Transcript Dynamical Mean Field Theory and Electronic Structure Calculations Gabriel Kotliar

Dynamical Mean Field
Theory and Electronic
Structure Calculations
Gabriel Kotliar
Center for Materials Theory
Rutgers University
Outline
Physics Today Vol 57, 53 (2004) Gabriel
Kotliar and Dieter Vollhardt


Incorporating electronic structure methods
in DMFT. C-DMFT. [M. Capone, M. Civelli ]
Why do we need k-sum to do optics.
Cerium puzzles. [K. Haule V. Udovenko ]
Why do we need functionals to do total
energies. Phonons and plutonium puzzles.
[X. Dai S. Savrasov ]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Two roads for ab-initio
calculation of electronic
structure of strongly
correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
DMFT ideas can be used in both cases.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M.
Katsnelson PRB 57, 6884 (1988).



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,
substract this out by shifting the heavy level
(double counting term)
The U matrix can be estimated from first principles
of viewed as parameters. Solve resulting model
using DMFT.
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Single site DMFT Impurity cavity construction:
A. Georges, G. Kotliar, PRB 45, 6497 (1992)]


i , j  ,
(tij   ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
S [Go] =
- 1
0
G
b
b
òò
0
co†s (t )[Go(t , t ')]cos (t ') + U ò no­ no¯
0
Weiss field
0
(iwn ) = iwn + m- D (iwn )
GL (iwn ) = ­ áco† (iwn )co (iwn )ñS (G0 )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+ S (iwn )
1
EDMFT [H. Kajueter Rutgers Ph.D Thesis 1995 Si and Smith
PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]


i , j  ,
b
D0-
1
i
  Vij ni n j
i , j 
b
òò
0
(tij   ij )(ci† c j  c †j ci )  U  ni  ni 
co†s ( t )Go( t , t ')cos ( t ') + no­ no¯U d( t , t ')
+ Do(t , t ')n0n0
0
é
(iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
Vk - P (iwn ) ú
ú
û
+ P (iwn )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
P (iwn )[G0 ] = D0- 1 (iwn ) + [án0 (iwn )n0(iwn )ñS () ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
1
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
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RUTGERS
+ S (iwn )
Realistic DMFT loop: matrix
inversion-tetrahedron method
iw ® iwOk
é H LL
ê
êH HL
ë
tk ® H LMTO (k ) ­ E
iG0- 1 = iwnO + e - D
b
é0
0 ù
ú
S=ê
ê0 S HH ú
ë
û
é0
0 ù
ú
D=ê
ê0 D HH ú
ë
û
HH
b
ò ò c (t )G
†
a
0
H LH ù
ú= H LMTO
H HH ú
û
0
(t , t ')ab cb (t ') + U abdcca†cb† cc cd
0
S HH (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 )
é
G0- 1 (iwn ) = êê
êë
å
k
ù -1
1
ú
+ S HH (iwn )
iwnOk - H LMTO (k ) - E - S (iwn ) ú
ú
ûHH
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Site Cell. Cellular DMFT. C-DMFT.
G. Kotliar,S.. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) is the hopping expressed in the
superlattice notations.
•Other cluster extensions (DCA, Katsnelson and Lichtenstein
periodized scheme, nested cluster schemes, PCMDFT ),
causality issues, O. Parcollet, G. Biroli and GK cond-matt
0307587
(2003)
RUTGERS
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N vs mu in one dimensional Hubbard model .
Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats,
[M. Capone M.Civelli C. Castellani V Kancharla and GK 2004]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Two roads for ab-initio
calculation of electronic
structure of strongly
correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional : Effective action construction R. Chitra G.
Kotliar PRB 62,12715. Kotliar Savrasov in New Theoretical Approaches to
Strongly Correlated Systems, A. M. Tsvelik ed. (2001) Kluwer Academic
Publishers. 259-301; cond-mat/0208241. S Savrasov G Kotliar condmat0308053.





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, caR(r-R)orbitals, and local GF
G(R,R)(i w) =
 dr ' dr c
R
(r ) *G (r , r ')(iw ) ca 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)(iw)]
Allows computation of total energy, phonons!!!!
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RUTGERS
LDA+DMFT Self-Consistency loop. See also S.
Savrasov and G. Kotliar cond-matt 0308053
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
G
0
Impurity
Solver
E
G
U

S.C.C.
DMFT
r (r) = T
å
G( r, r, iw)e
iw0+
nHH = T
å
iw
iw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
GHH ( r , r , iw)eiw 0
Impurity Solvers.

Hubbard I.
 Quantum Montecarlo.

Rational Approximations to the self energy, constructed with
slave bosons. cond-mat/0401539 V. Oudovenko, K. Haule, S.
Savrasov D. Villani and G. Kotliar.

Extensions of NCA. Th. Pruschke and N. Grewe, Z. Phys. B:
Condens. Matter 74, 439, 1989. SUNCA K. Haule, S.
Kirchner, J. Kroha, and P. W¨olfle, Phys. Rev. B 64, 155111,
(2001).
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RUTGERS
Application to Materials


Cerium: Alpha to Gamma Transition.
Plutonium : Alpha-Delta-Epsilon.
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Overview
 Various phases :
isostructural phase transition (T=298K, P=0.7GPa)
 (fcc) phase
[ magnetic moment
(Curie-Wiess law) ]
 a (fcc) phase
[ loss of magnetic
moment (Pauli-para) ]
with large
volume collapse
v/v  15
( -phase a  5.16 Å
a-phase a  4.8 Å)
volumes exp.
28Å3
a
34.4Å3

  -phase
LDA
24.7Å3
LDA+U
35.2Å3
(localized):
High T phase
Curie-Weiss law (localized
magnetic moment),
Large lattice constant
Tk around 60-80K
 a-phase (delocalized:Kondo-physics):
Low T phase
Loss of Magnetism (Fermi
liquid Pauli susceptibility) completely screened magnetic
moment
smaller lattice constant
Tk around 1000-2000K
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Qualitative Ideas.



B. Johansson, Philos. Mag. 30, 469 (1974). Mott
transition of the f electrons as a function of
pressure. Ce alpha gamma transition. spd electrons
are spectators.
Mathematical implementation, “metallic phase” treat
spdf electrons by LDA, “insulating phase” put f
electron in the core.
J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49,
1106 (1982); Kondo volume collapse picture. The
dominant effect is the spd-f hybridization.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Qualitative Ideas


alpha phase Kondo effect between spd and
f takes place. “insulating phase” no Kondo
effect (low Kondo temperature).
Mathematical implementation, Anderson
impurity model in the suplemented with
elastic terms. (precursor of realistic DMFT
ideas, but without self consistency
condition). J.W. Allen and L.Z. Liu, Phys.
Rev. B 46, 5047 (1992).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT:Ce spectra
M.B.Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov
J. Keller,Phys.Rev. Lett 87, 276403
(2001).
K. Held, A.K. McMahan, and R.T. Scalettar,
Phys. Rev.Lett. 87, 276404 (2001)
A.K.McMahan,K.Held,andR.T.Scalettar,Phys
Rev. B 67, 075108 (2003).
Successful calculations of
thermodynamics.
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RUTGERS
Unfortunately photoemission cannot decide
between the Kondo collapse picture and the
Mott transition picture.
Evolution of the spectra as a function of U ,
half filling full frustration, Hubbard model!!!!
X.Zhang M. Rozenberg
G. Kotliar (PRL 70,
1666(1993)).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
The schematic phase diagram of
cannot distinguish between the
two scenarios.



J.W. Allen and L.Z. Liu, Phys. Rev.
B 46, 5047 (1992). Kondo impurity
model + elastic terms.
DMFT phase diagram of a Hubbard
model at integer filling, has a region
between Uc1(T) and Uc2(T) where
two solutions coexist. A. Georges G.
Kotliar W. Krauth and M Rozenberg
RMP 68,13,(1996).
Coupling the two solutions to the
lattice gives a phase diagram akin to
alpha gamma cerium. Majumdar and
Krishnamurthy PRL 73 (1994).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission&experiment
•A. Mc Mahan K Held and R. Scalettar (2002)
•Zoffl et. al (2002)
THE STATE UNIVERSITY OF NEW JERSEY
•K. Haule V. UdovenkoRUTGERS
S. Savrasov and GK. (2004)
To resolve the conflict between the
Mott transition and the volume collapse
picture : Turn to Optics! Haule et.al.



Qualitative idea. The spd electrons have
much larger velocities, so optics will be
much more senstive to their behavior.
See if they are simple spectators (Mott
transition picture ) or wether a Kondo
binding unbinding takes pace (Kondo
collapse picture).
General method, bulk probe.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Optics formula
double pole
One divergence integrated out!
single pole
THE STATE UNIVERSITY OF NEW JERSEY
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Temperature dependence of
the optical conductivity.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Theory: Haule et. al. cond-matt 04
Expt: J.W. vanderEb PRL 886,3407 (2001)
The volume of alpha is 28.06°A and the temperature 580K. The volume of the gamma phase is
34.37°A and T = 1160K.
Experiments : alpha at 5 K and gamma phase at 300 K
.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Optical conductivity of Ce (expt. Van Der Eb et.al.
theory Haule et.al)
experiment
LDA+DMFT
•K. Haule et.al.
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RUTGERS
Origin of the features.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Conclusion: Cerium




Qualitatively good agreement with existing
experiment.
Some quantitative disagreement, see
however .
Experiments should study the temperature
dependence of the optics.
Optics + Theory can provide a simple
resolution of the Mott vs K-Collapse
conundrum.
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RUTGERS
Phases of Pu (A. Lawson LANL)
Los Alamos Science 26, (2000)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Delta phase of Plutonium: Problems with LDA
o


Many studies and implementations.(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).all give an equilibrium
volume of the  phase Is 35% lower than
experiment this is the largest discrepancy ever
known in DFT based calculations.
LSDA predicts magnetic long range (Solovyev et.al.)
Experimentally  Pu is not magnetic.
If one treats the f electrons as part of the core LDA
overestimates the volume by 30%
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu: DMFT total energy vs Volume
(Savrasov Kotliar and Abrahams 2001)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT studies of Pu.




Savrasov, S. Y., and G. Kotliar, 2003, Phys.
Rev. Lett. 90(5), 056401/1.
Savrasov, S. Y., and G. Kotliar, 2003, condmat/0308053 .
Savrasov, S. Y., G. Kotliar, and E.
Abrahams, 2001, Nature 410, 793
Dai X. Savrasov S.Y. Kotliar G. Migliori A.
Letbetter H, Abrahams A. Science 300, 953,
(2003)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DFT Studies of Pu


DFT in GGA predicts correctly the volume of the a
phase of Pu, when full potential LMTO (Soderlind
Eriksson and Wills) is used. This is usually taken
as an indication that a Pu is a weakly correlated
system
The shear moduli in the delta phase were
calculated within LDA and GGA by Bouchet et. al.
(2000) and c’ is negative!
.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Evolution of the spectra as a function of
U , half filling full frustration.
X.Zhang M. Rozenberg
G. Kotliar (PRL 70,
1666(1993)).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Alpha and delta Pu : Expt. Arko et.al. PRB
62, 1773 (2000). DMFT: Savrasov and Kotliar
THE STATE UNIVERSITY OF NEW JERSEY
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Phonon freq (THz) vs q in delta Pu X.
Dai et. al. Science vol 300, 953, 2003
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Expts’ Wong et. al. Science 301.
1078 (2003) Theory Dai et. al.
Science 300, 953, (2003)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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?

Having a functional, that computes total
energies opens the way to the computation
of phonon frequencies in correlated
materials (S. Savrasov and G. Kotliar 2002)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Phases of Pu (A. Lawson LANL)
Los Alamos Science 26, (2000)
THE STATE UNIVERSITY OF NEW JERSEY
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Epsilon Plutonium.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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 Ackland et. al. PRB.65, 184104
(2002); (and neglecting electronic entropy). TC ~
600 K.
RUTGERS
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Phonons epsilon
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RUTGERS
Summary


Incorporating electronic structure methods
in DMFT. C-DMFT. [M. Capone, M. Civelli ]
Why do we need k-sum to do optics.
Cerium puzzles. [K. Haule V. Udovenko ]
Why do we need functionals to do total
energies. Phonons and plutonium puzzles.
[X. Dai S. Savrasov ]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Why is optics calculation not
completely trivial?
Analytic tetrahedron method:
Integral is analytic and simple
(combination of logarithms)
Energies linearly interpolated
no simple analytic expression
Product of two energies linearly interpolated
ATM applicable
but numerically very unstable because of quadratic pole
1D example:
Parabola has 2 zeros (2poles)
Line has no zeros (no poles)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
LDA+DMFT functional
G LDA  DMFT [ r (r ) G a b VKS(r )ab]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ c * a R ( r )S a b c 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 [G ] - F DC
R
F Sum of local 2PI graphs with local U
matrix and local G
F DC [G ] = Un(n - 1)
1
2
n= T
å (G
abiw
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
+
i0
ab (iw)e
)
Schematic DMFT phase diagram one band
Hubbard model. Rozenberg et. al. 1996.
Introduce coupling to the lattice will cause a
volume jump across the first order
transition. (Majumdar and Krishnamurthy ).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Shear anisotropy. Expt. vs Theory



C’=(C11-C12)/2 = 4.78 GPa
GPa
C44= 33.59 GPa
GPa
C44=33.0
C44/C’ ~ 7 Largest shear anisotropy in any
element!
C44/C’ ~ 8.4
RUTGERS
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
C’=3.9
Benchmarking SUNCA, V.
Udovenko and K. Haule
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RUTGERS