Transcript Spectral Density Functional: a first principles approach to the electronic

Spectral Density Functional: a first
principles approach to the electronic
structure of correlated solids
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
2001 JRCAT-CERC Workshop on
Phase Control on Correlated Electron
Systems
Outline
Motivation. Some universal
aspects of simple DMFT the
Mott transition endpoint in
frustrated systems.
 Non universal physics requires
detailed material modeling.
Combining DMFT and band
structure a new functional for
electronic structure calculations
(S. Savrasov and GK)
 Results: d electrons Fe and Ni.
(Lichtenstein, Katsenelson and
GK, PRL in press)

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Outline


Results: f electrons delta Pu (
S. Savrasov G. K and E.
Abrahams,Nature (2001))
Conclusions: further extensions
the approach.
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Importance of Mott phenomena
Evolution of the electronic structure
between the atomic limit and the
band limit. Basic solid state problem.
Solved by band theory when the
atoms have a closed shell. Mott’s
problem: Open shell situation.
The “”in between regime” is ubiquitous
central them in strongly correlated
systems. Some unorthodox examples
Fe, Ni, Pu.
Solution of this problem and advances
in electronic structure theory (LDA
+DMFT)
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A time-honored example:
Mott transition in V2O3 under
pressure
or chemical substitution on V-site
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Phase Diag: Ni Se2-x Sx
G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)
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Mott transition in layered organic
conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al.
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(2001)
Theoretical Approach to the Mott
endpoint.



DMFT.Mean field approach to quantum
many body systems, constructing
equivalent impurity models embedded
in a bath to be determined self
consistently . Use exact numerical
techniques (QMC, ED ) as well as
semianalytical (IPT) approaches to
solve this simplified problem.
Study simple model Hamiltonians
(such as the one band model on simple
lattices)
Understand the results physically in
terms of a Landau theory :certain high
temperature aspects are independent
of the details of the model and the
approximations used.
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DMFT Review: A. Georges, G.
Kotliar, W. Krauth and M. Rozenberg
Rev. Mod. Phys. 68,13 (1996)]

 (t
i , j  ,
ij
†
j i
i
b
S[Go] =
  ij )(c c j  c c )  U  ni ni
†
i
b
b
†
c
ò ò os (t )[Go(t , t ')]cos (t ') + ò no­ no¯
0
0
0
G0- 1 (iwn ) = iwn + m- D (iwn )
Weiss field
GL (iwn ) = ­ áco† (iwn )co (iwn )ñS (G0 )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
é
- 1
G0 (iwn ) = êê
êë
- 1
å
k
ù
1
ú + S (iwn )
iwn - tk + m- S (iwn ) ú
ú
û
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DMFT Review: A. Georges, G.
Kotliar, W. Krauth and M. Rozenberg
Rev. Mod. Phys. 68,13 (1996)]

 (t
i , j  ,
ij
†
j i
i
b
S[Go] =
  ij )(c c j  c c )  U  ni ni
†
i
b
b
†
c
ò ò os (t )[Go(t , t ')]cos (t ') + ò no­ no¯
0
0
0
G0- 1 (iwn ) = iwn + m- D (iwn )
Weiss field
GL (iwn ) = ­ áco† (iwn )co (iwn )ñS (G0 )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
é
- 1
G0 (iwn ) = êê
êë
- 1
å
k
ù
1
ú + S (iwn )
iwn - tk + m- S (iwn ) ú
ú
û
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Schematic DMFT phase diagram
one band Hubbard model (half
filling, semicircular DOS, role of
partial frustration) Rozenberg et.al
PRL (1995)
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Landau Functional
G. Kotliar EPJB (1999)
  Fimp
 (i ) 2
FLG []  T  
 Fimp []
2
t
 Lloc [ f † , f ] 
f† ( i )  ( i ) f ( i )
†
 ,
 Log[  df dfe
]

Lloc [ f † , f ]   [ f † [
0
d
 e f ] f  Uf † f  f † f  ]d
d
Mettalic Order Parameter: (i )
 Fimp
(i )
 T    f† (i ) f (i )   2TG (i )[]  2
(i )
t
Spin Model Analogy:
h2
 FLG [h]  
 Log[[ch]2  h]
2J
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Functional Approach




The Landau functional offers a
direct connection to the atomic
energies
Allows us to study states away
from the saddle points,
All the qualitative features of
the phase diagram, are simple
consequences of the non
analytic nature of the functional.
Mott transitions and
bifurcations of the functional .
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Insights into the Mott
phenomena
The Mott transition is
driven by transfer of
spectral weight from
low to high energy as
we approach the
localized phase
Control parameters:
doping,
temperature,pressure…
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A time-honored example:
Mott transition in V2O3 under
pressure
or chemical substitution on V-site
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Evolution of the Spectral
Function with Temperature
Anomalous transfer of spectral weight
connected to the proximity to an Ising Mott
endpoint (Kotliar et.al.PRL 84, 5180
(2000))
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Ising character of Mott
endpoint



Singular part of the Weiss field is
proportional to h a Max{ (p-pc) (TTc)}1/ 3 in mean field and 5 in 3d
h couples to all physical quantities
which then exhibit a kink at the Mott
endpoint. Resistivity, double
occupancy,photoemission intensity,
integrated optical spectral weight, etc.
Divergence of the the compressibility ,in
particle hole asymmetric situations,
e.g. Furukawa and Imada
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Compressibility
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Mott transition endpoint
Rapid variation has been observed
in optical measurements in
vanadium oxide and nises
mixtures
Experimental questions: width of
the critical region. Ising
exponents or classical
exponents, validity of mean field
theory
Building of coherence in other
strongly correlated electron
systems.
condensation of doubly occupied
sites and onset of coherence .
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Insights from DMFT: think in
term of spectral functions ,
the density is not changing!
Resistivity near the metal insulator
endpoint ( Rozenberg et.al 1995) exceeds
the Mott limit
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Anomalous Resistivity and
Mott transition Ni Se2-x Sx
Miyasaka and Tagaki (2000)
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.
ARPES measurements on NiS2-xSex
Matsuura et. Al Phys. Rev B 58 (1998) 3690
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Two Roads for first principles calculations of
correlated materials using DMFT.
Correlation
functions etc..
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Insights from DMFT
 Low temperatures
several competing
phases . Their relative
stability depends on
chemistry and crystal
structure (ordered
phases)
High temperature
behavior around Mott
endpoint, more
universal regime,
captured by simple
models treated within
DMFT
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LDA+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, substract this
out by shifting the heavy level
(double counting term)
The U matrix can be estimated
from first principles of viewed as
parameters

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r  (r ) *G (r , r ')(i )  (r ')
DMFT +LDA : effective action
construction (Fukuda, Valiev and
Fernando , Chitra and GK).


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. GDFT[r(r)]
Introduce local orbitals,  R(r ) andf local
Greens function by projecting onto the
local orbitals.G(R,R)(i ) =
 dr ' dr 

R
(r ) *G (r , r ')(i ) a R( r ')
The exact free energy can be
expressed as a functional of the local
Greens function and of the density by
introducing sources for r(r) and G and
performing a Legendre transformation.
G[r(r),G(R,R)(i)]
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LDA+DMFT

The functional can be built in
perturbation theory in the interaction
(well defined diagrammatic rules )The
functional can also be constructed
from the atomic limit.

DFT is useful because e good
approximations to the exact density
functional GDFT[r(r)] exist, e.g. LDA….
A useful approximation to the exact
functional can be constructed, the
DMFT +LDA functional.

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LDA+DMFT functional
Glda + dmft[r , VKS , G, S ]
= ­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ S (iwn , r , r ')] ­
òV
KS
ò
(r )r (r )dr -
å
TrS (iwn )G (iwn ) +
iwn
Vext (r )r (r )dr +
1
2
ò
r (r )r (r ')
drdr '+ ExcLDA [r ] +
| r- r'|
F [G ] - F DC [G ]
Sum of local 2PI graphs with
local U matrix and local G
Double counting correction
1
F DC [G] = Un(n - 1) , n = T
2
å (
+
Gab (iw)ei 0
abiw
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)
Spectral density functional
Connection with atomic limit
Sat [G01 ]    ca ( )G01 ( ,  ')ca ( ')   U abcd ca ( )cb ( )cc ( )cd ( )
ab
F [G] = Wat [D ] - Tr[D G] - Tr log G + Tr Gat- 1G
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LDA+DMFT Self-Consistency loop
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
G

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw 0+
iw
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Realistic DMFT loop
iw ® iwOk
tk ® H LMTO (k ) ­ E
é H LL
ê
êH HL
ë
H LH ù
ú= H LMTO
H HH ú
û
iG0- 1 = iwnO + e - D
é0
0 ù
ú
S=ê
ê0 S HH ú
ë
û
é0
0 ù
ú
D=ê
ê0 D HH ú
ë
û
S HH (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 )
é
G (iwn ) = êê
êë
- 1
0
å
k
ù -1
1
ú
+ S HH (iwn )
iwnOk - H LMTO (k ) - E - S (iwn ) ú
ú
ûHH
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LDA functional
Glda + dmft[r , VKS , G, S ]
= ­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ S (iwn , r , r ')] ­
òV
KS
ò
(r )r (r )dr -
å
TrS (iwn )G (iwn ) +
iwn
Vext (r )r (r )dr +
1
2
ò
r (r )r (r ')
drdr '+ ExcLDA [r ] +
| r- r'|
F [G ] - F DC [G ]
Double counting correction
1
F DC [G] = Un(n - 1) , n = T
2
å (
+
Gab (iw)ei 0
abiw
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)
LDA+DMFT
References



V. Anisimov, A. Poteryaev, M.
Korotin, A. Anokhin and G.
Kotliar, J. Phys. Cond. Mat. 35,
7359-7367 (1997).
A Lichtenstein and M.
Katsenelson Phys. Rev. B 57,
6884 (1988).
S. Savrasov and G.Kotliar,
funcional formulation for full self
consistent implementation
(2001)
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Iron and Nickel: band
picture at low T,
crossover to real
space picture at high T
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Photoemission Spectra and
Spin Autocorrelation: Fe
(U=2, J=.9ev) (Lichtenstein,
Katsenelson,GK prl in press)
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Photoemission and
Spin Autocorrelation:
Ni (U=3, J=.9 ev)
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Iron and Nickel:mgnetic
properties (Lichtenstein,
Katsenelson,GK cond-mat
0102297)
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Ni and Fe: theory vs exp



( T=.9 Tc)/ B
ordered moment
Fe 1.5 ( theory)
Ni .3
(theory)
eff / B
1.55 (expt)
.35 (expt)
high T moment
Fe 3.09 (theory) 3.12 (expt)
Ni 1.50 (theory) 1.62 (expt)
Curie Temperature Tc


Fe 1900
Ni 700
( theory)
(theory)
1043(expt)
631 (expt)
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Fe and Ni
Spin wave stiffness controls the
effects of spatial flucuations, it is
about twice as large in Ni and in
Fe
 Classical calculations using
measured exchange constants
(Kudrnovski Drachl PRB 2001)
Weiss mean field theory gives
right Tc for Ni but overestimates
Fe , RPA corrections reduce Tc of
Ni by 10% only but reduce Tc of
Fe by nearly factor of 2.

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Delocalization-Localization
across the actinide series
o
o
o
o
o
f electrons in Th Pr U Np are
itinerant . From Am on they are
localized. Pu is at the boundary.
Pu has a simple cubic fcc
structure,the  phase which is easily
stabilized over a wide region in the
T,p phase diagram.
The  phase is non magnetic.
Many LDA , GGA studies ( Soderlind
et. Al 1990, Kollar et.al 1997,
Boettger et.al 1998, Wills et.al. 1999)
give an equilibrium volume of the 
phase Is 35% lower than
experiment
This is one of the largest discrepancy
ever known in DFT based
calculations.
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Small amounts of Ga
stabilize the  phase
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Problems with LDA
o
o
o
o
o
DFT in the LDA or GGA is a well
established tool for the calculation of
ground state properties.
Many studies (Freeman, Koelling
1972)APW methods
ASA and FP-LMTO Soderlind et. Al
1990, Kollar et.al 1997, Boettger et.al
1998, Wills et.al. 1999) give
an equilibrium volume of the 
phase Is 35% lower than
experiment
This is the largest discrepancy ever
known in DFT based calculations.
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Problems with LDA



LSDA predicts magnetic long range
order which is not observed
experimentally (Solovyev et.al.)
If one treats the f electrons as part of
the core LDA overestimates the
volume by 30%
LDA predicts correctly the volume of
the a phase of Pu, when full potential
LMTO (Soderlind and Wills). This is
usually taken as an indication that a
Pu is a weakly correlated system
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Pu: DMFT total energy vs
Volume (S. Savrasov )
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DOS, st./[eV*cell]
Lda vs Exp Spectra
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Pu Spectra DMFT(Savrasov)
EXP (Arko et. Al)
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Conclusion

The character of the localization
delocalization in simple( Hubbard)
models within DMFT is now fully
understood. (Rutgers –ENS), nice
qualitative insights.
This has lead to extensions to more
realistic models, and a beginning of a
first principles approach interpolating
between atoms and bands.
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Conclusions




Systematic improvements, short
range correlations.
Take a cluster of sites, include
the effect of the rest in a G0
(renormalization of the quadratic
part of the effective action).
What to take for G0:
DCA (M. Jarrell et.al) , CDMFT
( Savrasov and GK )
include the effects of the
electrons to renormalize the
quartic part of the action (spin
spin , charge charge
correlations) E. DMFT (Kajueter
and GK, Si et.al)
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Conclusions
Extensions of DMFT implemented on
model systems, carry over to more
realistic framework. Better
determination of Tcs.
First principles approach: determination
of the Hubbard parameters, and the
double counting corrections long
range coulomb interactions E-DMFT
Improvement in the treatement of
multiplet effects in the impurity
solvers.
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