Transcript Dynamical Mean Field Approach to the Electronic Structure

Towards a first Principles Electronic Structure
Method Based on Dynamical Mean Field
Theory
Gabriel Kotliar
Physics Department and Center for Materials
Theory Rutgers University
Montauk Long Island
September 13-17 2009
Outline
• Dynamical Mean Field Theory: Basic Ideas
• Dynamical Mean Field Theory and Electronic
Structure, LDA+ DMFT
• Illustrative Applications
Reviews:
G. Kotliar et. al. Reviews of Modern Physics
78, 865-951, (2006).
K. Held Advances in Physics 56, 829 (2007)
Correlated Electron Systems Pose Basic
Questions in CMT
• FROM ATOMS TO SOLIDS
• How to describe electron
from localized to itinerant
?
• How do the physical
properties evolve ?
• Spectra and Total Energies
DMFT Local Physics of a solid as atom in a medium


i , j ,
(tij   ij )(ci† c j  c†j ci )  U  nini
H Anderson Imp   (V c0† A +c.c). 
 ,
i
 A A  



†
,
c0† c0  Uc0†c0c0† c0
,
10
1
G(k , i) 
i  t (k )  (i)
Spectra=- Im G(k,)
iwn + m-
å
a
é
Va Va
= ê
w - ea êë
*
- 1
å
k
ù
1
ú + S (iwn ) [ea ,V a ]
iwn + m- t (k ) - S (iwn ) ú
û
Self consistency for V and 
• Simple extensions to phases with LRO
• Locality: simple extensions to cluster of
sites.
• Rapid advances in impurity solvers
Early Review: Georges Kotliar Krauth Rozenberg RMP 68, 13 (1996)
12
But how accurate is it ?
Important tests in Cold Atom
Traps
Cluster DMFT
S latt (k , w) = S 11 + (1/ 2)S 23 (cos kx + cos ky
.
+ (1/ 4)S 24 cos kx cos ky
Reviews:
T. Maier et. al. Rev. Mod. Phys. 77, 1027,
(2005).
G. Kotliar et. al. Rev. of Mod. Phys. 78,
865, (2006).
A.M Tremblay B. Kyung D. Senechal JLT
Phys. 32, 424-451 (2006)
(w + m- S latt (k , w))- 1 = ( w + m- S 11 )- 1
1
(w + m- S 23 )- 1 (cos kx + cos ky )
2
1
+ ( w + m- S 24 )- 1 cos kx cos ky
4
+
Cluster DMFT Difficulties
•2x2 cluster DMFT equations are considerably harder to
solve and to interpret than single site DMFT.
•Uniqueness: No unique formulation of cluster DMFT.
• Reconstruction of k dependence of quantities.
•Multiplicity of Solutions.
CDMFT vs BA in the 1D Hubbard Model
density n vs chemical potential μ
two site cluster
Gap vs U at half filling
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) ]
Outline
• Dynamical Mean Field Theory: Basic Idea
• Dynamical Mean Field Theory and Electronic
Structure and LDA+
•
•
•
•
DMFT
Applications to 3d Materials
Applications to 4f Materials
Applications to 5f Materials
Outlook
Functional formulation. Chitra and Kotliar Phys.
Rev. B 63, 115110 (2001) Ambladah et. alInt. Jour
Mod. Phys. B 13, 535 (1999) .
+
Ir>=|R, r>
1
f ( x)VC - 1 ( x, x ')f ( x ') +
2òò
Gloc r ' r = - < y ( Rr ') y † ( Rr ') >
ò if ( x)y
†
( x)y ( x)
< f ( R ' r ')f ( Rr ) > - < f ( R ' r ') > < f ( Rr ) > = W
1
1
[G,W ]  TrLnG  Tr[G01  G 1 ]G  TrLnW  Tr[VC1  W 1 ]W  Ehartree  [G ,W ]
2
2
[G,W ]   EDMFT [Gloc ,Wloc , Gnonloc  0,Wnonloc  0]
Double loop in Gloc and Wloc
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
EDMFT loop
Chitra and Kotliar Phys. Rev. B 63, 115110 (2001). G. Kotliar and S. Savrasov in New
Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer
Academic Publishers. 259-301 . cond-mat/0208241
S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)
Gloc  
k
Wloc  
q
1

  H (k ) 
Gloc ,Wloc 
G
1

1
VC ( q) 
Gloc ,Wloc 
W
•Full implementation in the context of a a one orbital lattice model. P Sun and G.
Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop one can treat the
graphs involving Gnonloc Wnonloc in perturbation theory. . Phys. Rev. Lett. 92,
196402 (2004)
Limiting case (perturbation theory as solvers) Zeyn and Antropov. N. E. Zein and
V. P. Antropov, J. Appl. Phys. 89, 7314 (2001), Phys. Rev. Lett. 89, 126402 (2002)
• Application to semiconductors N. Zeyn S. Savrasov and G. Kotliar PRL 96,
226403, 2006
LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A.
Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).
S
®
æ0
ç
ç
ç
è0
ö
÷
÷
+
÷
÷
- Edcø
0
S
ff
æ[Wloc ]spd ,sps [Wloc ]spd , f
Wloc (iw) = ççç
çè [Wloc ] f ,spd
[Wloc ] ff
æVxc[k ]spd ,sps
ç
ç
ç Vxc[k ] f ,spd
è
é0
ù
0
ö
÷
ê
ú
÷
®
÷
÷
ø
ê0 Uabcd ú
ë
û
Vxc[k ]spd , f ö
÷
÷
÷
÷
Vxc[k ] ff ø
U is parametrized in
terms of Slater
integrals F0 F2 F4 ….
Determine energy and and  self consistently from extremizing a functional : the spectral
density functional . Chitra and Kotliar (2001) . R. Chitra and G. Kotliar, Phys. Rev. B 63,
115110
(2001). Savrasov and Kotliar (2001) Full self consistent implementation . Review: Kotliar
et.al. RMP (2006)
Gdft[r ] ¾ ¾
® Glda +
t[Gloc , r ,U ]
dmf
12
Effective interaction among electrons.
Constrained RPA (cRPA) Ferdi Ariasetiwan ,A, M Imada, A
Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104
(2004)
Identity:
W  [1  vP]1 v,
P  Ploc  Prest
1
 [1  Wr Pd ] Wr
Wrest ( )  [1  vPrest ( )]1 v
Wrest ( )  U ( )
Wr
energy-dependent effective
interaction between the 3d electrons
Can be used to extract a screened U
LDA+DMFT functional
 LDA  DMFT [ r (r ) G a b V (r ) ab]
KS
- Tr log[iwn + Ñ 2 / 2 - VKS - c *a R ( r )S a b Rc b R ( r )] -
ò
VKS ( r )r ( r )dr -
ò
å
Vext ( r )r ( r )dr +
å
TrS (iwn )G (iwn ) +
iwn
1
2
ò
r ( r )r ( r ')
LDA
drdr '+ E xc
[r ] +
|r- r'|
F [Ga b R ] - F DC
R
 Sum of local 2PI graphs with local U
matrix and local G
F DC [G ] = Un(n - 1)
1
2
n= T
å (G
+
i0
ab (iw)e
)
abiw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT Self-Consistency loop
c ka | - Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
Edc
G
U

S.C.C.
DMFT
r (r) = T
å
iw
G ( r , r , i w) e
iw 0+
nHH = T
å
iw
+
GHH ( r , r , iw)eiw 0
Practical Matters
• Choice of the projector, in the simplest case
choice of orbital. (i.e. Projective LMTO’s )
• Basis in which to truncate the Kohn Sham
Hamiltonian.
c k a | - Ñ 2 + Vxc (r ) | c kb
• Implementation
of charge self consistency
•Impurity Solvers: slave bosons, NCA, OCA, CTQMC,
Hubbard I, etc. tradeoff between speed and accuracy.
• Choice of U and double counting.
Total Energy as a function of volume for Pu. Wrest(iw)
vs
(ev)
(a.u. 27.2 ev)
Pu
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
N, Zein , Following Aryasetiwan Imada
Georges Kotliar Bierman and
Lichtenstein. PRB 70 195104. (2004)
DMFT Phonons in fcc -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)
Main DMFT Concepts
Local Self Energies and Correlated Bands
1
G(k , i) 
i  H k  (i)
Weiss Weiss field, collective
hybridizationfunction, quantifies the
degree of localization
ab
D (w) =
å
a
V V
w - ea
Valence Histograms. Describes the
history of the “atom” in the solid,
multiplets!
Functionals of density and
spectra give total energies
Gdft[r ] ¾ ¾
® Glda +
[Gloc, r ,U ]
dmft
a *
b
a a
Qualitative Phase diagram :frustrated Hubbard model,
integer filling M. Rozenberg et.al. 75, 105 (1995)
CONCEPT:
(orbitally
resolved)
spectral
function.
Transfer of
spectral
weight.
CONCEPT:
Mott
transition.
T/W
DMFT view of Pu: adding
orbitals and coupling to
strucure to this “bare bones
phase diagram “
10
What is the valence in the late actinides ?
Plutonium has an unusual form of MIXED VALENCE
Finding the f occupancyTobin et. al. PRB 72, 085109 2005 K. Moore and G. VanDerLaan RMP
(2009). Shim et. al. Europhysics Lett (2009)
LDA results
Looking for moments. Pu under (negative ) pressure. C
Marianetti, K Haule GK and M. Fluss
Phys. Rev. Lett. 101, 056403 (2008)
Application to
Electron and Hole
Doped Cuprates :
Review: Armitage Fournier Green
(arXiv:0906.2931 )
Single Site vs 2 site CDMFT Phase
Diagram
Doping NCCO
.03 ev
.2 ev
N. L. Wang, G. Li, D. Wu, X. H. Chen, C. H.
Wang, and H. Ding, Phys. Rev. B 73, 184502
(2006). Y. Onose et al., Phys. Rev. B, 69, 024504
(2004)
Optical Spectral Weights. C. Weber et. al.
Not a very sensitive probe of the strength of correlations around the intermediate
correlation regime. Expt points :
Y. Onose et al., Phys. Rev. B, 69, 024504 (2004). S. Uchida et al., Phys. Rev. B 43, 7942
(1991).
Underdoped vs Overdoped T=0
Phys. Rev. B 74, 125110 (2006)
arXiv:cond-mat/0508302T.
Stanescu and G. Kotliar Phys. Rev.
B 74, 125110 (2006)
M. Civelli PRB 79,195113 (2009)
F. F. Balakirev et. al. arXiv.org:0710.4612
(2007).
Avoided Quantum Criticality : QCP under
the dome . arXiv:cond-mat/0605149K.
Haule and GK Phys. Rev. B 76, 092503
(2007). Coherence vanishes
underdoped
scattering
at Tc
optimally
overdoped
Real Space Picture
Singlet formation.
S,T N=2 singlet, triplet
E N=0
1+ states with 1 electron
in + orb
• Momentum Space Picture: High T
Underdoped region: arcs shrink as T is reduced. Overdoped
region FS sharpens as T is reduced.
Conclusion
• Dynamical Mean Field Theory:
Locality as a Basic Idea
• Dynamical Mean Field Theory
and Electronic Structure.
• Some Interesting Applications
• Many others taking place, many
groups working in this area all
over the world.
Thanks for your
Attention!!
Kinetic energy
Comparison of 2 and 4 sites