The Mott Transition and the Challenge of Strongly Correlated

Electron Systems. G. Kotliar

Physics Department and Center for Materials Theory

Rutgers

PIPT Showcase Conference UBC Vancouver May 12th 2005

Outline• Correlated Electron Materials.

• Dynamical Mean Field Theory.

• The Mott transition problem: qualitative insights from DMFT.

• Towards first principles calculations of the electronic structure of correlated materials. Pu Am and the Mott transition across the actinide series.

The Standard Model of Solids

• Itinerant limit. Band Theory. Wave picture of the electron in momentum space. . Pauli susceptibility.

• Localized model. Real space picture of electrons bound to atoms. Curie susceptibility at high temperatures, spin-orbital ordering at low temperatures.

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, colossal magneto-resistance, huge volume collapses……………..

Large Metallic Resistivities

21 1 1( ) (100 )Mott

F 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

Breakdown of the standard model of solids.

• Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200 ohm cm

• Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics.

• The quantitative tools of the standard model fail.

21 ( )F Fe k k l

h

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

A k G kk

MODEL HAMILTONIAN AND OBSERVABLES

Limiting case itinerant electrons( ) ( )kk

A

( ) ( , )k

A A k

( ) ( ) ( )B AA

Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

† †

, ,

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

t c c c c U n n

Parameters: U/t , T, carrier concentration, frustration :

Limit of large lattice coordination1~ d ij nearest neighborsijtd

† 1~i jc cd

†

,

1 1~ ~ (1)ij i jj

t c c d Od d

~O(1)i iUn n

Metzner Vollhardt, 891( , )

( )k

G k ii i

Muller-Hartmann 89

Mean-Field Classical vs Quantum

Classical case Quantum case

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

†

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( )( )[ ]

[ ]n

kn k

n

G ii t

G i

ww m

w

D =D - - +D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh0 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

10G-

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).

• The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting the average energy already contained in LDA.

• Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, treat with DMFT. In this method U is either a parameter or is estimated from constrained LDA

• • Describes the excitation spectra of many strongly correalted solids. .

Spectral Density Functional• Determine the self energy , the density and the

structure of the solid self consistently. By extremizing a functional of these quantities. (Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, PRB 2005). Coupling of electronic degrees of freedom to structural degrees of freedom. Full implementation for Pu. Savrasov and Kotliar Nature 2001.

• Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens functions.

Mott transition in V2O3 under pressure

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

The Mott transition and Universality

Same behavior at high tempeartures, completely

different at low T

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling.  M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .

COHERENCE INCOHERENCE CROSSOVER

V2O3:Anomalous transfer of spectral weight

Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593

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.

Single-site DMFT and expts

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.

Mott Transition in the Actinide Series

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 %

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­PUPU

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

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 ]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

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.

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

First Principles DMFT Studies of Pu

• 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. Qualitative Insights and quantitative studies. Double well. Alpha and Delta Pu.

Approach the Mott point from the right Am under Approach the Mott point from the right Am under pressurepressure

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.

Realization in Am ??

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

0

1 2

( , ) ( )( )(cos cos ) ( )(cos .cos ) .......

latt kkx ky kx ky

Cluster Extensions of Single Site DMFT

Conclusions Future Directions• DMFT: Method under development, but it already gives

new insights into materials…….• Exciting development: cluster extensions. Allows us to

see to check the accuracy of the single site DMFT corrections, and obtain new physics at lower temperatures and closer to the Mott transition where the single site DMFT breaks down.

• Captures new physics beyond single site DMFT , i.e. d wave superconductivity, and other novel aspects of the Mott transition in two dimensional systems.

• Allow us to focus on deviations of experiments from DMFT.

• DMFT and RG developments

Some 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)

Am Equation of State: LDA+DMFT Predictions Am Equation of State: LDA+DMFT Predictions (Savrasov Kotliar Haule Murthy 2005)(Savrasov Kotliar Haule Murthy 2005)

LDA+DMFT predictions: Non­magnetic­f6­ground­state­with­J=0­(7F0)

Equilibrium­Volume:­­­­­­Vtheory/Vexp=0.93 Bulk­Modulus:­Btheory=47­GPa

­­­­­Experimentally­B=40-45­GPa

Theoretical­P(V)­using­LDA+DMFT­

Self-consistent­evaluations­of­total­energies­with­LDA+DMFT­.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 AmericiumLDA+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

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

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

Cluster Extensions of DMFT

Pu is not MAGNETIC, alpha and delta have comparable

susceptibility and specifi heat.

More important, one would like to be able to evaluate from the theory itself when the approximation is reliable!! And captures new fascinating aspects of the

immediate vecinity of the Mott transition in two dimensional systems…..

0

1 2

( , ) ( )( )(cos cos ) ( )(cos .cos ) .......

latt kkx ky kx ky

Cluster Extensions of Single Site DMFT

Some 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)

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)

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)

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

Anomalous Resistivity

PRL 91,061401 (2003)

• Approach the Mott transition, if the localized configuration has an OPEN shell the mass increases as the transition is approached.

Consistent theory, entropy increases monotonically as U Uc .

• Approach the Mott transition, if the localized configuration has a CLOSED shell. We have an apparent paradox. To approach the Mott transitions the bands have to narrow, but the insulator has not entropy.. SOLUTION: superconductivity intervenes.

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

???

•BACKUPS

C. Urano et. al. PRL 85, 1052 (2000)

Strong Correlation Anomalies cannot be understood within the standard model of solids, based on a RIGID BAND PICTURE,e.g.“Metallic “resistivities that rise without sign of saturation beyond the Mott limit, temperature dependence of the integrated optical weight up to high frequency

RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).

2

0( ) , ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

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).

Am under pressure. Lindbaum et.al. PRB 63,2141010(2001)

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

and Phys. Rev.B (2001) . 

1 †1 ( ) ( , ') ( ') ( ) ( ) ( ) 2

Cx 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 Spetral Density Functional of Gloc and Wloc.

Model Hamiltonians and Observables

† †

, ,

( )( )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 temperature

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?

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.

Further Approximations. o 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) .

o Truncate the W operator act on the H sector only. i.e.

• Replace W() by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g.

M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

( , ', ) ( ') ( ) ( )( ( ) ) ( ')dcxc R H R Rr r r r V r r E rabe a ab bw d f w fS = - - S S -

( , ', ) ( ) ( ) ( ) ( ') ( ')R H R R R RW r r r r W r rabgde a b abgd g dw ff wff=S

or the U matrix can be adjusted empirically.• At this point, the approximation can be derived

from a functional (Savrasov and Kotliar 2001)

• FURTHER APPROXIMATION, ignore charge self consistency, namely set

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also . A­Lichtenstein­and­M.­Katsnelson­PRB­57,­6884­(1988).

Reviews:Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. �McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65.

• Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428.

• Georges, A., 2004, Electronic Archive, .lanl.gov, condmat/ 0403123 .

loc[ ]G

[ ] [ ]LDAVxc Vxc

LDA+DMFT Self-Consistency loop

G0 G

Im p u rityS olver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w += å

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

DMFT

U

Edc

0( , , )HHi

HHi

n T G r r i e w

w

w += å

Realistic DMFT loop( )k LMTOt H k E® - LMTO

LL LH

HL HH

H HH

H Hé ùê ú=ê úë û

ki i Ow w®

10 niG i Ow e- = + - D

0 00 HH

é ùê úS =ê úSë û0 00 HH

é ùê úD=ê úDë û

0

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

110

1( ) ( )( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w

--é ùê ú= +Sê ú- - - Sê úë ûå

kj il ijklU Udd ®

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

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]2 | ' |

[ ]

R R

n

n KS

KS n ni

LDAext xc

DCR

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr Er r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -- S +

+ + +-F - F

åòò òå

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

1[ ] ( 1)2DC G Un nF = - ( )0( ) i

ababi

n T G i ew

w += å

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

Anomalous Resistivity

PRL 91,061401 (2003)

The Mott Transiton across the Actinides Series.

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltoniancluster exterior exteriorH H

Medium of free electrons :

impurity model.

Solve for the medium using

Self Consistency

G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)

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

Insulatinganion layer

-(ET)2X are across Mott transition

ET =

X-1

[(ET)2]+1conducting ET layer

t’t

modeled to triangular lattice

t’t

modeled to triangular lattice

Single-site DMFT as a zeroth order picture ?

Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, 226402. (2004))

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 t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

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

( , 0)Ek

kk

A k

Evolution of the k resolved Spectral Function at zero frequency. (QMC

study Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) ( 0, )vs k A k

Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

U/D=2 U/D=2.25

Momentum Space Differentiation the high

temperature story T/W=1/88

Actinies , role of Pu in the periodic table

CMDFT Studies of the Mott Transition

• cond-mat/0308577 [PRL, 92, 226402. (2004) ]• Cluster Dynamical Mean Field analysis of the Mott transition• : O. Parcollet, G. Biroli, G. Kotliar

• cond-mat/0411696 [abs, ps, pdf, other] :• Dynamical Breakup of the Fermi Surface in a doped Mott Insulator M. Civelli (1), M. Capone (2), S. S. Kancharla (3), O. Parcollet (4), G.

Kotliar • cond-mat/0502565 • Title: Short-Range Correlation Induced Pseudogap in Doped Mott

Insulators• B. Kyung, S. S. Kancharla, D. Sénéchal, A. -M. S. Tremblay, M.

Civelli, G. Kotliar

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.

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory+Perturbation

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

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

Mean-Field Classical vs Quantum

Quantum case

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

†

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( )( )[ ]

[ ]n

kn k

n

G ii t

G i

ww

w

D =D - -D

å

† †

, ,

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

t c c c c U n n

1( )] ( )( )[ ]

1( )[ ]( )]

[

[[ ]

n n nn

nk n n k

i i iG i

G ii i t

w m w ww

ww m w

+ - S =D - D

D = + - S -å

Phase Diag: Ni Se2-x Sx

Mott transition in systems with close shell.

• Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side.

• DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science 2002.

• Mechanism is relevant to Americium.

Mott transition in systems with close shell.

• Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side.

• DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science 2002.

• Mechanism is relevant to Americium.

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455