splitting of Kondo peak
Gouder , Havela PRB
2002, 2003
Photoemission and valence in Pu
|ground state > = |a f5(spd)3>+ |b f6 (spd)2>
Af(w)
approximate decomposition
f5<->f6
f5->f4
f6->f7
Conclusions





DMFT can describe crossover from local moment regime to heavy
fermion state in heavy fermions. The crossover is very slow.
Width of heavy quasiparticle bands is predicted to be only ~3meV
. We predict a set of three heavy bands with their dispersion.
Mid-IR peak of the optical conductivity in 115’s is split due to pr
esence of two type’s of hybridization
Ce moment is more coupled to out-of-plane In then
in-plane In which explains the sensitivity of 115’s to substitution
of transition metal ion
DMFT predicts Pu to be nonmagnetic (heavy fermion like) and Cm
to be magnetic
Thank you!
Fermi surfaces
Increasing temperature from 10K to 300K:
 Gradual decrease of electron FS
Most of FS parts show similar trend
Big change might be expected in the G plane –
small hole like FS pockets (g,h) merge into
electron FS e1 (present in LDA-f-core but not in LDA)
Fermi surface a and c do not appear in DMFT results
ARPES of CeIrIn5
Fujimori et al. (2006)
Ce 4f partial spectral functions
LDA+DMFT (10K)
LDA+DMFT (400K)
Blue lines : LDA bands
Hole fermi surface at z=p
LDA
A
R
A
R
Z
R
A
R
A
LDA+DMFT (10 K)
No Fermi surfaces
LDA+DMFT (400 K)
dHva freq. and effective mass
Analytic impurity solvers (summing certain types of diagrams),
expansion in terms of hybridization
K.H. Phys. Rev. B 64, 155111 (2001)
Fully dressed atomic propagators
hybridization
SUNCA
D
•Allows correct treatment of multiplets
•Very precise at high and intermediate
frequencies and high to intermediate
temperatures
Complementary to CTQMC
(imaginary axis -> low energy)