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