1. Electronic structure of Strongly Correlated Materials: A Dynamical Mean Field Perspective.
Kristjan Haule,
Physics Department and
Center for Materials Theory
Rutgers University
Collaborators: Ji-Hoon Shim, S.Savrasov, G.Kotliar
ES 07 - Raleigh

2. Standard theory of solids
Band Theory: electrons as waves: Rigid non-dipersive 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 ……

3. 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.

4. Crossover: bad insulator to bad metal
Universality of the Mott transition
Crossover: bad insulator to bad metal
Critical point
First order MIT
Ni2-xSex
V2O3
k organics
1B HB model (DMFT):

5. Localization
Delocalization

6. Basic questions
How to describe the physics of strong correlations close to the Mott boundary?
How to computed spectroscopic quantities (single particle spectra, optical conductivity phonon dispersion…) from first principles?
New concepts, new techniques….. DMFT maybe simplest approach to meet this challenge

7. DMFT + electronic structure method
Basic idea of DMFT: reduce the quantum many body problem to a problem
of an atom in a conduction band, which obeys DMFT self-consistency condition
(A. Georges et al., RMP 68, 13 (1996)).
DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional
Basic idea of DMFT+electronic structure method (LDA or GW):
For less correlated bands (s,p): use LDA or GW
For correlated bands (f or d): with DMFT add all local diagrams
(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).
Effective (DFT-like) single particle
spectrum consists of delta like peaks
DMFT stectral function contains renormalized
quasiparticles and Hubbard bands

8. Generalized Q. impurity problem!
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!

9. Exact “QMC” impurity solver, expansion in terms of hybridization
K.H. Phys. Rev. B 75, (2007)
P. Werner, Phys. Rev. Lett. 97, (2006)
General impurity problem
Diagrammatic expansion in terms of hybridization D
+Metropolis sampling over the diagrams k
Exact method: samples all diagrams!
Allows correct treatment of multiplets

10. Volume of actinides Trivalent metals with nonbonding f shell
Partly localized,
partly delocalized
f’s participate in bonding

11. Anomalous Resistivity
Maximum metallic
resistivity:
s=e2 kF/h
Fournier & Troc (1985)

12. Dramatic increase of specific heat
Heavy-fermion behavior
in an element

13. Pauli-like from melting to lowest T
NO Magnetic moments!
Pauli-like from melting to lowest T
No curie Weiss up to 600K

14. 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
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
No magnetic moments,
large mass
Large specific heat,
Many phases, small or large volume

15. 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~7.6mB)
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?

16. Very strong multiplet splitting
Atomic multiplet splitting crucial -> splits Kondo peak N
Atom F2 F4 F6 x 92 U
8.513
5.502
4.017
0.226 93 Np
9.008
5.838
4.268
0.262 94 Pu
8.859
5.714
4.169
0.276 95 Am
9.313
6.021
4.398
0.315 96 Cm
10.27
6.692
4.906
0.380
Increasing F’s an SOC
Used as input to DMFT calculation - code of R.D. Cowan

17. 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
J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).

18. 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
Magnetization of Cm:

19. Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell
(Probability for atomic configurations)
Pu partly f5 partly f6
f electron fluctuates between these
atomic states on the time scale t~h/Tk
(femtoseconds)
Probabilities:
5 electrons 80%
6 electrons 20%
4 electrons <1%
One dominant atomic state – ground state of the atom
J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).

20. Fingerprint of atomic multiplets - splitting of Kondo peak
Gouder , Havela PRB
2002, 2003

21. Photoemission and valence in Pu
|ground state > = |a f5(spd)3>+ |b f6 (spd)2>
approximate decomposition
Af(w)
f5<->f6
f6->f7
f5->f4

22. Probe for Valence and Multiplet structure: EELS&XAS
Electron energy loss spectroscopy (EELS) or
X-ray absorption spectroscopy (XAS)
core
valence
4d3/2
4d5/2
5f5/2
5f7/2
Excitations from 4d core to 5f valence
A plot of the X-ray absorption
as a function of energy
Energy loss [eV]
Core splitting~50eV
4d5/2->5f7/2 &
4d5/2->5f5/2
4d3/2->5f5/2 hv
Core splitting~50eV
Measures unoccupied valence 5f states
Probes high energy Hubbard bands!
Branching ration B=A5/2/(A5/2+A3/2)
B=B0 - 4/15<l.s>/(14-nf)

23. LDA+DMFT 2/3<l.s>=-5/2(B-B0) (14-nf)
One measured quantity B, two unknowns
Close to atom (IC regime)
Itinerancy tends to decrease <l.s>
[a] G. Van der Laan et al., PRL 93, (2004).
[b] G. Kalkowski et al., PRB 35, 2667 (1987)
[c] K.T. Moore et al., PRB 73, (2006).
[d] K.T. Moore et al., PRL in press

24. Specific heat Could Pu be close to f6 like Am?
(Shick, Anisimov, Purovskii) (5f)6 conf
Purovskii et.al. cond-mat/ :
f6 configuration gives smaller g
in d Pu than a Pu

25. A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse,
Americium
f6 -> L=3, S=3, J=0
Mott Transition?
"soft" phase
f localized
"hard" phase
f bonding
A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse,
Phys. Rev. B 63, (2001)
J.-C. Griveau, J. Rebizant, G. H. Lander, and G.Kotliar Phys. Rev. Lett. 94, (2005)

26. Large multiple effects:
Am within LDA+DMFT
Large multiple effects:
F(0)=4.5 eV
F(2)=8.0 eV
F(4)=5.4 eV
F(6)=4.0 eV
S. Y. Savrasov, K.H., and G. Kotliar Phys. Rev. Lett. 96, (2006)

27. Comparisson with experiment
Am within LDA+DMFT
from J=0 to J=7/2
Comparisson with experiment
V=V0 Am I
V=0.76V0 Am III
V=0.63V0 Am IV
nf=6.2
nf=6
Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. Müller Phys. Rev. Lett. 52, (1984)
“Soft” phase not in local moment regime
since J=0 (no entropy)
Theory: S. Y. Savrasov, K.H., and G. Kotliar Phys. Rev. Lett. 96, (2006)
"Hard" phase similar to d Pu,
Kondo physics due to hybridization, however,
nf still far from Kondo regime

28. What is captured by single site DMFT?
Captures volume collapse transition (first order Mott-like transition)
Predicts well photoemission spectra, optics spectra,
total energy at the Mott boundary
Antiferromagnetic ordering of magnetic moments,
magnetism at finite temperature
Branching ratios in XAS experiments, Dynamic valence fluctuations,
Specific heat
Gap in charge transfer insulators like PuO2

29. Beyond single site DMFT
What is missing in DMFT?
Momentum dependence of the self-energy m*/m=1/Z
Various orders: d-waveSC,…
Variation of Z, m*,t on the Fermi surface
Non trivial insulator (frustrated magnets)
Non-local interactions (spin-spin, long range Columb,correlated hopping..)
Present in cluster DMFT:
Quantum time fluctuations
Spatially short range quantum fluctuations
Present in DMFT:
Quantum time fluctuations

30. Plaquette DMFT for the Hubbard model as relevant for cuprates
Small next-nearest neighbor component
(except in the underdoped regime)
Large onsite
component
anomalous SE-SC

31. Complicated Fermi surface evolution with temperature
underoped phase
“fermi arcs”
Superconducting phase-
banana like Fermi surface
“arcs” decrease
with T

32. Conclusion
Pu and Am (under pressure) are unique strongly correlated elements. Unique mixed valence.
They require, new concepts, new computational methods, new algorithms, DMFT!
Cluster extensions of DMFT can describe many features of cuprates including superconductivity and gapping of fermi surface (pseudogap)

33. Many strongly correlated compounds await
the explanation:
CeCoIn5, CeRhIn5, CeIrIn5

34. Photoemission of CeIrIn5

35. Photoemission of CeIrIn5
LDA+DMFT DOS
Comparison
to experiment

36. Optics of CeIrIn5 LDA+DMFT K.S. Burch et.al., Experiment:
cond-mat/
Experiment:

38. Optimal doping: Coherence scale seems to vanish
underdoped
scattering
at Tc
optimally
overdoped Tc

39. Multiplets correctly treated
New continuous time QMC, expansion in terms of hybridization
General impurity problem
Diagrammatic expansion in terms of hybridization D
+Metropolis sampling over the diagrams k
Contains all: “Non-crossing” and all crossing diagrams!
Multiplets correctly treated

40. Conclusions
LDA+DMFT can describe interplay of lattice and electronic structure near Mott transition. Gives physical connection between spectra, lattice structure, optics,....
Allows to study the Mott transition in open and closed shell cases.
In actinides and their compounds, single site LDA+DMFT gives the zero-th order picture
2D models of high-Tc require cluster of sites. Some aspects of optimally doped regime can be described with cluster DMFT on plaquette:
Large scattering rate in normal state close to optimal doping

41. How does the electron go from being localized to itinerant.
Basic questions
How does the electron go from being localized to itinerant.
How do the physical properties evolve.
How to bridge between the microscopic information (atomic positions) and experimental measurements.
New concepts, new techniques….. DMFT simplest approach to meet this challenge

42. Coherence incoherence crossover in the 1B HB model (DMFT)
Phase diagram of the HM with partial frustration at half-filling
M. Rozenberg et.al., Phys. Rev. Lett. 75, 105 (1995).

43. Singlet-type Mott state (no entropy) goes mixed valence under pressure
-> Tc enhanced (Capone et.al, Science 296, 2364 (2002))

44. Overview
DMFT in actinides and their compounds (Spectral density functional approach).
Examples:
Plutonium, Americium, Curium.
Compounds: PuAm
Observables:
Valence, Photoemission, and Optics, X-ray absorption

45. Why is Plutonium so special?
Heavy-fermion behavior
in an element
Typical heavy fermions
(large mass->small Tk
Curie Weis at T>Tk)
No curie Weiss up to 600K

46. Why is Plutonium so special?
Heavy-fermion behavior
in an element

47. Overview of actinides Many phases
25% increase in volume between a and d phase
Two phases of Ce, a and g
with 15% volume difference

48. f-sumrule for core-valence conductivity
Similar to optical conductivity:
Current:
Expressed in core valence orbitals:
The f-sumrule:
can be expressed as
Branching ration B=A5/2/(A5/2+A3/2)
Energy loss [eV]
Core splitting~50eV
4d5/2->5f7/2
4d3/2->5f5/2
B=B0 - 4/15<l.s>/(14-nf)
B0~3/5
Branching ratio depends on:
average SO coupling in the f-shell <l.s>
average number of holes in the f-shell nf
A5/2 area under the 5/2 peak
B.T. Tole and G. van de Laan, PRA 38, 1943 (1988)

49. Optical conductivity
2p->5f
5f->5f
Pu: similar to heavy fermions (Kondo type conductivity)
Scale is large MIR peak at 0.5eV
PuO2: typical semiconductor with 2eV gap, charge transfer

50. Generalized Q. impurity problem!
Spectral density functional theory
(G. Kotliar et.al., RMP 2006).
observable of interest is the "local“ Green's functions (spectral function)
Currently feasible approximations: LDA+DMFT:
Variation gives st. eq.:
Generalized Q. impurity problem!

51. Pu-Am mixture, 50%Pu,50%Am
Lattice expands -> Kondo collapse is expected
Our calculations suggest
charge transfer
Pu d phase stabilized by shift to
mixed valence nf~5.2->nf~5.4
Hybridization decreases, but nf increases,
Tk does not change significantly!
f6: Shorikov, et al., PRB 72, (2005);
Shick et al, Europhys. Lett. 69, 588 (2005).
Pourovskii et al., Europhys. Lett. 74, 479 (2006).