Transcript ISSP-Kashiwa 2001 Tokyo 1 -5 October

Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
ISSP-Kashiwa 2001
Tokyo 1st-5th October
the Mott phenomena
Evolution of the electronic structure between
the atomic limit and the band limit in an open
shell situation.
The “”in between regime” is ubiquitous central
them in strongly correlated systems, gives
rise to interesting physics.
New insights and new techniques from the
solution of the Mott transition problem within
dynamical mean field of simple model
Hamiltonians
Use the ideas and concepts that resulted from
this development to give physical insights into
real materials.
Steps taken to turn the technology developed to
solve the toy models into a practical
electronic structure method.
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Outline




Background: DMFT study of
the Mott transition in a toy
model. Behavior of the
compressibility near the Mott
transition endpoint.
DMFT as an electronic structure
method. From Lda to LDA+U to
LDA+ DMFT.
DMFT results for delta Pu, and
some qualitative insights into the
“Mott transition across the
actinide series”
Fe and Ni, a new look at the
classic itinerant ferromagnets
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Goal of the talk

Describe some recent steps
taken to make DMFT into an
electronic structure tool.

model Hamiltonian review see
A. Georges talk in this workshop
and consult reviews:

Prushke T. Jarrell M. and Freericks J.
Adv. Phys. 44,187 (1995)

A. Georges, G. Kotliar, W. Krauth and
M. Rozenberg Rev. Mod. Phys. 68,13
(1996)]
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Outline:





Choice of Basis.
Realistic self consistency
condition
Brief Comment on Impurity
Solvers
Integration with LDA. Effective
action formulation. Comparison
with LDA and LDA+U
Some examples in real materials,
transition metals and actinides.
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Acknowledgements:
Collaborators, Colleagues,
Support for realistic
work………….




S. Lichtenstein (Nijmeigen), E
Abrahams (Rutgers)
G. Biroli (Rutgers), R. Chitra (RutgersJussieux), V. Udovenko (Rutgers), S.
Savrasov (Rutgers-NJIT)
G. Palsson, I. Yang (Rutgers)
NSF, DOE and ONR
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DMFT Impurity cavity
construction: A. Georges, G. Kotliar,
PRB, (1992)]

 (t
i , j  ,
ij
  ij )(c c j  c c )  U  ni ni
†
i
i
b
S[Go] =
- 1
0
†
j i
b
b
†
c
ò ò os (t )[Go(t , t ')]cos (t ') +U ò no­ no¯
0
0
Weiss field
0
G (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 ) ]- 1
é
- 1
G0 (iwn ) = êê
êë
- 1
å
k
ù
1
ú + S (iwn )
iwn - tk + m- S (iwn ) úúû
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Good method to study
the 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.
Strategy, look electronic structure problems
where this physics is absolutely essential ,
Fe, Ni, Pu …………….
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Elements of the Dynamical Mean
Field Construction and C-DMFT.



Definition of the local degrees of
freedom
Expression of the Weiss field in
terms of the local variables (I.e.
the self consistency condition)
Expression of the lattice self
energy in terms of the cluster self
energy.
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Cellular DMFT : Basis selection. Exact
spectra is basis independent DMFT
results are not.
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Lattice action
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Elimination of the medium
variables
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Determination of the effective
medium.
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Connection between cluster and lattice
self energy.
The estimation of the lattice self
energy in terms of the cluster energy
has to be done using additional
information Ex. Translation invariance
•C-DMFT is manifestly causal: causal
impurity solvers result in causal self
energies and Green functions (GK S.
Savrasov G. Palsson and G. Biroli)
•Improved estimators for the lattice self
energy are available (Biroli and Kotliar)
•In simple cases C-DMFT converges
faster than other causal cluster
schemes.
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Convergence of CDMFT, test in a
soluble problem (G. Biroli and G.
Kotliar)
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Realistic DMFT self
consistency 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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Realistic implementation
of the self consistency
condition
é
G (iwn ) = êê
êë
- 1
0
å
k
ù -1
1
ú
+ S HH (iwn )
ú
iwnOk - H LMTO (k ) - E - S (iwn ) ú
ûHH
•H and S, do not
commute
•Need to do k sum for
each frequency
•DMFT implementation of
Lambin Vigneron
tetrahedron integration V.
Anisimov, A. Poteryaev, M. Korotin,
A. Anokhin and G. Kotliar, J. Phys.
Cond. Mat. 35, 7359-7367 (1997).
•Transport Coeff (G. Palsson V.
Udovenko and G. Kotliar)
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Solving the DMFT
equations
G
0
Impurity
Solver
G0
Impurity
Solver
G

S.C.C.
S.C.C.
•Wide variety of
computational tools (QMC,
NRG,ED….)
•Semi-analytical Methods
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G

DMFT+QMC (A.
Rozenberg)
Lichtenstein, M.
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Solving the impurity



Multiorbital situation and several
atoms per unit cell considerably
increase the size of the space H
(of heavy electrons).
QMC scales as [N(N-1)/2]^3 N
dimension of H
Fast interpolation schemes
(Slave Boson at low frequency,
Roth method at high frequency, +
1st mode coupling correction),
match at intermediate
frequencies. (Savrasov et.al
2001)
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Schematic DMFT phase diagram
one band Hubbard model (half
filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL
(1995)
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Recent QMC phase diagram of
the frustrated Half filled Hubbard
model with semicircular DOS (
Joo and Udovenko).
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Case study: IPT half filled
Hubbard one band

(Uc1)exact = 2.1 (Exact diag, Rozenberg,
Kajueter, Kotliar PRB 1996) , (Uc1)IPT =2.4

(Uc2)exact =2.97+_.05(Projective self consistent
method, Moeller Si Rozenberg Kotliar Fisher
PRL 1995 ) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg
Chitra and Kotliar PRL 1999), (TMIT )IPT =.5

(UMIT )exact =2.38 +- .03 (QMC Rozenberg
Chitra and Kotliar PRL 1991), (UMIT )IPT =2.5
For realistic studies errors due to other
sources (for example the value of U, are at
least of the same order of magnitude).

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Compressibility near a
Mott transition
Interaction driven Mott transition
Brinkman Rice . k ~ (Uc –U)
 Doping driven Mott transition
(Gutzwiller, Brinkman Rice, Slave
Boson method) . k is non
singular
 Numerical simulations T=0 QMC
, . k diverges
As 1/  (Furukawa and Imada)

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The Mott transition as
a bifurcation
At different points in the phase
diagram, different behaviors. k
vanishes at Uc2
(interaction driven Mott transition)
At zero temperature k is non
singular, at the doping driven
Mott transition
Behavior at UMIT TMIT ?

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
The Mott transition as
a bifurcation in
effective action
[G,  ]
[G,  ]
0
G
 [G, c ]
0
 G G
2
Zero mode with S=0 and p=0,
couples generically
Divergent compressibility
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Qualitative phase
diagram in the U, T , 
plane (Murthy Rozenberg
and Kotliar 2001)
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QMC calculationof n
vs  (Murthy Rozenberg
and Kotliar 2001, 2 band
model, U=3.0)
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QMC n vs  (Murthy
Rozenberg and Kotliar
2001, 2 band, low T
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Compresibility vs T
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Two Roads for calculations of
the electronic structure of
correlated materials
Crystal Structure
+atomic positions
Model
Hamiltonian
Correlation functions
Total energies etc.
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LDA[  (r )]
LDA functional
Conjugate field, VKS(r)
LDA[  (r ), VKS (r)]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ] ­
ò
1
Vext (r )r (r )dr +
2
ò
òV
KS
(r )r (r )dr
r (r )r (r ')
drdr '+ ExcLDA [r ]
| r- r'|
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Minimize LDA
functional
VKS (r ) = Vext (r ) +
e kj )y kj
*
ò
dExcLDA [r ]
r (r )
dr '+
| r- r'|
dr (r )
(r )y kj(r ) =
å e
wn
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iwn 0+
tr
[iwn +
LDA+U
functional
 LDA  U [  (r), m (r), n ab ]
 LDA  U [  (r), m (r), n a b, VKS(r), BKS(r),  ab]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ s .BKS ­
å
f
*
aR
(r )l abf bR (r )] ­
R
-
òV
KS
(r )r (r )dr 1
ò
ò V (r )r (r )dr + 2 ò
å F [G] - F
ext
BKS (r )m(r )dr -
å
Trl n +
iwn
r (r )r (r ')
drdr '+ ExcLDA [r ] +
| r- r '|
DC
R
1
   nabU abcd ncd
2
1
F DC [G ] = Un(n - 1)
2
n= T
å (G
ab
abiw
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(iw)e
i 0+
)
Double counting term
(Lichtenstein et.al)
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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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Spectral Density Functional :
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. DFT[(r)]
Introduce local orbitals, cR(r-R)orbitals, and
local GF
G(R,R)(i
dr ' w)
dr=c R (r ) *G (r , r ')(iw ) c 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)
and G and performing a Legendre
transformation, [(r),G(R,R)(iw)]
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Spectral Density
Functional

The exact functional can be built in
perturbation theory in the interaction
(well defined diagrammatic rules )The
functional can also be constructed
from the atomic limit, but no explicit
expression exists.

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.

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LDA+DMFT
functional
 LDA  DMFT [  (r ) m (r ) G a b VKS(r) BKS(r) ab]
­ Tr log[iwn + Ñ 2 / 2 ­ VKS ­ c *a R(r )S a bc b R(r )] ­
òV
KS
ò
å
(r )r (r )dr -
å
TrS (iwn )G (iwn ) +
iwn
1
Vext (r )r (r )dr +
2
ò
r (r )r (r ')
drdr '+ ExcLDA[r ] +
| r- r '|
F [G ] - F DC
R
 Sum of local 2PI graphs
with local U matrix and local G
1
F DC [G ] = Un(n - 1)
2
n= T
å (G
ab
abiw
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(iw)e
i 0+
)
Comments on
LDA+DMFT
•
•
•
•
Static limit of the LDA+DMFT
functional , with = HF reduces to
LDA+U
Removes inconsistencies of this
approach,
Only in the orbitally ordered
Hartree Fock limit, the Greens
function of the heavy electrons is
fully coherent
Gives the local spectra and the
total energy simultaneously,
treating QP and H bands on the
same footing.
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LDA+DMFT
Connection with atomic limit

c ( )G ( ,  ')ca ( ')   U

a
1
0
c ( )c

abcd a
ab
W
[

]


Log
e
at

G G 
1
0
1
at
 Wat
[[G]]  G

 Sat
Weiss field
F [G] = Wat [D ] - Tr[D G] - Tr log G + Tr Gat- 1G
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
b
LDA+DMFT Self-Consistency loop
E
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO (k )
Impurity
Solver
G
0
G
U

S.C.C.
DMFT
r (r) = T
å
G(r, r, iw)e
iw0+
nHH = T
å
iw
iw
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+
GHH (r , r , iw)eiw0
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+DMFT
References



V. Anisimov, A. Poteryaev, M.
Korotin, A. Anokhin and G. Kotliar,
J. Phys. Cond. Mat. 35, 73597367 (1997).
A Lichtenstein and M.
Katsenelson Phys. Rev. B 57,
6884 (1988).
S. Savrasov and G.Kotliar,
funcional formulation for full self
consistent implementation of a
spectral density functional( condmat 2001)
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Functional Approach




The functional approach offers a
direct connection to the atomic
energies. One is free to add
terms which vanish quadratically
at the saddle point.
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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Functional
Approach
G. Kotliar EPJB (
(iw )2
FLG []  T w 2  Fimp []
t
†
†
†  Lloc [ f , f ]  w , f ( iw )  ( iw ) f ( iw )
  Fimp  Log[ df dfe
]

d
Lloc [ f , f ]   [ f [  e f ] f  Uf† f f† f ]d
d
0
†
†
Mettalic Order Parameter: (iw )
 Fimp
(iw )
†
 T w  f (iw ) f (iw )   2TG (iw )[]  2
(iw )
t
Spin Model Analogy:
h2
 FLG [h]    Log[[ch]2  h]
2J
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Case study in f electrons,
Mott transition in the
actinide series
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Pu: Anomalous thermal
expansion (J. Smith LANL)
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Small amounts of Ga
stabilize the  phase
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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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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%
Notice however that LDA predicts
correctly the volume of the  phase of
Pu, when full potential LMTO
(Soderlind Eriksson and Wills). This is
usually taken as an indication that 
Pu is a weakly correlated system
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Conventional
viewpoint



Alpha Pu is a simple metal, it can
be described with LDA +
correction. In contrast delta Pu is
strongly correlated.
Constrained LDA approach
(Erickson, Wills, Balatzki,
Becker). In Alpha Pu, all the 5f
electrons are treated as band
like, while in Delta Pu, 4 5f
electrons are band-like while one
5f electron is deloclized.
Same situation in LDA + U
(Savrasov andGK Bouchet et. al.
[Bouchet’s talk]) .Delta Pu has
U=4,Alpha Pu has U =0.
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Problems with the
conventional viewpoint of
Pu




The specific heat of delta Pu, is
only twice as big as that of alpha
Pu.
The susceptibility of alpha Pu is
in fact larger than that of delta
Pu.
The resistivity of alpha Pu is
comparable to that of delta Pu.
Only the structural and elastic
properties are completely
different.
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Pu Specific Heat
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Anomalous Resistivity
J. Smith LANL
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MAGNETIC
SUSCEPTIBILITY
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Dynamical Mean Field
View of Pu
(Savrasov Kotliar and Abrahams,
Nature 2001)



Delta and Alpha Pu are both
strongly correlated, the DMFT
mean field free energy has a
double well structure, for the
same value of U. One where the f
electron is a bit more localized
(delta) than in the other (alpha).
Is the natural consequence of the
model hamiltonian phase diagram
once electronic structure is about
to vary.
This result resolves one of the
basic paradoxes in the physics of
Pu.
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Pu: DMFT total energy vs
Volume
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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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PU: ALPHA AND DELTA
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Case study Fe and Ni




Archetypical itinerant
ferromagnets
LSDA predicts correct low T
moment
Band picture holds at low T
Main challenge, finite T
properties (Lichtenstein’s talk).
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Iron and Nickel: crossover to
a real space picture at high T
(Lichtenstein, Katsnelson
and GK)
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However not everything
in low T phase is OK as
far as LDA goes..



Magnetic anisotropy puzzle. LDA
predicts the incorrect easy axis
for Nickel .(instead of 111)
LDA Fermi surface has features
which are not seen in DeHaas
Van Alphen ( Lonzarich)
Use LDA+ U to tackle these
refined issues, (WE cannot be
resolved with DMFT, compare
parameters with Lichtenstein’s )
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Some Earlier Work:
Kondorskii and E Straube Sov Phys.
JETP 36, 188 (1973)
G. H Dallderop P J Kelly M Schuurmans
Phys. Rev. B 41, 11919 (1990)
Trygg, Johansson Eriksson and Wills
Phys. Rev. Lett. 75 2871 (1995)
Schneider M Erickson and Jansen J.
Appl Phys. 81 3869 (1997)
I Solovyev, Lichenstein Terakura Phys.
Rev. Lett 80, 5758 (LDA+U +SO
Coupling)…….
Present work : Imseok Yang, S Savrasov
and GK
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Origin of Magnetic
Anisotropy





Spin orbit coupling L.S
L is a variable which is sensitive
to correlations, a reminder of the
atomic physics
Crystal fields quench L,
interactions enhance it,
T2g levels carry moment, eg
levels do not any redistribution of
these no matter how small will
affect L.
Both J and U matter !
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Magnetic anisotropy of Fe and Ni
LDA+ U
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Surprise correct Ni Fermi
Surface!
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Conclusion


The character of the localization
delocalization in simple(
Hubbard) models within DMFT
is now fully understood, nice
qualitative insights.
This has lead to extensions to
more realistic models, and a
beginning of a first principles
approach interpolating between
atoms and band, encouraging
results for simple elements
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DMFT Review: A. Georges, G.
Kotliar, W. Krauth and M. Rozenberg
Rev. Mod. Phys. 68,13 (1996)]

 (t
i , j  ,
b
S[Go] =
  ij )(c c j  c c )  U  ni ni
†
i
ij
†
j i
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 ) úúû
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Outlook




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 Palsson 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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Outlook
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, phonon entropies, ………
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Ni moment
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Fe moment
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Magnetic anisotropy
vs U , J=.95 Ni
1
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3
Magnetic anisotropy
Fe J=.8
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