Electronic Structure of Correlated Materials : a DMFT Perspective

Gabriel Kotliar

Physics Department andCenter for Materials Theory

Rutgers Universityand KITP

Institute for Theoretical PhysicsUCSB Santa Barbara

Brookhaven National Laboratory

September 12th 2002

Supported by the NSF DMR 0096462

THE STATE UNIVERSITY OF NEW JERSEY

RUTGERS

Outline

The Mott transition problem and electronic structure.

Dynamical Mean Field Theory Model Hamiltonian Studies of the Mott

transition. Universal aspects. System specific studies of materials.

LDA+DMFT. Some case studies. Outlook

RUTGERS

Weakly correlated electrons:band theory. Simple conceptual picture of the ground

state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

A methods for performing quantitative calculations. (Density functional theory, in various approximations).

RUTGERS

Momentum Space (Sommerfeld)

Standard model of solids (Bloch, Landau) Periodic potential, waves form bands , k in Brillouin zone . Interactions renormalize away.

2 ( )F Fe k k l

The electron in a solid: wave picture

Maximum metallic resistivity 200 ohm cm

RUTGERS

Standard Model of Solids Qualitative predictions: low temperature dependence of

thermodynamics and transport.

Optical response, transition between the bands. Qualitative predictions: filled bands give rise to

insulting behavior. Compounds with odd number of electrons are metals.

Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.

~H constR~ const S T ~VC T

RUTGERS

Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )

KS ext

ErV r r V r dr

dr= + +

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse

Tremendous progress in ab initio modelling of lattice dynamics& electron-phonon interactions has been achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

(Savrasov, PRB 1996)

RUTGERS

The electron in a solid: particle picture.Ba

Array of hydrogen atoms is insulating if a>>aB.

Mott: correlations localize the electron

e_ e_ e_ e_

Superexchange

Think in real space , solid collection of atoms

High T : local moments, Low T spin-orbital order

RUTGERS

Mott : 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….)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

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

RUTGERS

Photoemission spectroscopy.

Measures density of states for (BIS)adding and (PES) removing electrons

RUTGERS

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 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.

RUTGERS

Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Transfer of spectral weight which is non local in frequency.

Dramatic failure of DFT based approximations in predicting physical properties.

RUTGERS

Correlated Materials do big things

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.

RUTGERS

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) )

RUTGERS

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

RUTGERS

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

RUTGERS

Failure of the Standard Model: NiSe2-xSx

Miyasaka and Takagi (2000)

RUTGERS

Mean-Field : Classical vs Quantum

Classical case Quantum case

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

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

s st m t t tt ¯

¶+ - D - +

¶òò ò

†( )( ) ( )

MFL o n o n HG c i c iw w D=- á ñ

( )[ ][ ]

=D - -

,ij i j i

J S S h S- -å å

MF eff oH h S=-

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

RUTGERS

Limit of large lattice coordination

1~ d ij nearest neighborsijt

† 1~i jc c

1 1~ ~ (1)ij i j

t c c d Od d

~O(1)i i

Metzner Vollhardt, 89

1( , )

G k ii i

Muller-Hartmann 89

RUTGERS

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1( ) ( )

V ( )n nk nk

D i ii

-é ùê ú= +Pê ú- Pê úë ûå

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb b

s st t t t d t t ¯ ¯+òò

† †

t c c c c U n n

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

( , ')Do t t+

RUTGERS

E-DMFT references

H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis).

Q. Si and Smith PRL [analysis of quantum critical points]

RUTGERS

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

RUTGERS

Insights from DMFT Low temperature Ordered phases . Stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.

RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration)

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

RUTGERS

Kuwamoto Honig and Appell PRB (1980)M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

RUTGERS

Phase Diag: Ni Se2-x Sx

RUTGERS

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…

RUTGERS

Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

RUTGERS

. ARPES measurements on NiS2-xSex

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

RUTGERS

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

RUTGERS

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)

RUTGERS

Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

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

0( ) ,

ned P J

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy

RUTGERS

. ARPES measurements on NiS2-xSex

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

RUTGERS

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

RUTGERS

Anomalous Resistivity and Mott transition Ni Se2-x Sx

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

RUTGERS

Qualitative phase diagram in the U, T , plane (two band Kotliar Murthy Rozenberg PRL (2002).

Coexistence regions between localized and delocalized spectral functions.

RUTGERS

QMC calculationof n vs (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0)

diverges at generic Mott endpoints

RUTGERS

Insights from DMFT Mott transition as a bifurcation of an

effective action

Important role of the incoherent part of the spectral function at finite temperature

Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space.

RUTGERS

Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

RUTGERS

Realistic Calculationsof the Electronic Structure of Correlated materials

Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials.

Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997)

Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

RUTGERS

Combining 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, subtract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles or viewed as parameters

RUTGERS

Materials……

Pu Fe, Ni, La1-x Srx TiO3 NiO …………….

RUTGERS

Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

Problems with density functional treatements of PuPu

•DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.•Many studies (APW Freeman, Koelling 1972, ASA and FP-LMTO, Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) show•an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment•This is the largest discrepancy ever known in DFT based calculations.•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%•Weak correlation picture for alpha phase.

RUTGERS

Pu DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 410, 793 (2001)

RUTGERS

Lda vs Exp Spectra (Joyce et.al.)

RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Joyce , Arko et.al)

RUTGERS

Alpha and delta are strongly correlated,

The coexistence in the toy model, give rise to

two distinct phases in the realistic calculation.

In progress: phonon spectrum, epsilon phase……

RUTGERS

LaxSr1-xO3

Adding holes to a Mott insulator in three dimensions.

Canonical example of a Brinkman Rice system.

RUTGERS

(Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3

RUTGERS

DMFT calculation U near the Mott transition, Rozenberg et.al 94

RUTGERS

Hall Coefficient, electron like.

RUTGERS

La1-xSrxTiO3 photoemission

RUTGERS

Evolution of spectra with doping U=4

RUTGERS

Optical conductivity

RUTGERS

Realistic Computation of Optical Properties : La1-xSrxTiO3

RUTGERS

Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures has a

Curie Weiss law with a moment larger than the ordered moment.

RUTGERS

Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001)

RUTGERS

Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

0 3( )q

RUTGERS

Ni and Fe: theory vs exp eff 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)

RUTGERS

Fe and Ni Consistent picture of Fe (more localized) and Ni

(more correlated) 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

RUTGERS

Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 2001)

RUTGERS

Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

Failures of lda for ground state properties. Failures of lda for ground state properties.

NiO dielectric constant. LSDA:35.7 Exp:5.7

Lattice dynamics cannot be predicted:

• Optical G-phonon in MnO within LSDA: 3.04 THz, Experimentally: 7.86 THz (Massidda, et.al, PRL 1999)

• Bulk modulus for metallic Plutonium is one order of magnitude too large within LDA (214 GPa vs. 30 GPa) Also elastic constants are off.(Bouchet, et.al, J.Phys.C, 2001)

RUTGERS

Functional approach allows computation of linear response.(S. Savrasov and GK 2002

Apply to NiO, canonical Mott insulator.

U=8 ev, J=.9ev

Simple Impurity solver Hubbard 1.

Results for NiO: PhononsResults for NiO: Phonons

Solid circles – theory, open circles – exp. (Roy et.al, 1976)

RUTGERS

NiO U=8ev, J=1ev, Savrasov and GK (2002)

RUTGERS

Summary

Introduction to strongly correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal

aspects insights from DMFT System specific studies: LDA+DMFT Outlook

RUTGERS

Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) phase of Pu: S. Savrasov et al (Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al

PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al

1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..

RUTGERS

Acknowledgements: Development of DMFT

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

Support: National Science Foundation.

Work on Pu: Departament of Energy and LANL.

Work on Fe and Ni: Office of Naval Research

RUTGERS

Challenges

Short Range Magnetic Correlations without magnetic order.

Single Site DMFT does not capture these effects

1~ 0ij i j

J S S dd

RUTGERS

Pu: Anomalous thermal expansion (J. Smith LANL)

RUTGERS

Problems with LDA

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods

o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% Is 35% lower than experimentlower than experiment

o This is the largest discrepancy ever known in DFT based calculations.

RUTGERS

Problems 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%

Notice however that LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that Pu is a weakly correlated system

RUTGERS

Conventional 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. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.

RUTGERS

Problems 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.

RUTGERS

Pu Specific Heat

RUTGERS

Anomalous ResistivityJ. Smith LANL

RUTGERS

MAGNETIC SUSCEPTIBILITY

RUTGERS

Dynamical 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 electronic structure is about to vary.

This result resolves one of the basic paradoxes in the physics of Pu.

RUTGERS

Pu: DMFT total energy vs Volume

RUTGERS

Lda vs Exp Spectra

RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)

RUTGERS

PU: ALPHA AND DELTA

RUTGERS

Case study Fe and Ni

Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T But at high temperatures, they resemble

more a collection of atoms with reduced moment.

The crossover from low T to high T requires DMFT.

RUTGERS

Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK)

RUTGERS

However not everything in low T phase is OK as far as LDA goes..

Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel .(instead of 111)

LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)

Use LDA+ U to tackle these refined issues, (WE cannot be resolved with DMFT, compare parameters with Lichtenstein’s )

RUTGERS

Some Earlier Work:

Kondorskii and E Straube Sov Phys. JETP 36, 188 (1973)

G. H Dallderop P J Kelly M Schuurmans Phys. Rev. B 41, 11919 (1990)

Trygg, Johansson Eriksson and Wills Phys. Rev. Lett. 75 2871 (1995) Schneider M Erickson and Jansen J. Appl Phys. 81 3869 (1997)

I Solovyev, Lichenstein Terakura Phys. Rev. Lett 80, 5758 (LDA+U +SO Coupling)…….

Present work : Imseok Yang, S Savrasov and GK

RUTGERS

Origin of Magnetic Anisotropy Spin orbit coupling L.S L is a variable which is sensitive to

correlations, a reminder of the atomic physics Crystal fields quench L, interactions enhance

it, T2g levels carry moment, eg levels do not

any redistribution of these no matter how small will affect L.

Both J and U matter !

RUTGERS

Magnetic anisotropy of Fe and Ni LDA+ U

RUTGERS

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 interpolating between atoms and band, encouraging results for simple elements

RUTGERS

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c n nb b b

s st t t t ¯= +òò ò

† †

t c c c c U n n

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

Weiss field

RUTGERS

Resistivities

RUTGERS

Resistivities

RUTGERS

Thermodynamics LiVO4

RUTGERS

Resistivity saturation

RUTGERS

Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998

Title:

Creator:gnuplotPreview:

was not saved

a preview included in it.Comment:

cript printer, but not toother types of printers.

IPTNCA

RUTGERS

Anomalous Resistivities: DopedHubbard Model (QMC)

Prushke and Jarrell 1993

RUTGERS

Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest

in a G0 (renormalization of the quadratic part of the effective action). What to take for G0:

DCA (M. Jarrell et.al) , CDMFT ( Savrasov Palsson and GK )

include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)

RUTGERS

Outlook

Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs…………

First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT

Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ………

RUTGERS

Ni moment

RUTGERS

Fe moment

RUTGERS

Magnetic anisotropy vs U , J=.95 Ni

RUTGERS

Magnetic anisotropy Fe J=.8

RUTGERS

Delocalization-Localization across the actinide series

o f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.

o Pu has a simple cubic fcc structure,the phase which is easily stabilized over a wide region in the T,p phase diagram.

o The phase is non magnetic.o Many LDA , GGA studies ( Soderlind et. Al 1990,

Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment

o This is one of the largest discrepancy ever known in DFT based calculations.

RUTGERS

Pu: Anomalous thermal expansion (J. Smith LANL)

RUTGERS

Problems with LDA

RUTGERS

Problems 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%

Notice however that LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that Pu is a weakly correlated system

RUTGERS

Problems 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.

RUTGERS

Pu Specific Heat

RUTGERS

Anomalous ResistivityJ. Smith LANL

RUTGERS

MAGNETIC SUSCEPTIBILITY

RUTGERS

Dynamical 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 electronic structure is about to vary.

This result resolves one of the basic paradoxes in the physics of Pu.

RUTGERS

LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

KS n n

LDAext xc

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

a b ba

r rr r

- +Ñ - - S -

+ + +-

ò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

n T G i ew

KS ab [ ( ) G V ( ) ]LDA DMFT a br r

RUTGERS

Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with = HF

reduces to LDA+U• 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.

RUTGERS

Anomalous Resistivities: DopedHubbard Model (QMC)

Prushke and Jarrell 1993

RUTGERS

Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998

Title:

Creator:gnuplotPreview:

was not saved

a preview included in it.Comment:

cript printer, but not toother types of printers.

IPTNCA

RUTGERS

DMFT: Methods of Solution

RUTGERS

LDA functional

2log[ / 2 ] ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r rr r

- +Ñ - -

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

Conjugate field, VKS(r)

RUTGERS

Minimize LDA functional

[ ]( )( ) ( ) '

| ' | ( )

KS ext

ErV r V r dr

d rrdr

= + +-ò

( ) { )[ / 2 ]

( ) ( ) n

ikj kj kj

n KSkj

r f tri V

r r ew

r e yw

+Ñ -=å å

Kohn Sham eigenvalues, auxiliary quantities.

RUTGERS

Anomalous transfer of spectral weight heavy fermions

RUTGERS

Anomalous transfer of spectral weight

RUTGERS

Anomalous transfer of spectral weigth heavy fermions

RUTGERS

V2O3 resistivity

RUTGERS

LDA+DMFT Self-Consistency loop

Im puritySo lver

S .C .C .

0( ) ( , , ) i

r T G r r i e w

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

0( , , )HHi

n T G r r i e w

RUTGERS

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

† †

t c c c c U n n

10 ( ) ( )n n nG i i iw w m w- = + - D

Weiss field

RUTGERS

Landau Functional

† †,

[ , ] ( ) ( ) ( )†

† † † †

Mettalic Order Para

( )[ ] [ ]

[ , ] [ [ ] ]

( )( )

r: ( )

( ) 2 ( )[ ]( )

LG imp

L f f f i i f i

iF T F

F Log df dfe

dL f f f e f Uf f f f d

F iT f i f i TG i

Spin Model An

[ ] [[ ]2 ]

alogy:

hF h Log ch h

G. Kotliar EPJB (1999)

RUTGERS

Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment

Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)

eff 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)

RUTGERS

Problems with LDA

RUTGERS

Spectral 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 DFT(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

RUTGERS

LDA functional

2log[ / 2 ] ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r rr r

- +Ñ - -

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

RUTGERS

Failure of the StandardModel: Anomalous Spectral Weight TransferOptical Conductivity o of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993)

Neff depends on T

RUTGERS

Degenerate Hubbard model

† †

t c c c c U n n

Doping or chemical potential

Frustration (t’/t)

T temperatureMott transition as a function of doping, pressure temperature etc.

RUTGERS

Outlook

The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR

Strategy: advancing our understanding scale by scale

Generalized cluster methods to capture longer range magnetic correlations

New structures in k space. Cellular DMFT

RUTGERS

Small amounts of Ga stabilize the phase

RUTGERS

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

Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit

RUTGERS

res85520219st9v9t9es

RUTGERS

LDA functional

2log[ / 2 ] ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r rr r

- +Ñ - -

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

RUTGERS

Hubbard model

† †

t c c c c U n n

Doping or chemical potential

Frustration (t’/t)

T temperatureMott transition as a function of doping, pressure temperature etc.

RUTGERS

Dynamical Mean Field Theory, cavity constructionA. Georges G. Kotliar 92

( )[ ( ')] ( ')eff o o o oc c U n nSb b b

s st m t t tt ¯

¶= + - D - +

¶òò ò ( )wD

†( )( ) ( )

MFo n o n SG c i c iw w D=- á ñ 1( )

( )[ ][ ]

=D - -

† †

t c c c c U n n

RUTGERS

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

† †

t c c c c U n n

10 ( ) ( )n n nG i i iw w m w- = + - D

1 † 10 0 0 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c iw w w w- -S = + á ñ

Weiss field

RUTGERS

Solving the DMFT equations

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,ED….)Analytical Methods•Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

Im puritySo lver

S .C .C .

RUTGERS

Single site DMFT, functional formulation. Construct a functional of the observable ,local Greens function. Express in terms of Weiss field, (semicircularDOS) [G. Kotliar EBJB 99]

† †,

[ , ] ( ) ( ) ( )†

( )[ ] [ ]

[ ]loc

L f f f i i f i

iF T F

F Log df dfe

Ising analgoy

[ ] [ [2 ]]2LG

hF h Log ch h

RUTGERS

Compressibilty divergence : One band case (Kotliar Murthy and Rozenberg 2002)

RUTGERS

X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration

RUTGERS

Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar).

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. DFT(r)]

Introduce local orbitals, R(r-R)orbitals, and local GFG(R,R)(i ) = The exact free energy can be expressed as a functional

of the local Greens function and of the density by introducing (r),G(R,R)(i)]

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

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

RUTGERS

Optical spectral weight

RUTGERS

LDA+DMFT Self-Consistency loop

Im puritySo lver

S .C .C .

0( ) ( , , ) i

r T G r r i e w

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

0( , , )HHi

n T G r r i e w

RUTGERS

Parallel development: Fujimori et.al

RUTGERS

C-DMFT

C:DMFT The lattice self energy is inferred from the cluster self energy.

0 0cG G ab¾¾® c

abS ¾¾®Sij ijt tab¾¾®

Electronic Structure of Correlated Materials : a DMFT Perspective

Documents

Transcript of Electronic Structure of Correlated Materials : a DMFT Perspective

IJS Strongly correlated materials from Dynamical Mean Field Perspective. Thanks to: G.Kotliar, S. Savrasov, V. Oudovenko DMFT(SUNCA method) two-band Hubbard.

ALGORITHMIC PERSPECTIVE ON STRONGLY CORRELATED …ALGORITHMIC PERSPECTIVE ON STRONGLY CORRELATED SYSTEMS Lecture: Introduction to Matrix Product States Fall 2015 Professor: Bryan Clark

Realistic investigations of correlated electron systems ...scalettar.physics.ucdavis.edu/publications/lda+dmft.pdf · Article LDA LDA+DMFT LDA+U 0U=W∞ metal insulator Fig. 1 With

DFT+DMFT and its Discontents: towards a first-principles theory of correlated electron ...sites.bu.edu/icambu/files/2013/10/MillisICAMBU.pdf · 2013-10-01 · DFT+DMFT and its Discontents:

Developing the bosonic DMFT formalismonline.itp.ucsb.edu/online/compqcm10/pollet/pdf/BDMFT.pdfDMFT 101 A. Georges, Lectures on the Physics of Highly Correlated Electron Systems VIII

DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

Strongly Correlated Electron Materials: Some DMFT Concepts and Applications Strongly Correlated Electron Materials: Some DMFT Concepts and Applications.

Electronic Structure of Elemental Plutonium: A Dynamical Mean Field Perspective (DMFT)

Quantum-cluster theories from a variational perspectiveDIA cellular DIA variational CA oo DMFT cellular DMFT 2 1 Lb,-. / Michael Potthoff Quantum-cluster methods for correlated materials,

Dynamical mean-field theory (DMFT)susi.theochem.tuwien.ac.at/events/ws2017/notes/Tomczak_DMFT.pdf · Dynamical mean-field theory (DMFT) ELECTRONIC STRUCTURE OF CORRELATED MATERIALS

Mikhail Katsnelson - Radboud Universiteit · 3. Exchange interactions from first principles 4. Beyond DFT: correlated systems and LDA+DMFT 5. Itinerant-electron magnets: Fe, Co, Ni

DMFT-GODDAdmftgodda.in/pdf/DMFT_General_Body_Dated_21.03.2020.pdf2020/03/21 · DMFT PMU Swaniti Initiative, New Delhi 05 Swaniti Initiative, New Delhi 15. DMFT 3ž-

Extensions of Single Site DMFT and its Applications to Correlated Materials

Cellular-DMFT approach to the electronic structure of correlated solids. Application to the sp, 3d,4f and 5f electron systems. Collaborators, N.Zein K.

Introduction to correlated materials, DFT, DMFT DFT +DMFT · Determine energy and and self consistently from extremizingextremizing a a functional ChitraChitra and and Kotliar(()2001)

Understanding Heavy Fermion Systems: a DMFT perspective Understanding Heavy Fermion Systems: a DMFT perspective Gabriel Kotliar and Center for Materials.

Electronic Structure of Correlated Materials: LDA+U, DMFT, and …t-ozaki.issp.u-tokyo.ac.jp/winter-school16/1-Corr-Han.pdf · 2016-12-17 · Electronic Structure of Correlated Materials:

Electronic Structure of Strongly Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University.

Dynamical mean-field theory (DMFT)wien2k.at/events/ws2017/notes/Tomczak_DMFT.pdf · 2017-09-22 · Dynamical mean-field theory (DMFT) ELECTRONIC STRUCTURE OF CORRELATED MATERIALS

DMFT Original Artical