Strongly Correlated Electron Systems: a DMFT Perspective

Strongly Correlated Electron Systems: a DMFT

Perspective Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Colloquium UBC September (2004)

THE STATE UNIVERSITY OF NEW JERSEY

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Outline

Introduction to the strong correlation problem.

Essentials of DMFT The Mott transition problem: some insights

from studies of models. Towards an electronic structure method:

applications to materials: Ce, Pu Outlook


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Momentum Space (Sommerfeld)

Standard model of solids Periodic potential, waves form bands , k in Brillouin zone

2 ( )F Fe k k l

h

The electron in a solid: wave picture

Maximum metallic resistivity 200 ohm cm

2

2k

k

m

Landau: Interactions renormalize away


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Standard Model of Solids

~ const ~H constR~VC T

RIGID BAND PICTURE. Optical response, transitions between bands.

Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport


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The electron in a solid: particle picture.

NiO, MnO, …Array of atoms is insulating if a>>aB. Mott: correlations localize the electron

e_ e_ e_ e_

•Think in real space , solid collection of atoms•High T : local moments, Low T spin-orbital order

1

T •Superexchange


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Mott : Correlations localize the electronLow densities, electron behaves as a particle,use

atomic physics, work in 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….)

• Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

•H H H+ H H H motion of H+ forms the lower Hubbard band

•H H H H- H H motion of H_ forms the upper Hubbard band


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Localization vs Delocalization Strong Correlation Problem

• A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem.

•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA –GW or LDA+U or Hartree Fock work well.•Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. New reference point, to replace the Kohn Sham system.


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DFT+GW program has been less succesful in correlated situations.

Strong interactions localize the particles. Atoms with open shells are not easily connected to band theory.

The spectrum in this case, contain Hubbard bands which are NOT simply perturbatively connected to the Kohn Sham orbitals.

Need an alternative reference point for doing perturbation theory! Situation is worse “in between the atomic and the localized limit”

DMFT!


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Correlated Materials do “big” things

Mott transition.Huge resistivity changes V2O3.

Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .

Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000

(La1-xSrx)MnO3 Colossal Magneto-resistance.


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Strongly Correlated Materials.

Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999).

Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )

Huge volume collapses, Ce, Pu……


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Breakdown of standard model

LDA+GW program fails badly. Large metallic resistivities exceeding the

Mott limit. [Anderson, Emery and Kivelson]

Breakdown of the rigid band picture. Need new ways to think about the excitations.

Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki]


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Strongly correlated systems are usually treated with model Hamiltonians

In practice other methods (eg constrained LDA are used)


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Strongly correlated systems are usually treated with model Hamiltonians

They are hard to derive and hard to solve.

In practice other methods (eg. constrained LDA are used)


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Outline Introduction to the strong correlation problem

and to the Mott transition. DMFT ideas Applications to the Mott transition problem:

some insights from studies of models. Towards an electronic structure method:

applications to materials: Pu………. Outlook


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Mean-Field : Classical vs Quantum

Classical case Quantum case

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n


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Insert transparency from nijmeigen About infinite dimensions, and about Greens functions.


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DMFT: Effective Action point of view.

Identify observable, A. Construct an exact functional of <A>=a, [a] which is stationary at the physical value of a.

Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.

DMFT, build functionals of the LOCAL spectral function.[Density of states for adding or removing and electron] Exact functionals exist. We also have good

approximations! Extension to an ab initio method.


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LDA+DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and

G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B

57, 6884 (1988). S. Savrasov and G.Kotliar and Abrahams

funcional formulation for full self consistent Nature {\bf 410}, 793(2001).

Reviews: Held et.al. , Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and and Kotliar cond-mat/0211076:


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How good is the LOCAL approximation?


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C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


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N vs mu in one dimension.Compare 2+8 vs exact Bethe Anzats, [M. Capone and M.Civelli]


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Outline


Essentials of DMFT Applications to the Mott transition problem:

some insights from studies of models. Towards an electronic structure method:

applications to materials Outlook


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The Mott transition

Electronically driven MIT. Forces to face directly the localization

delocalization problem. Relevant to many systems, eg V2O3 Techniques applicable to a very broad

range or problems.


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Mott transition in V2O3 under pressure or chemical substitution on V-site


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Resistivity. Limelette et. al.


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How good is the local approximation ? Single site DMFT study of the Mott transition,

based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions.

New experiments and reexamination of old ones give credence to that the local picture is quite good.

DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally , NON RIGID BAND picture, highly resistive states, etc….


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Insight

Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling.

At high temperatures, the phase diagram is generic, insensitive to microscopic details.

At low temperatures, details matters.


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Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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Insight, in the strongly correlated region the one particle density of states has a three peak structurelow energy quasiparticle peak plus Hubbard bands.


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DMFT has bridged the gap between band theory and atomic physics. Delocalized picture, it

should resemble the density of states, (perhaps with some additional shifts and satellites).

Localized picture. Two peaks at the ionization

and affinity energy of the atom.


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One electron spectra near the Mott transition, three peak structure.


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)


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QP in V2O3 was recently found Mo et.al


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Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…


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Evolution 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)


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)


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Anomalous metallic resistivities

In the “ in between region “ anomalous

resistivities are the rule rather than the exception.


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Failure of the Standard Model: NiSe2-xSx

Miyasaka and Takagi (2000)


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Anomalous Resistivity and Mott transition (Rozenberg et. Al. ) Ni Se2-x Sx

Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )


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More recent work, organics, Limelette et. al.(PRL 2003)


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Anomalous Resistivities when wave picture does not apply. Doped Hubbard model

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Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight. Consequences for optics.


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Anomalous transfer of spectral weight in v2O3


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Anomalous 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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Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]


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Anomalous transfer of spectral weight heavy fermions Rozenberg Kajueter Kotliar (1996)


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Anomalous transfer of optical weight [A. Damascelli D. Van der Marel ]


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Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy


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DMFT and the strong correlation anomalies: crossover from momentum space to real space picture

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Three peak structure of DOS Transfer of spectral weight which is non

local in frequency.

Dramatic failure of DFT based approximations in predicting physical properties.


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Outline

Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some

insights from studies of models. Towards an electronic structure method:

applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d organics, magnetic semiconductors,SrRuO4,V2O3………….

Outlook


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Generalized phase diagram

T

U/WRelax Structure, bands,

orbitals


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Pu in the periodic table

actinides


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Electronic Physics of Pu


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DFT studies.

Underestimates the volume by 35 % Predicts Pu to be magnetic. Largest quantitative failure of DFT-LDA-

GA Fail to predict a stable delta phase.


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Phonon 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.


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Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


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Inelastic X ray scattering. Wong et. al. Science 301, 1078 (2003).


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Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)


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Alpha and delta Pu


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Outline


Essentials of DMFT The Mott transition problem: some insights

from studies of models. Towards an electronic structure method:

applications to materials: Pu Outlook


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What do we want from materials theory?

New concepts , qualitative ideas Understanding, explanation of existent

experiments, and predictions of new ones. Quantitative capabilities with predictivepower.

Notoriously difficult to achieve in strongly correlated materials. DMFT is delivering on both fronts.


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Outlook

Local approach to strongly correlated electrons.

Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials.


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Conclusion

The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.

This has lead to extensions to more realistic models, and a beginning of a first principles approach to the electronic structure of correlated materials.


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Outlook Systematic improvements, short range

correlations, cluster methods, improved mean fields.

Improved interfaces with electronic structure.

Exploration of complex strongly correlated materials. Correlation effects on surfaces,

large molecules, systems out of equilibrium, illumination, finite currents, aeging.


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Acknowledgements: Development of DMFT

Collaborators: V. Anisimov,G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet , G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang

Support: NSF DMR 0096462

Support: Instrumentation. NSF DMR-0116068

Work on Fe and Ni: ONR4-2650

Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02


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High Performance Computing ProjectDepartment of Physics and Astronomy

National Science Foundation - NSF0116068: Acquisition of a

Network Cluster of Advanced Workstations for First Principles Electronic Structure

Calculations of Complex Materials

High Performance Computing

http://beowulf.rutgers.edu


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TOP 500 (ICL-UT)


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TOP 500


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Shear anisotropy fcc Pu (GPa)

C’=(C11-C12)/2 = 4.78

C44= 33.59

C44/C’ ~ 8 Largest shear anisotropy in any element!

LDA Calculations (Bouchet et. al.) C’= -48


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Dai et. al.


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Epsilon Plutonium.


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Anomalous transfer of spectral weight heavy fermions


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Anomalous transfer of spectral weight


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Anomalous transfer of spectral weigth heavy fermions


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V2O3 resistivity


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Example: DMFT for lattice model (e.g. single band Hubbard).Muller Hartman 89, Chitra and Kotliar 99.

Observable: Local Greens function Gii ().

Exact functional [Gii () DMFT Approximation to the functional.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få


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Spectral Density Functional : effective action construction (Chitra and GK).

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

Approximate functional using DMFT insights.

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r

Strongly Correlated Electron Systems: a DMFT Perspective

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