Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel...
-
date post
20-Dec-2015 -
Category
Documents
-
view
220 -
download
4
Transcript of Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel...
Correlation Effects in Itinerant Magnets : Towards a realistic
Dynamical Mean Field Approach
Gabriel Kotliar
Physics Department
Rutgers University
In Electronic Structure and Computational Magnetism
July 15-17 (2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outline
Dynamical Mean Field Theory: a tool for treating correlations in model Hamiltonians.
Towards Realistic implementations of DMFT.
Applications to Fe and Ni.
Conclusions and outlook.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Acknowledgements Collaborators and References:
A. Lichtenstein M. Katsnelson and G. Kotliar Phys. Rev Lett. 87, 067205 (2001).
I Yang S. Savrasov and G. Kotliar Phys. Rev. Lett. 87, 216405 (2001).
Useful Discussions K. Hathaway and G. Lonzarich
Support NSF and ONR
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Strong Correlation Problem
•Two limiting cases of the electronic structure problem are well understood. The high density limit ( spectrum of one particle excitations forms bands) and the low density limit (spectrum of atomic like excitations, Hubbard bands).•Correlated compounds: electrons in partially filled shells. Not close to the well understood limits . Non perturbative regime. Standard approaches (LDA, HF ) do not work well.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Motivations for going beyond density functional theory. DFT is a theory for ground state properties. Its
Kohn Sham spectra can be taken a starting point for perturbative (eg. GW ) calculations of the excitation spectra and transport.
This does not work for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. Character of the spectra (QP bands + Hubbard bands ) is not captured by LDA.
LDA –GGA is less accurate in determining some ground state properties in correlated materials.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT DMFT simplest many body technique which
describes correctly the open shell atomic limit and the band limit . Exact in the limit of large lattice coordination.
Band physics (i.e. kinetic energy) survive in the atomic limit (superexchange). Some aspects of atomic physics survive even in itinerant systems (J, U, Hubbard bands, satellites, L)
Computations of one electron spectra, transport properties…
Spectral density functional. Connects the one electron spectral function and the total energy.
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
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
( , ')o o oD n nt t ¯+
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
C-DMFT
C:DMFT The lattice self energy is inferred from the cluster self energy.
0 0cG G ab¾¾® c
abS ¾¾®Sij ijt tab¾¾®
Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc=2 CDMFT
vs Nc=1
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 Reviews: 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
From model hamiltonians to realistic calculations. DMFT as a method to be incorporated in electronic
structure calculations. Important in regimes where local moments are
present (e.g. NiO above its Neel temperature) Incorporate realistic structure and orbital
degeneracy information in many body studies. Combination of electronic structure(LDA,GGA,GW)
and many body methods (DMFT)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Interface with electronic structure.
Derive model hamiltonians, solve by DMFT
(or cluster extensions). Total energy? Full many body aproach, treat light electrons byt
GW or screend HF, heavy electrons by DMFT [GK and Chitra, GK and S. Savrasov, P.Sun and GK cond-matt 0205522]
Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Combining LDA and DMFT The light, SP electrons well described by LDA The heavier D electrons treat by model DMFT. LDA already contains an average interaction of the heavy
electrons, subtract this out by shifting the heavy level (double counting term, Lichtenstein et.al.)
Atomic physics parameters . U=F0 cost of double occupancy irrespectively of spin, J=F2+F4, Hunds energy favoring spin polarization .F2/F4=.6
Calculations of U, Edc, study as a function of these parameters.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Combine Dynamical Mean Field Theory with Electronic structure methods. Single site DMFT made correct qualitative
predictions. Make realistic by: Incorporating all the electrons. Add realistic orbital structure. U, J….. Add realistic crystal structure. Allow the atoms to move.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Two roads for ab-initio calculation of electronic structure of strongly correlated materials
Correlation Functions Total Energies etc.
Model Hamiltonian
Crystal structure +Atomic positions
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).
Lichtenstein and Katsenelson PRB (1998) Savrasov Kotliar and Abrahams Nature 410, 793
(2001)) Kotliar, Savrasov, in Kotliar, Savrasov, in New Theoretical New Theoretical approaches to strongly correlated systemsapproaches to strongly correlated systems , , Edited by A. Tsvelik, Kluwer Publishers, 2001)Edited by A. Tsvelik, Kluwer Publishers, 2001)
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 (Gunnarson and Anisimov, Mc Mahan et. Al. Hybertsen et.al) or viewed as parameters
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Density functional theory and Dynamical Mean Field Theory DFT: Static mean field, electrons in an
effective potential. Functional of the density.
DMFT: Promote the local (or a few quasilocal Greens functions or observables) to the basic quantities of the theory.
Express the free energy as a functional of those quasilocal quantities.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar).
DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]
Introduce local orbitals, R(r-R)orbitals, and local GFG(R,R)(i ) = The exact free energy can be expressed as a functional
of the local Greens function and of the density by introducing (r),G(R,R)(i)]
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
Full self consistent implementation.
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT-outer loop relax
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
U
Edc
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
ff &
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Realistic DMFT loop
( )k LMTOt H k E® -LMTO
LL LH
HL HH
H HH
H H
é ùê ú=ê úë û
ki i Ow w®
10 niG i Ow e- = + - D
0 0
0 HH
é ùê úS =ê úSë û
0 0
0 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
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
Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) phase of Pu: S. Savrasov et al (Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al
PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al
1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT
Developed initially to treat correlation effects in model Hamiltonians.
Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
Extension to realistic setting [V. Anisimov, A. Poteryaev, M. Korotin, Anokhin and G. Kotliar, J. Phys. Cond. Mat 9, 7359 (1997). S. Savrasov, G. Kotliar and E. Abrahams, Nature 410, 793 (2001). ] Lichtenstein and Katsnelson [Phys.Rev. B 57, 6884(1998) ]
Unlike DFT, DMFT computes both free energies and one electron (photoemission ) spectra and many other physical quantities at finite temperatures.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT Spectral Density Functional (Fukuda, Valiev and Fernando , Chitra and GK, Savrasov and GK).
DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the local density by Legendre transformation.
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 sources for (r) and G and performing a double Legendre transformton
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional
Formal construction of a functional of the d spectral density
DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.
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
ATOM 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
Atom =Sum of all local 2PI graphs build with local Coulomb interaction matrix, parametrized by Slater integrals F0, F2 and F4 and local G.Express in terms of AIM model.
KS [ ( ) G( ) V ( ) ( ) ]LDA DMFT a b abn nr i r i
( ) ( )G i iw w¾¾®D
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outer loop relax
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 + =
U
Edc
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
ff &
Impurity Solver
SCC
G,G0
DMFTLDA+U
Hartree-Fock
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Outer loop relax
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 + =
U
Edc
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
ff &
Impurity Solver
SCC
G,G0
DMFTLDA+U
Imp. Solver: Hartree-Fock
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
U
E
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT and LDA+U
• Static limit of the LDA+DMFT functional ,
• with = HF reduces to the LDA+U functionalof Anisimov et.al.
• Crude approximation. Reasonable in ordered situations.
, ( ) nab ab cd cdni U
( )0( ) iab ab
abi
n T G i ew
w+
= å
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT
If the self energy matrix is weakly k dependent is the physical self energy.
Since is a matrix, DMFT changes the shape of the Fermi surface
DMFT is absolutely necessary in the high temperature “local moment”regime. LDA+U with an effective U is OK at low energy.
DMFT is needed to describe spectra with QP and Hubbard bands or satellites.
( )ni
( )abni
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Applications of LDA+DMFT
Organics Alpha-Gamma Cerium V2O3 Volume collapse in Pu Photoemission of ruthenates Doping driven Mott transition in LaSrTiO3 Itinerant Ferromagnetism Bucky Balls
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Applications: Itinerant Ferromagnetism, Ni Fe
Compromise in the resources used for the solution of the one electron problem, and the many body problem.
Goal: obtain an overall approximate but consistent picture of how correlations affect physical properties. Estimate sensitivity on parameters.
Tc, spectra, susceptibility, [QMC- impurity solver] [ASA, relatively small number of k points]
Magnetic anisotropy [HF-impurity solver][full potential LMTO, large number of k points, non collinear magnetization]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Case study Fe and Ni
Archetypical itinerant ferromagnets
LSDA predicts correct low T moment
Band picture holds at low T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Iron and Nickel: crossover to a real space picture at high T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Other aspects that require to treate correlations beyond LDA
Magnetic anisotropy. L.S effect. LDA predicts the incorrect easy axis(100) for Nickel .(instead of the correct one (111) )
LDA Fermi surface in Nickel has features which are not seen in DeHaas Van Alphen ( G. Lonzarich)
Photoemission spectra of Ni : 6 ev satellite 30% band narrowing, reduction of exchange splitting.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT-QMC: Numerical Details
256 k points 105 - 106 QMC sweeps Analytic continuation via maximum entropy. Tight binding LMTO-ASA
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 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
Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)
2
0 3( )q
Meff
T Tc
c
T
T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni and Fe: theory vs exp
eff high T moment
Fe 3.1 (theory) 3.12 (expt)
Ni 1.5 (theory) 1.62 (expt)
Curie Temperature Tc
Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
MAEissmall(1eV/Atom)
Ni:2.8eV/Atomeasyaxis111Fe:1.4eV/Atomeasyaxis100LongstandingproblemEarlypapers
•VanVleck(PR1937)•Brooks(PR1940)
Magnetic anisotropy
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Trygget.al(1995);SCFTotalenergywithlarge#ofk-points;WrongeasyaxisforNi.
Otherrelatedworks:
Halilovetal.(1998)G.Schneideretal.(1997)Wangetal.(1993)Beidenet.al.(1998)
LDA calculations
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Full-potentialmultiplekappaLMTOmethod.
Paulitreatmentofrelativisticeffects.
Non-collinearintraatomicmagnetismincluded.
ExploredifferentEdc.(DetailsIYangPh.Dthesis)GeneralizedrelativisticLDA+Uwithoccupanciesn’
MethodMethod
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
WorkofTrygget.al.provesequivalenceofspecialpointsandtetrahedra.Confirmed.(broadening0.15Ry.)ConvergentEtotneeds15000k’s.Weuse28000k’s.Convergencycheckedto100000k’s.SUNE10Kwith64processorsused.LDAresultsofTrygget.al.reproduced:Ni0.5eV001,exp.2.8eV111,Fe0.5eV001,exp.1.4eV001.
Numerical Considerations
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
StudiesofMAEasfunctionofUandJ.
BothUandJinfluencemagneticmomentwhichisOKinLDA:0.6BforNiand2.2BforFe.
HowtofixmomentinLDA+U:
FindM(U,J)andtracepathforwhichmomentdoesnotchange.
LDA+U ResultsLDA+U Results
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
MagneticmomentasfunctionofUandJforNi
NN i - M(U,J)i - M(U,J)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
MagneticmomentasfunctionofUandJforFe
Fe - M(U,J)Fe - M(U,J)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
MAEasafunctionofU(J)
U=1.9eV,J=1.2eV
U=1.2eV,J=0.8eV
Ni
Fd
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
egformingX2pocket
eg
LDA vs LDA+U for NiLDA vs LDA+U for Ni
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni U=2,J=.1 PT (Katsenelson and Lichtenstein)cond-matt 2002
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusions Satellite in majority band at 6 ev, 30 % reduction
of bandwidth, exchange splitting reduction from band theory value (.6ev) to .3 ev
Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe. Single site should work for Ni, and overestimate Tc for Fe.
Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
OverallconsistentpictureoftheeffectsofcorrelationsonitinerantmagnetsusingDMFT.CanreproducecorrecteasyaxisandMAEofFeandNi.CancorrecttheFermisurfaceofNi.
Conclusions
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Work in progress
With existing techniques, derive practical formulae for the magnetic anisotropy of systems containing partially localized and itinerant electrons.
Further tests of DMFT on interesting materials.
Incorporate extensions of DMFT to incorporate frequency dependent interations (GW+DMFT) and to larger clusters.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
NochangesofFermisurfacefound
LDA and LDA+U bands for Fe
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDAelectronicstructureforNiE(k) for NiE(k) for Ni
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
CalculatedFermisurfaceforNiusingLDA+U.NoartificialX2pocket
Fermi Surface for Ni
Impurity Solver
SCC
G,G0
DMFTLDA+U
Hartree-Fock
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
GW+DMFT functional.
S. Savrasov and GK. P. Sun and GK. (cond matt).
Realistic Theories of Correlated Materials
ITP, Santa-Barbara
July 20 – December 20 (2002)
O.K. Andesen, A. Georges,
G. Kotliar, and A. Lichtenstein
http://www.itp.ucsb.edu/activities/future/
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, NRG,ED….)
•Analytical Methods
G0 G
Im puritySo lver
S .C .C .
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT
Construction is easily extended to states with broken translational spin and orbital order.
Large number of techniques for solving DMFT equations for a review see
A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Minimize LDA functional
[ ]( )( ) ( ) '
| ' | ( )
LDAxc
KS ext
ErV r V r dr
r r r
d rrdr
= + +-ò
0*2
( ) { )[ / 2 ]
( ) ( ) n
n
ikj kj kj
n KSkj
r f tri V
r r ew
w
r e yw
y +=
+Ñ -=å å
Kohn Sham eigenvalues, auxiliary quantities.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA functional
2log[ / 2 ] ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
n KS KS
LDAext xc
Tr i V V r r dr
r rV r r dr drdr E
r r
w r
r rr r
- +Ñ - -
+ +-
ò
ò ò
[ ( )]LDA r
[ ( ), ( )]LDA KSr V r
Conjugate field, VKS(r)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Double counting term (Lichtenstein et.al)
subtracts average correlation
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle. LDA predicts
the incorrect easy axis(100) for Nickel .(instead of the correct one (111)
LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)
Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )
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 Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]
†
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
Single site DMFT, functional of local Greens function G.
Express in terms of Weiss field (semicircularDOS)
[ , ] log[ ] ( ) ( ) [ ]ijn n nG Tr i t Tr i G i Gw w w-GS =- - S - S +F
† †,
2
2
[ , ] ( ) ( ) ( )†
( )[ ] [ ]
[ ]loc
imp
L f f f i i f i
imp
iF T F
t
F Log df dfe
[ ]DMFT atom ii
i
GF = Få Local self energy (Muller Hartman 89)
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 Coulomb interaction matrix and local G
1[ ] ( 1)
2DC G Un nF = - ( )0( ) iab ab
abi
n T G i ew
w+
= å
KS [ ( ) G( ) V ( ) ( ) ]LDA DMFT a b abn nr i r i
Calculated MAE for Ni and FeCalculated MAE for Ni and Feusing LDA+U methodusing LDA+U method
S. Y. SavrasovNew Jersey Institute of Technology
In collaboration with:Imseok Yang (Ph.D Thesis, RU)Gabriel Kotliar (RU)
Sponsored by Office of Naval Research
Grant No: ONR 4-2650Phys. Rev. Lett. 87, 216405 (2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
TotalEnergyDFTjobwithhugek-pointsummationproblem.
Eckardet.al(1987);Rightorder;WrongeasyaxisforFe.
Daalderlopet.al(1990);Forcetheorem;WrongeasyaxisforNi.
VaryingpositionofFermilevel,artificialX2pocketinfluenceseasyaxis.
Calculations
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment
Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)
eff high T moment
Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt)
Curie Temperature Tc
Fe 1900 5 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)
CorrelationsCorrelations
Many-bodyHubbardinteractionsareimportant(notcapturedbyLDA)
DMFT:onsitecorrelationsaretreatedexactly,bothatomicandbandlimitareOK.
StaticlimitofDMFT:LDA+Umethod:
Self-energy()->(static)
Solutionofimpuritymodelcollapsestodeterminationofn
Problemcanbesolvednow.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
The Strong Correlation ProblemTwo limiting cases of the electronic structure of
solids are understood:the high density limit and the limit of well separated atoms.
Many materials have electron states that are in between these two limiting situations and require the development of new electronic structure methods to predict some of its properties (spectra, energy, transport,….)
DMFT simplest many body technique which treats simultaneously the open shell atomic limit and the band limit .
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