Title: Strongly Correlated Electron Systems a Dynamical Mean Field Perspective
1 Strongly Correlated Electron Systems a
Dynamical Mean Field Perspective
- G. Kotliar
- Physics Department and Center for Materials
Theory - Rutgers
5th International Conference on Inelastic X-Ray
Scattering. Argonne National Labs Chicago
September 20 2004
2Outline
- Introduction to the concepts of the dynamical
mean field method. - Application The temperature driven Mott
transition. Theoretical predictions, and
experiments. IXS ? - Application elemental Pu. DMFT predictions, and
a key IXS experiment.
3Electronic states in weakly and strongly
correlated materials
- Simple metals, semiconductors. Fermi Liquid
Description Quasiparticles and quasiholes, (and
their bound states ). Computational tool
Density functional theory perturbation theory
in W, GW method. - Correlated electrons. Atomic states. Hubbard
bands. Narrow bands. Many anomalies. - Need tool that treats Hubbard bands, and
quasiparticle bands, real and momentum space on
the same footing. DMFT!
4Strongly Correlated Electron Systems Display
remarkable phenomena, that cannot be understood
within the standard model of solids.
Resistivities that rise without sign of
saturation beyond the Mott limit, (e.g. H.
Takagis work on Vanadates), temperature
dependence of the integrated optical weight up
to high frequency (e.g. Vandermarels work on
Silicides).
THE WHY
Correlated electrons do big things, large
volume collapses, colossal magnetoresitance, high
temperature superconductivity . Properties are
very sensitive to structure chemistry and
stoichiometry, and control parameters large non
linear susceptibilites,etc.
5C. Urano et. al. PRL 85, 1052 (2000)
6THE HOW
Need non perturbative tool.
How to think about their electronic states ? How
to compute their properties ? Mapping onto
connecting their properties, a simpler reference
system. A self consistent impurity model living
on SITES, LINKS and PLAQUETTES......
- DYNAMICAL MEAN FIELD THEORY.
- "Optimal Gaussian Medium " " Local Quantum
Degrees of Freedom " "their interaction " - is a good reference frame for understanding,
and predicting physical properties - of correlated materials. Focus on local
quantities, construct functionals of those
quantities, similarities with DFT.
7Two paths for ab-initio calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
DMFT ideas can be used in both cases.
8 Dynamical Mean Field Theory (DMFT) Cavity
Construction A. Georges and G. Kotliar PRB 45,
6479 (1992). C.DMFT G. Kotliar et. al. Phys. Rev.
Lett 87,186401 (2001).
9One dimensional Hubbard model 2 site (LINK)
CDMFT compare with Bethe Anzats, V. Kancharla
C. Bolech and GK PRB 67, 075110
(2003)M.CaponeM.Civelli V Kancharla
C.Castellani and GK P. R B 69,195105 (2004)
A rapidly convergent algorithm ?
U/t4.
10Functional formulation. Chitra and Kotliar
(2001), Savrasov and Kotliarcond- matt0308053
(2003).
IrgtR, rgt
Double loop in Gloc and Wloc
11Impurity model representability of spectral
density functional.
12LDADMFT V. Anisimov, A. Poteryaev, M. Korotin,
A. Anokhin and G. Kotliar, J. Phys. Cond. Mat.
35, 7359 (1997). A Lichtenstein and M. Katsnelson
PRB 57, 6884 (1988).
- The light, SP (or SPD) electrons are extended,
well described by LDA .The heavy, D (or F)
electrons are localized treat by DMFT. - LDA Kohn Sham Hamiltonian already contains an
average interaction of the heavy electrons,
subtract this out by shifting the heavy level
(double counting term) - Kinetic energy is provided by the Kohn Sham
Hamiltonian (sometimes after downfolding ). The U
matrix can be estimated from first principles of
viewed as parameters. Solve resulting model
using DMFT.
13What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
14T
15How do we know there is some truth in this
picture ? Qualitative Predictions Verified
- Two different features in spectra. Quasiparticles
bands and Hubbard bands. - Transfer of spectral weight which is non local in
frequency. Optics and Photoemission. - Two crossovers, associated with gap closure and
loss of coherence. Transport. - Mott transition endpoint, is Ising like, couples
to all electronic properties. - Recently numerical approaches in two dimensions
found the first order line(M. Imada), C-DMFT 4
site studies (Parcollet et. al.).
16Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
17Anomalous transfer of optical spectral weight V2O3
- M Rozenberg G. Kotliar and H. Kajuter Phys. Rev.
B 54, 8452 (1996). - M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
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19Optical transfer of spectral weight , kappa
organics. Eldridge, J., Kornelsen, K.,Wang,
H.,Williams, J., Crouch, A., and Watkins, D.,
Sol. State. Comm., 79, 583 (1991).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
20Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
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22Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi 2000
23Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
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25Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
26Why does it work Energy Landscape of a
Correlated Material and a top to bottom approach
to correlated materials.
Single site DMFT. High temperature universality
vs low temperature sensitivity to detail for
materials near a temperature-pressure driven
Mott transition
Energy
T
Configurational Coordinate in the space of
Hamiltonians
27What did we gain?
- Conceptual understanding of how the electronic
structure evolves when the electron goes from
localized to itinerant. - Uc1 Uc2, transfer of spectral weight, .
- A general methodology which was extended to
clusters (non trivial!) and integrated into an
electronic structure method, which allows us to
incorporate structure and chemistry. Both are
needed away from the high temperature universal
region.
28- Mott transition across the 5fs, a very
interesting playground for studying correlated
electron phenomena. - DMFT ideas have been extended into a framework
capable of making first principles first
principles studies of correlated materials. Pu
Phonons. Combining theory and experiments to
separate the contributions of different energy
scales, and length scales to the bonding - In single site DMFT , superconductivity is an
unavoidable consequence when we try to go move
from a metallic state to a Mott insulator
where the atoms have a closed shell (no entropy).
Realization in Am under pressure ?
29DMFT Phonons in fcc d-Pu connect bonding to
energy and length scales.
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
30Where do we go now ?
- One can study a large number of experimentally
relevant problems within the single site
framework. - Continue the methodological development, we need
tools! - Solve the CDMFT Mott transition problem on the
plaquette problem, hard, but it is a significant
improvement, the early mean field theories while
keeping its physical appeal. - Study material trends, make contact with
phenomenological approaches, doped semiconductors
(Bhatt and Sachdev), heavy fermions ,
115s(Nakatsuji, Pines and Fisk )
31Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
32Pu in the periodic table
actinides
33 Mott transition in the actinide series (Smith
Kmetko phase diagram)
34Electronic Physics of Pu
35Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
36Elastic Deformations
Uniform compressionDp-B DV/V
Volume conserving deformations
F/Ac44 Dx/L
F/Ac Dx/L
In most cubic materials the shear does not depend
strongly on crystal orientation,fcc Al,
c44/c1.2, in Pu C44/C 6 largest shear
anisotropy of any element.
37Anomalous Resistivity
Maximum metallic resistivity
38Specific heat and susceptibility.
39Delta phase of Plutonium Problems with LDA
- Many studies and implementations.(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).all give
an equilibrium volume of the d phase Is 35
lower than experiment this is the largest
discrepancy ever known in DFT based calculations. - LSDA predicts magnetic long range (Solovyev
et.al.) Experimentally d Pu is not magnetic. - If one treats the f electrons as part of the core
LDA overestimates the volume by 30
40 DFT Studies of a Pu
- DFT in GGA predicts correctly the volume of the a
phase of Pu, when full potential LMTO (Soderlind
Eriksson and Wills) is used. This is usually
taken as an indication that a Pu is a weakly
correlated system. Experimentally, there are
clear signs of electron correlation in a Pu . - .
41Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
42Total Energy as a function of volume for Pu W
(ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001) Non
magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
43Lda vs Exp Spectra
44Alpha and delta Pu
45Phonon Spectra
- Electrons are the glue that hold the atoms
together. Vibration spectra (phonons) probe the
electronic structure. - Phonon spectra reveals instablities, via soft
modes. - Phonon spectrum of Pu had not been measured until
recently.
46Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
47Inelastic X Ray. Phonon energy 10 mev, photon
energy 10 Kev.
E Ei - Ef Q ki - kf
48Expt. Wong et. al.
49Expts Wong et. al.
50DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
51Conclusion
- DMFT. Electronic Structure Method in Development.
a) quantitative results b) qualitative
understanding by linking real materials to
impurity models. Concepts to think about
correlate materials. - System specific. Many materials to be studied,
realistic matrix elements for each spectroscopy.
Optics. IXS. - Interplay of theory and experiment. DMFT can
enhance joint theoretical- experimental advances
in the field of correlated electron materials.
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53Epsilon Plutonium.
54Phonon entropy drives the epsilon delta phase
transition
- Epsilon is slightly more delocalized than delta,
has SMALLER volume and lies at HIGHER energy than
delta at T0. But it has a much larger phonon
entropy than delta. - At the phase transition the volume shrinks but
the phonon entropy increases. - Estimates of the phase transition following
Drumont and G. Ackland et. al. PRB.65, 184104
(2002) (and neglecting electronic entropy).
TC 600 K.
55Transverse Phonon along (0,1,1) in epsilon Pu in
self consistent Born approximation.