Title: Gabriel Kotliar
1 - Gabriel Kotliar
- Physics Department and
- Center for Materials Theory
- Rutgers University
2The Strong Correlation Problem
- Two limiting cases of the electronic structure of
solids are understoodthe high density limit and
the limit of well separated atoms. - High densities, the is electron be a wave, use
band theory, k-space - One particle excitations quasi-particle,quasi-hol
e bands, collective modes. - Density Functional Theory with approximations
suggested by the Kohn Sham formulation, (LDA GGA)
is a successful computational tool for the total
energy, and a good starting point for
perturbative calculation of spectra, GW.
3Mott Correlations localize the electron
Low densities, electron behaves as a particle,use
atomic physics, real space One particle
excitations Hubbard Atoms sharp excitation
lines corresponding to adding or removing
electrons. In solids they broaden by their
incoherent motion, Hubbard bands (eg. bandsNiO,
CoO MnO.) Magnetic and Orbital Ordering at low
T Quantitative calculations of Hubbard bands and
exchange constants, LDA U, Hartree Fock. Atomic
Physics.
4Localization vs Delocalization Strong Correlation
Problem
- 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 LDAU or Hartree Fock works well
- Dynamical Mean Field Theory Simplest approach to
the electronic structure, which interpolates
correctly between atoms and bands
5Outline
- Motivation plutonium puzzles.
- Review of Dynamical Mean Field Theory an
Extension to realistic systems . DMFT and DFT. - A case study of system specific properties f
.electrons DMFT Results for d Pu. - A case study of system specific properties d
electrons in Fe and Ni.
6Collborators and References
- Reviews of DMFT A. Georges G. Kotliar W krauth
and M . Rozenberg Rev Mod Pnys 68, 13 (1996).
DMFT and LDA R. Chitra and G. Kotliar Phys. Rev.
B 62,, 12715 (2000). - S. Savrasov and G. Kotliar cond-mat cond-mat
0106308. - DMFT study of Plutonium. S. Savrasov, G. Kotiar
and E. Abrahams, Nature 410, 793 (2001). S.
Savrasov and G. Kotliar - DMFT study of Iron and Nickel. S. Lichtenstein M
Katsenelson and G. Kotliar Phys. Rev. Lett 87,
(2001).
7Outline
- Motivation plutonium puzzles.
- Review of Dynamical Mean Field Theory an
Extension to realistic systems . DMFT and DFT. - A case study of system specific properties f
.electrons DMFT Results for d Pu. - A case study of system specific properties d
electrons in Fe and Ni.
8 Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
9Pu Complex Phase Diagram (J. Smith LANL)
10Small amounts of Ga stabilize the d phase
11Problems with LDA
- DFT in the LDA or GGA is a well established tool
for the calculation of ground state properties. - Many studies (Freeman, Koelling 1972)APW methods
- ASA and FP-LMTO Soderlind et. Al 1990, Kollar
et.al 1997, Boettger et.al 1998, Wills et.al.
1999) give - an equilibrium volume of the d phase Is 35
lower than experiment - This is the largest discrepancy ever known in DFT
based calculations.
12Problems with LDA
- LSDA predicts magnetic long range order which is
not observed experimentally (Solovyev et.al.) - If one treats the f electrons as part of the core
LDA overestimates the volume by 30 - LDA predicts correctly the volume of the a phase
of Pu, using full potential LMTO (Soderlind and
Wills). This is usually taken as an indication
that a Pu is a weakly correlated system.
13Other Methods
- LDA U (Savrasov and Kotliar Phys. Rev. Lett.
84, 3670, 2000, Bouchet et. al 2000) predicts
correct volume of Pu with the constrained LDA
estimate of U4 ev. However, it predicts spurious
magnetic long range order and a spectra which is
very different from experiments. - Requires U0 to treat the alpha phase, which has
many physical properties in common with the delta
phase. - Similar problems with the constrained (4 of the
5f electrons are treated as core ) LDA approach
of Erikson and Wills.
14Conventional viewpoint
- Alpha Pu is a simple metal, it can be described
with LDA correction. In contrast delta Pu is
strongly correlated. - Constrained LDA approach (Erickson, Wills,
Balatzki, Becker). In Alpha Pu, all the 5f
electrons are treated as band like, while in
Delta Pu, 4 5f electrons are band-like while one
5f electron is deloclized. - Same situation in LDA U (Savrasov andGK
Bouchet et. al. ) .Delta Pu has U4,Alpha Pu
has U 0.
15Problems with the conventional viewpoint of Pu
- The specific heat of delta Pu, is only twice as
big as that of alpha Pu. - The susceptibility of alpha Pu is in fact larger
than that of delta Pu. - The resistivity of alpha Pu is comparable to that
of delta Pu. - Only the structural and elastic properties are
completely different.
16MAGNETIC SUSCEPTIBILITY
17Pu Specific Heat
18Anomalous ResistivityJ. Smith LANL
19Outline
- Motivation plutonium puzzles.
- Review of Dynamical Mean Field Theory an
Extension to realistic systems . DMFT and DFT. - A case study of system specific properties f
.electrons DMFT Results for d Pu. - A case study of system specific properties d
electrons in Fe and Ni.
20DMFT Impurity cavity construction A. Georges,
G. Kotliar, PRB, (1992)
Weiss field
21Single site DMFT, functional formulation
- Express in terms of Weiss field (semicircularDOS)
- The Mott transition as bifurcation point in
functionals oGG or FD, (G. Kotliar EPJB 99)
Local self energy (Muller Hartman 89)
22Solving the DMFT equations
- Wide variety of computational tools (QMC,
NRG,ED.) - Analytical Methods
23DMFT
- Construction is easily extended to states with
broken translational spin and orbital order. - Large number of techniques for solving DMFT
equations for a review see - A. Georges, G. Kotliar, W. Krauth and M.
Rozenberg Rev. Mod. Phys. 68,13 (1996)
24Schematic DMFT phase diagram one band Hubbard
(half filling, semicircular DOS, role of partial
frustration) Rozenberg et.al PRL (1995)
25Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to an Ising Mott endpoint
(Kotliar et.al.PRL 84, 5180 (2000))
26Localization Delocalization
- The Mott transition/crossover is driven by
transfer of spectral weight from low to high
energy as we approach the localized phase - Control parameters doping, temperature,pressure
- Intermediate U region is NOT perturbatively
accessible. DMFT a new starting point to access
this regime.
27Qualitative phase diagram in the U, T , m plane
(two band Kotliar and Rozenberg (2001))
- Coexistence regions between localized and
delocalized spectral functions.
28QMC calculationof n vs m (Murthy Rozenberg and
Kotliar 2001, 2 band, U3.0)
k diverges at generic Mott endpoints
29Combining LDA and DMFT
- The light, SP (or SPD) electrons are extended,
well described by LDA - The heavy, D (or F) electrons are localized,treat
by DMFT. - LDA already contains an average interaction of
the heavy electrons, substract this out by
shifting the heavy level (double counting term) - The U matrix can be estimated from first
principles or viewed as parameters
30Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and GK).
- DFT, consider the exact free energy as a
functional of an external potential. Express the
free energy as a functional of the density by
Legendre transformation. GDFTr(r) - Introduce local orbitals, caR(r-R)orbitals, and
local GF - G(R,R)(i w)
- The exact free energy can be expressed as a
functional of the local Greens function and of
the density by introducing sources for r(r) and G
and performing a Legendre transformation,
Gr(r),G(R,R)(iw)
31Spectral Density Functional
- The exact functional can be built in perturbation
theory in the interaction (well defined
diagrammatic rules )The functional can also be
constructed expanding around the the atomic
limit. No explicit expression exists. - DFT is useful because good approximations to the
exact density functional GDFTr(r) exist, e.g.
LDA, GGA - A useful approximation to the exact functional
can be constructed, the DMFT LDA functional.
32LDADMFT functnl
F Sum of local 2PI graphs with local U matrix and
local G
33LDADMFT Self-Consistency loop
E
U
DMFT
34Comments on LDADMFT
- Static limit of the LDADMFT functional , with F
FHF reduces to LDAU - Removes inconsistencies and shortcomings of this
approach. DMFT retain correlations effects in
the absence of orbital ordering. - Only in the orbitally ordered Hartree Fock limit,
the Greens function of the heavy electrons is
fully coherent - Gives the local spectra and the total energy
simultaneously, treating QP and H bands on the
same footing.
35Outline
- Motivation plutonium puzzles.
- Review of Dynamical Mean Field Theory an
Extension to realistic systems . DMFT and DFT. - A case study of system specific properties f
.electrons DMFT Results for d Pu. - A case study of system specific properties d
electrons in Fe and Ni.
36Pu DMFT total energy vs Volume Savrasov Kotliar
Abrahams to appear in Nature
37Lda vs Exp Spectra
38Pu Spectra DMFT(Savrasov) EXP (Arko et.al)
39PU ALPHA AND DELTA
40Dynamical Mean Field View of Pu(Savrasov Kotliar
and Abrahams, Nature 2001)
- Delta and Alpha Pu are both strongly correlated,
the DMFT mean field free energy has a double
well structure, for the same value of U. One
where the f electron is a bit more localized
(delta) than in the other (alpha). - Is the natural consequence of the model
Hamiltonian phase diagram once the structure is
about to vary. - This result resolves one of the basic paradoxes
in the physics of Pu.
41Minimum of the melting point
- Divergence of the compressibility at the Mott
transition endpoint. - Rapid variation of the density of the solid as a
function of pressure, in the localization
delocalization crossover region. - Slow variation of the volume as a function of
pressure in the liquid phase
42Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
43Cerium melting T vs p
44Pu Anomalous thermal expansion (J. Smith LANL)
45Double well structure and d Pu
- Qualitative explanation
of negative thermal expansion - Sensitivity to impurities which easily raise the
energy of the a -like minimum.
46Double well structure and d Pu
- negative thermal expansion
- Sensitivity to impurities which easily raise the
energy of the a -like minimum.
47Future directions
- Including short range correlations. Less local
physics, C-DMFT. - Life without U, including the effects of long
range Coulomb interactions, E-DMFT and GW. - Applications are just beginning, many surprises
ahead
48Outline
- Motivation plutonium puzzles.
- Review of Dynamical Mean Field Theory an
Extension to realistic systems . DMFT and DFT. - A case study of system specific properties f
.electrons DMFT Results for d Pu. - A case study with d electrons in Fe and Ni.
49Case study Fe and Ni
- Archetypical itinerant ferromagnets
- LSDA predicts correct low T moment
- Band picture holds at low T
- Main puzzle at high temperatures c has a Curie
Weiss law with a moment much larger than the
ordered moment. - Magnetic anisotropy
-
50Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
GK)
51Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
52Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
53Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
54Ni and Fe theory vs exp
- m( T.9 Tc)/ mB ordered moment
- Fe 1.5 ( theory) 1.55 (expt)
- Ni .3 (theory) .35 (expt)
- meff / mB high T moment
- Fe 3.1 (theory) 3.12 (expt)
- Ni 1.5 (theory) 1.62 (expt)
- Curie Temperature Tc
- Fe 1900 ( theory) 1043(expt)
- Ni 700 (theory) 631 (expt)
55Fe and Ni
- Satellite in minority band at 6 ev, 30
reduction of bandwidth, exchange splitting
reduction .3 ev - Spin wave stiffness controls the effects of
spatial flucuations, it is about twice as large
in Ni and in Fe - Mean field calculations using measured exchange
constants(Kudrnovski Drachl PRB 2001) right Tc
for Ni but overestimates Fe , RPA corrections
reduce Tc of Ni by 10 and Tc of Fe by 50.
56However not everything in low T phase is OK as
far as LDA goes..
- Magnetic anisotropy puzzle. LDA predicts the
incorrect easy axis(100) for Nickel .(instead of
the correct one (111) - LDA Fermi surface has features which are not seen
in DeHaas Van Alphen ( Lonzarich) - Use LDA U to tackle these refined issues, (
compare parameters with DMFT results )