Post on 24-Jan-2016
description
Dynamical Mean Field Approach to Strongly Correlated Electrons
Gabriel Kotliar
Physics Department andCenter for Materials Theory
Rutgers University
Field Theory and Statistical Mechanics
Rome 10-15 June (2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline Correlated Electrons and the Mott transition
problem. Dynamical Mean Field Theory. Cavity
construction. Effective action construction.[G Jona-Lasinio, Nuovo Cimento 34, (1964),
De Dominicis and Martin, Fukuda ] Model Hamiltonian Studies of the Mott
transition in frustrated systems. Universal aspects.
Application to itinerant ferromagnets: Fe,Ni. Outlook
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Standard model of solid
(Sommerfeld)
(Bloch )Periodic potential, waves form bands , k in Brillouin zone .
(Landau) Interactions renormalize away.
Justification: perturbative RG (Benfatto Gallavotti)
2 ( )F Fe k k l
h
The electron in a solid: wave picture
Consequences: Maximum metallic resistivity 200 ohm cm
2
2k
k
m
THE STATE UNIVERSITY OF NEW JERSEY
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
Ba
Think in real space , solid collection of atoms
High T : local moments, Low T spin-orbital order
1
T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Evolution of the spectra from localized to itinerant Low densities. Electron as particle bound to
atom. High densities. Electrons are waves spread
thru the crystal. Mott transition problem: evolution between
the two limits, in the open shell case. Non perturbative problem. Key to understanding many interesting
solids.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in V2O3 under pressure or chemical substitution on V-site
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Failure of the Standard Model: NiSe2-xSx
Miyasaka and Takagi (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Limit of large lattice coordination
1~ d ij nearest neighborsijt
d
† 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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Missing in this limit
Short Range Magnetic Correlations without magnetic order. Long wavelength modes.
Trust more in frustrated situations and at high temperatures.
2
1~ 0ij i j
j
J S S dd
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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 cavity construction A. Georges G. Kotliar 92
†
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
0
1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ
Weiss field
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
†( )( ) ( )
MFL o n o n HG c i c iw 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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Solving the DMFT equations
G 0 G
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, many models……….. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
G0 G
Im puritySo lver
S .C .C .
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Different Extensions
Take larger clusters in the cavity construction, e.g. cellular DMFT.[Kotliar Savrasov Palsson and Biroli], DCA[Jarrell and Krishnamurthy]
Take into account approximately the renormalization of the quartic coupling, e.g. extended DMFT. [Sachdev and Ye, Kajueter Kotliar, Si and Smith]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Single site DMFT, functional formulation. Construct a functional of the local Greens function
Expressed in terms of Weiss field (semicircularDOS) [G. Kotliar EBJB 99]
[ , ] log[ ]
( ) ( ) [ ]
DMFT ijn
n n atom ii
i
G Tr i t
Tr i G i G
w
w w
-G S =- - S
- S + Få
† †,
2
2
[ , ] ( ) ( ) ( )†
( )[ ] [ ]
[ ]loc
imp
L f f f i i f i
imp
iF T F
t
F Log df dfe
2
Ising analgoy
[ ] [ [2 ]]2LG
hF h Log ch h
J
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
C-DMFT functional formulation. Construct a functional of the restriction of the Greens function to the cluster and its supercell translations.
[ , ] log[ ]
( ) ( ) [ ]pcluster
CDMFT ijn
n n
i
G Tr i t
Tr i G i G
w
w w
-G S =- - S
- S + FåSigma and G are non zero on the
selected cluster and its supercell translations and are non zero otherwise.
Lattice quantities are inferred or projected out from the local quantities.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
C-DMFT: test in one dimension. (Bolech, Kancharla and Kotliar 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc=2 CDMFT
vs Nc=1
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Results: Schematic DMFT phase diagram Hubbard model (partial frustration)
THE STATE UNIVERSITY OF NEW JERSEY
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.
THE STATE UNIVERSITY OF NEW JERSEY
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Qualitative phase diagram in the U, T , plane,full frustration ( GK Murthy and Rozenberg 2002)
Shaded regions :the DMFT equations have a metallic-like and an insulating-like solution).
THE STATE UNIVERSITY OF NEW JERSEY
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).ForeshadowedbyCastellaniDiCastroFeinbergRanninger(1979).
THE STATE UNIVERSITY OF NEW JERSEY
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…
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
. ARPES measurements on NiS2-xSex
Matsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Spectral Weight Transfer: Optics
0( ) ,eff effd P J
iV
2
0( ) ,
ned P J
iV m
, ,H hamiltonian J electric current P polarization
, ,eff eff effH J PBelow energy
THE STATE UNIVERSITY OF NEW JERSEY
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi]
THE STATE UNIVERSITY OF NEW JERSEY
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 )
THE STATE UNIVERSITY OF NEW JERSEY
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.
THE STATE UNIVERSITY OF NEW JERSEY
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))
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional : effective action construction ( Chitra and GK PRB 2001).
DFT, exact free energy as a functional of an external potential. Legendre transform to obtain a functional of the density DFT(r)]. [Hohenberg and Kohn, Lieb, Fukuda]
Introduce local orbitals, R(r-R)orbitals, and local GF
G(R,R)(i ) = The exact free energy can be expressed as a functional
of the local Greens function and of the density by introducing (r),G(R,R)(i)]
A useful approximation to the exact functional can be constructed.
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
THE STATE UNIVERSITY OF NEW JERSEY
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT Self-Consistency loop
G0 G
Im puritySo lver
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
EdcU
E
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Case study Fe and Ni
Band picture holds at low T. LSDA predicts correct low T moment
At high temperatures has a Curie Weiss law with a (fluctuating) moment larger than the T=0 ordered moment.
Localization delocalization crossover as a function of T.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Iron and Nickel:magnetic properties (Lichtenstein, Katsnelson,GK PRL 01)
0 3( )q
Meff
T Tc
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni and Fe: theory vs exp / ordered moment
Fe 2.5 ( theory) 2.2(expt) Ni .6 (theory) .6(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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Fe and Ni Consistent picture of Fe (more localized) and Ni
(more itinerant but 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, twice as large in Ni and in Fe Cluster methods.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outlook Many open problems!
Strategy: advancing our understanding scale by scale.
New local physics in plaquettes.
Cluster methods to capture longer range magnetic correlations. New structures in k space. Cellular DMFT
Many applications to real materials.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT functional2 *log[ / 2 ( ) ( )]
( ) ( ) ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
[ ]
R R
n
n KS
KS n n
i
LDAext xc
DC
R
Tr i V r r
V r r dr Tr i G i
r rV r r dr drdr E
r r
G
a b ba
w
w c c
r w w
r rr r
- +Ñ - - S -
- S +
+ + +-
F - F
åò
ò òå
Sum of local 2PI graphs with local U matrix and local G
1[ ] ( 1)
2DC G Un nF = - ( )0( ) iab
abi
n T G i ew
w+
= å
KS ab [ ( ) G V ( ) ]LDA DMFT a br r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)