Electronic Structure of Strongly Correlated Materials:Insights from Dynamical Mean Field

Theory (DMFT).

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers UniversityCenter for Materials Theory Rutgers University

CPTH Ecole Polytechnique Palaiseau, and CPTH CEA Saclay , France

$upport : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.

REUNIÓN NACIONAL DE FÍSICA DEL ESTADO SÓLIDO. GEFES IV Alicante Spain. February 1-3 (2006)

Band Theory: electrons as waves.

Landau Fermi Liquid Theory. At low energies the electrons behave as non interacting quasiparticles.

Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory+ GW

Perturbation Theory. 2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

Rigid bands , optical transitions , thermodynamics, transport………

1 10 GW[ - ]KS KS crystG G V V

LDA+GW: semiconducting gaps. Reviews J. Wilkins, M.

VanSchilfgaarde

Success story : Density Success story : Density Functional linear responseFunctional linear response

Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001

Correlated Electron Materials

• Are not well described by either the itinerant or the localized framework .

• Compounds with partially filled f and d shells. Need new starting point for their description. Non perturbative problem. New reference frame for computing their physical properties.

• Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, huge resistivity changes across the MIT, colossal magneto-resistance, huge volume collapses, large masses in heavy Fermions, ……………..

Breakdown of the Standard Model :Large Metallic Resistivities

21 1 1( )

(100 )MottF Fe k k l

cmh

Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Vander Marel

(2005) Takagi (2003 ) Neff depends on T

Localization vs Delocalization Strong Correlation Problem

• Many interesting compounds do not fit within the “Standard Model”.

• Tend to have elements with partially filled d and f shells. Competition between kinetic and Coulomb interactions.

• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).

• Non perturbative problem.• Require a framework that combines both

atomic physics and band theory. DMFT.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

1( , )

( )k

G k ii i

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

0

1 2

( , ) ( )

( )(cos cos ) ( )(cos .cos ) .......latt k

kx ky kx ky

Cluster Extensions of Single Site DMFT

Many Techniques for solving the impurity model: QMC, (Fye-

Hirsch), NCA, ED(Krauth –Caffarel),

IPT, …………For a review see Kotliar et. Al to appear in RMP

(2006)

For reviews of cluster methods see: Georges et.al. RMP (1996) Maier et.al

RMP (2005), Kotliar et.al cond-mat 0511085. to appear in RMP (2006) Kyung

et.al cond-mat 0511085

Weiss­FieldAlternative (T. Stanescu and

G. K. ) periodize the cumulants rather than the self energies.

Parametrizes the physics in terms of a few functions .

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.

Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

Mott transition in V2O3 under pressure

or chemical substitution on V-site. How does the electron go from localized to itinerant.

Pressure Driven Mott transition

How does the electron go from the localized to

the itinerant limit ?

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site

DMFT. High temperature universality

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

V2O3:Anomalous transfer of spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

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

2000]

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

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

Single site DMFT and kappa organics

Ising critical endpoint! In V2O3

P. Limelette et.al. Science 302, 89 (2003)

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

Conclusions.

• Three peak structure, quasiparticles and Hubbard bands.

• Non local transfer of spectral weight.• Large metallic resistivities.• The Mott transition is driven by transfer of

spectral weight from low to high energy as we approach the localized phase.

• Coherent and incoherence crossover. Real and momentum space.

• Theory and experiments begin to agree on a broad picture.

† †

, ,

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

t c c c c U n n

Cuprate superconductors and the Hubbard Model . PW Anderson 1987

RVB physics and Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc.

• Slave boson approach. <b> coherence order parameter. singlet formation order parameters.Baskaran Zhou Anderson , Ruckenstein et.al (1987) .

Other states flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) have point zeros.

RVB phase diagram of the Cuprate Superconductors. Superexchange.

• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero.

• Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria

N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

Problems with the approach.• Neel order. How to continue a Neel insulating state ?

Need to treat properly finite T.• Temperature dependence of the penetration depth [Wen

and Lee , Ioffe and Millis ] . Theory:s[T]=x-Ta x2 , Exp: [T]= x-T a.

• Mean field is too uniform on the Fermi surface, in contradiction with ARPES.

• No quantitative computations in the regime where there is a coherent-incoherent crossover which compare well with experiments. [e.g. Ioffe Kotliar 1990]

The development of CDMFT solves may solve many of these problems.!!

Photoemission spectra near the antinodal direction

in a Bi2212 underdoped sample. Campuzano et.al

EDC along different parts of the zone, from Zhou et.al.

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site

DMFT. High temperature universality

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

The development of CDMFT solves may solve many of the problems of the early slave bosons RVB

theory .!! Theoretical approach: study the plaquette CDMFT

equations. • Ignore inhomogeneities and phase

separation.

• Follow separately each mean field state.

• Focus on the physics results from the proximity to a Mott insulating state and to which extent it accounts for the experimental observations.

Competition of AF and SC M. Capone M. Civelli and GK (2006)

Superconductivity in the Hubbard model role of the Mott transition and influence of the super-

exchange. ( M. Capone et.al. V. Kancharla et. al. CDMFT+ED, 4+ 8 sites t’=0) .

Pd

Order Parameter and Superconducting Gap do not always scale! ED study in the SC state Capone Civelli

Parcollet and GK (2006)

Evolution of DOS with doping U=8t. Capone et.al. : Superconductivity is driven by transfer of spectral weight ,

slave boson b2 !

Anomalous Self Energy. (from Capone et.al 2006.) Notice the remarkable increase with decreasing doping! True superconducting pairing!! U=8t

Significant Difference with Migdal-Eliashberg.

Follow the “normal state” with doping. Civelli et.al. PRL 95, 106402 (2005)

Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k U=16 t, t’=-.3

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

Different for electron doped!

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

K.M. Shen et.al. 2004

2X2 CDMFT

Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K cond-matt 0508302

Conclusion

• CDMFT delivers the spectra. • Path between d-wave and insulator. Dynamical RVB!• Lines of zeros. Connection with other work. of A. Tsvelik

and collaborators. (Perturbation theory in chains , see however Biermann et.al). T.Stanescu, fully self consistent scheme.

• Weak coupling RG (T. M. Rice and collaborators). Truncation of the Fermi surface.

CDMFT presents it as a strong coupling instability that begins FAR FROM FERMI SURFACE.

Realistic Descriptions of Materials and a First Principles Approach to

Strongly Correlated Electron Systems.

• Incorporate realistic band structure and orbital degeneracy.

• Incorporate the coupling of the lattice degrees of freedom to the electronic degrees of freedom.

• Predict properties of matter without empirical information.

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • Realistic band structure and orbital degeneracy.

Describes the excitation spectra of many strongly correlated solids. .

Spectral Density Functionals. Chitra and Kotliar PRB 2001 Savrasov et. al. Nature (2001) Savrasov and Kotliar PRB (2005)

•Determine the self energy , the density and the structure of the solid by extremizing a functional of these quantities. Coupling of electronic degrees of freedom to structural degrees of freedom.

Mott Transition in ActinidesMott Transition in Actinides

This­regime­is­not­well­described­by­traditional­techniques­of­electronic­structure­techniques­and­require­new­methods­which­take­into­account­the­itinerant­and­the­localized­character­of­the­electron­on­the­same­footing.

after G. Lander, Science (2003).

The­f­electrons­in­Plutonium­­are­close­to­a­localization-delocalization­transition­(Johansson, 1974) .­Modern­understanding­of­the­phenomena­with­DMFT­(Savrasov­and­Kotliar­2002-2003)

after J. Lashley et.al, cond-mat (2005).

Mott Transition

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Summary• Review the standard model of solids.• Introduced some of the problems posed by strongly correlated

electron materials.• Dynamical Mean Field Theory (DMFT). New reference frame to

think about the physics of these materials and compute its properties.

• The Mott Transition in 3d frustrated transition metal oxides and in high temperature superconductors.

• Future Directions. The field of correlated electrons is at the brink of a revolution.(C) DMFT : Rapid development of conceptual tools and computational abilities. Theoretical Spectroscopy in the making.

Prelude to theoretical material design using strongly correlated elemenets.

• Focus on the deviations between CDMFT and experiments to elucidate the role of long wavelength non Gaussian fluctuations.

Gracias por invitarme y por vuestra atencion!

Functional formulation to achieve more realistic calculations For a review see

Kotliar et.al. to appear in RMP..

Spectral Density Functional

LDA+DMFT dc lda loc[ , E ; ]lda dmft U G

loc [ ]cdmft locW G

loc [ Edc ; ] sdft U G

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) generalizing LDA+ U

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

Final Goal [ ]dft

Approaching the Mott transition: CDMFT Picture

• Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• D wave gapping of the single particle spectra as the Mott transition is approached. Real and Imaginary part of the self energies grow approaching half filling. Unlike weak coupling!

• Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004) . Both real and imaginary parts of the self energy get larger. Strong Coupling instability.

Two paths for the calculation of electronic structure of materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

† †

, ,

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

t c c c c U n n

Hubbard Model

ARPES spectra for La2−xSrxCuO4 at doping x =0.063, 0.09, 0.22. From Zhou et al

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc.

n=1

Order in Perturbation Theory

Order in PT

Range of the clusters

Basis set size. DMFT

GW

r site CDMFT

l=1

l=2

l=lmax

r=1

r=2

n=2

GW+ first vertex correction

Spectral Density Functional

LDA+DMFT dc lda loc[ E ; ]lda dmft U G

loc loc [ ]cdmft loc locW G

xc loc dc[ ] V Elocsdft U G

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)

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

Conclusions sp systems.

• Not well described by single site DMFT. But very well described by first principles cdmft with relatively small clusters. [2 or 3 coordination spheres]

• Weakly correlated materials. Use cheap impurity solvers.

• Fast, self consistent way of getting first principles electronic structure without LDA. Good trends for semiconducting gaps and band withds.

Earlier approximations as limiting cases;sinlge site DMFT for models A.

Georges G. Kotliar PRB (1992)

Spectral Density Functional

LDA+DMFT dc lda loc[ E ; ]lda dmft U G

loc loc [ ]cdmft loc locW G

xc loc dc[ ] V Elocsdft U G

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)

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

Spectral shapes. Large Doping Stanescu and GK cond-mat

0508302

Small Doping. T. Stanescu and GK cond-matt 0508302

Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K cond-matt 0508302

Lines of Zeros and Spectral Shapes. Stanescu and GK cond-matt 0508302

Two paths for calculation of electronic structure of

strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

• The combination of realistic band theory and many body physics, is a very broad subject.

• Having a practical and tractable non perturbative method for solving many body Hamiltonians, the next step is to bring more realistic descriptions of the materials Orbital degeneracy and realistic band structure.

• LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

• The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT.

• Functional formulation similar to DFT: Total Energy as functional of local Green function: Spectral Density Functional Theory(Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, Abrahams, Nature 2001).

• Focus on the “local “ spectral function A() (and of the local screened Coulomb interaction W() ) of the solid.

• Write a functional of the local spectral function such that its stationary point, give the energy of the solid.

• No explicit expression for the exact functional exists, but good approximations are available. LDA+DMFT.

• The spectral function is computed by solving a local impurity model in a medium .Which is a new reference system to think about correlated electrons.

• Add non local perturbative corrections “GW+DMFT”.• Explosion of papers, refining the techniques and applying

it to many different materials.

Actinies , role of Pu in the periodic table

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%.

Within LDA fcc Pu has a negative shear modulus.

LSDA predicts Pu to be magnetic with a 5 b moment. Experimentally it is not.

Treating f electrons as core overestimates the volume by 30 %

Pu is not MAGNETIC, alpha and delta have comparable

susceptibility and specifi heat.

DMFT : What is the dominant atomic configuration ,what is the fate of the atomic moment ?

• Snapshots of the f electron :Dominant configuration:(5f)5

• Naïve view Lz=-3,-2,-1,0,1, ML=-5 B, ,S=5/2 Ms=5 B . Mtot=0

• More realistic calculations, (GGA+U),itineracy, crystal fields ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000)

• This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).

Contrast Am:(5f)6

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Double well structure and Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low

Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]

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.

. • AFunctional of the cluster Greens function. Allows the investigation of the normal

state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. T. Maier et. al. (2000) . ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8t )

• Use exact diag ( Krauth Caffarel 1995 ) as a solver to reach larger U’s and smaller Temperature and CDMFT as the mean field scheme. • Recently (K. Haule and GK ) the region near the superconducting –normal state

transition temperature near optimal doping was studied using NCA + DCA .• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS -(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • b--------> b(k), ----- (), k• Extends the functional form of self energy to finite T and higher frequency.

CDMFT study of cuprates

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

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

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 earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

• Pu strongly correlated element, at the brink of a Mott instability.

• Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu.

• Clues to understanding other Pu anomalies.

Outline• Introduction to strongly correlated electrons.• Introduction to Dynamical Mean Field Theory

(DMFT)• The Mott transition problem. Theory and

experiments.• More realistic calculations. Pu the Mott

transition across the actinide series. • Conclusions . Current developments and future

directions.

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2.

S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressureAmericium under pressure

Density­functional­based­electronic­structure­calculations: Non­magnetic­LDA/GGA­predicts­volume­50%­off.­ Magnetic­GGA­corrects­most­of­error­in­volume­but­gives­m~6B

(Soderlind et.al., PRB 2000). Experimentally,­Am­has non­magnetic­f6­ground­state­with­J=0­(7F0)

Experimental­Equation­of­State­(after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Mott transition in open (right) and closed (left) shell systems.

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

Am under pressure: J.C. GriveauJ. Rebizant G. Lander G. Kotliar PRL (2005)

Collaborators References

• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996).

• Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Conclusion• DMFT. Electronic Structure Method under development.

Local Approach. Cluster extensions. • Quantitative results , connection between electronic

structure, scales and bonding. • Qualitative understanding by linking real materials to

impurity models. Concepts to think about correlated materials.

• Closely tied to experiments. System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics.……

• Role of loclity.• Material design using strongly correlated systems.

Anomalous Resistivity

PRL 91,061401 (2003)

The delta –epsilon transition

• The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

• What drives this phase transition?

• LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

Phonon 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 T=0. 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.

Pu in the periodic table

actinides

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene

before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J

= 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

J. C. Griveau et. al. (2004)

H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.

Superconductivity due to valence fluctuations ?

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)

Epilogue, the search for a quasiparticle peak and its demise,

photoemission, transport. Confirmation of the DMFT

predictions ARPES measurements on NiS2-xSex

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

S.-K. Mo et al., Phys Rev. Lett. 90, 186403 (2003).

Limelette et. al. [Science] G. Kotliar [Science].

. ARPES measurements on NiS2-xSex

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

One Particle Local Spectral Function and

Angle Integrated Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef,

f() A() M2

• Probability of absorbing an electron and transfering energy =Ei-Ef,

(1-f()) A() M2

• Theory. Compute one particle greens function and use spectral function.

e

e

Dynamical Mean Field Theory

• Focus on the local spectral function A() of the solid.• Write a functional of the local spectral function such that

its stationary point, give the energy of the solid.• No explicit expression for the exact functional exists,

but good approximations are available.• The spectral function is computed by solving a local

impurity model. Which is a new reference system to think about correlated electrons.

• Ref: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod

Phys 68,1 (1996) . Generalizations to realistic electronic structure. (G. Kotliar and S. Savrasov in )

Evolution of the spectral function at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

[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) ]

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.

Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension.

Site Cell. Cellular DMFT. C-DMFT. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other cluster extensions (DCA Jarrell Krishnamurthy, M

Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes , causality

issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 .

Searching for a quasiparticle peak

Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3

from optics.

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

. ARPES measurements on NiS2-xSex

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

QP in V2O3 was recently found Mo et.al

organics• ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene

(ET)2 X

Increasing pressure ----- increasing t’ ------------

X0 X1 X2 X3• (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2• Spin liquid Mott transition

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

2204­(2000) )

More examples

• LiCoO2• Used in batteries,

laptops, cell phones

Large thermoelectric power in a metal with a large number of carriers NaCo2O4

TS

V

Vanadium Oxide Transport under pressure. Limelette etal

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

Expt. Wong et. al.

[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) ]

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.

Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension.

Two paths for ab-initio calculation of electronic

structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

Failure of the standard model : Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

2

4 3

1

A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar,S. Savrasov, G. Palsson and G. Biroli, PRL 87, 186401 (2001) .

Mott Transition in ActinidesMott Transition in Actinides

This­regime­is­not­well­described­by­traditional­techniques­of­electronic­structure­techniques­and­require­new­methods­which­take­into­account­the­itinerant­and­the­localized­character­of­the­electron­on­the­same­footing.

after G. Lander, Science (2003).

The­f­electrons­in­Plutonium­­are­close­to­a­localization-delocalization­transition­(Johansson, 1974) .

after J. Lashley et.al, cond-mat (2005).

Mott Transition

Resistivity in AmericiumResistivity in Americium

Resistivity­behavior(after Griveau et.al, PRL 2005)

Superconductivity

•­Under­pressure,­resistivity­of­Am­raises­almost­an­order­of­magnitude­and­reaches­its­value­of­500­m*cm

•­Superconductivity­in­Am­is­observed­with­Tc­~­0.5K

Photoemission in Am, Pu, SmPhotoemission in Am, Pu, Sm

after J. R. Naegele, Phys. Rev. Lett. (1984).

Atomic multiplet structure­emerges­from­measured­­photoemission­spectra­in­Am (5f6),­Sm(4f6)­-­­Signature­for­f­electrons­localization.

Am Equation of State: LDA+DMFT PredictionsAm Equation of State: LDA+DMFT Predictions

LDA+DMFT predictions: Non­magnetic­f6­ground­

state­with­J=0­(7F0) Equilibrium­Volume:­­­­­­Vtheory/Vexp=0.93

Bulk­Modulus:­Btheory=47­GPa

­­­­­Experimentally­B=40-45­GPa

Theoretical­P(V)­using­LDA+DMFT­

Self-consistent­evaluations­of­total­energies­with­LDA+DMFT­using­matrix­Hubbard­I­method.

Accounting­for­full­atomic­multiplet­structure­using­Slater­integrals:F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eVNew­algorithms­allow­studies­of­complex­structures.­

Predictions­for­Am­II

Predictions­for­Am­IV

Predictions­for­Am­III

Predictions­for­Am­I

Photoemission Spectrum from Photoemission Spectrum from 77FF00 Americium Americium

LDA+DMFT­Density­of­States

Experimental­Photoemission­Spectrum(after J. Naegele et.al, PRL 1984)

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+DMFT­Density­of­States

Exact­Diag.­for­atomic­shell

F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV

Matrix­Hubbard­I­Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Alpha and delta Pu

Failure of the StandardModel: Anomalous Spectral Weight Transfer

Optical Conductivity o of FeSi for T=20,40, 200 and 250 K from Schlesinger et.al (1993)

0( )d

Neff depends on T

1

10

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

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

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

†

0 0

( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b

s st t t t d t t­ ¯+òò

† †

, ,

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

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

i j

V n n

0 0( , ')Do n nt t+

A. C. Lawson et. al. LA UR 04-6008

F(T,V)=Fphonons+Finvar

=125 K =.5 = 1400 KD

Invar model A. C. Lawson et. al. LA UR 04-6008

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Breakdown of standard model

• Large metallic resistivities exceeding the Mott limit.• Breakdown of the rigid band picture.• Anomalous transfer of spectral weight in photoemission

and optics. • LDA+GW loses its predictive power.

• Need new reference frame, to think about and compute the properties of correlated materials.

• Need new starting point to do perturbation theory.

Limit of large lattice coordination1

~ d ij nearest neighborsijtd

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

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

Ba

Think in real space , solid collection of atoms

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

1

T

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site

DMFT. High temperature universality

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

Optical 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).

RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Treatement needs refinement

• The kinetic energy of the Hubbard model contains both the kinetic energy of the holes, and the superexchange energy of the spins.

• Physically they are very different.

• Experimentally only measures the kinetic energy of the holes.

Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping or chemical potential

Frustration (t’/t)

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

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

1 1( ) [ ( )]

( )n loc nloc n

i R G iG i

w ww

-D = +1

[ ][ ]k k

R zz e

=-å

†

0 0 0

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

s st t t t ­ ¯= +òò ò

† †

, ,

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

t c c c c U n n

0

†( )( ) ( ) ( )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

2( ) ( )n loc ni t G iw wD =Semicircular density of states. Behte lattice.

1( , ) Im[ ( , )] Im[ ]

( , )k

A k G kk

Photoemission and the Theory of Electronic Structure

Limiting case itinerant electrons( ) ( )kk

A

( ) ( , )k

A A k

( ) ( ) ( )B AA Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

Cellular DMFT studies of the doped Mott insulator : the plaquette as a reference frame. Dynamical RVB

Collaborators: M. Civelli, K. Haule, M. Capone, O. Parcollet, T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrook) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke)

.

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)

One Particle Spectral Function and Angle

Integrated Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

• Probability of absorbing an electron and transfering energy =Ei-Ef, and momentum k

(1-f()) A() M2

• Theory. Compute one particle greens function and use spectral function.

e

e