Magnetism: from Stoner to Hubbard - cond

Magnetism:
from Stoner to Hubbard
Alexander Lichtenstein
University of Hamburg
In collaboration with
A. Rubtsov, M. Katsnelson
Outline
—  Heisenberg, Stoner, and Hubbard
—  Many-body approach: D(M)FT functionals
—  Correlation effects in electronic structure
—  Magnetism of correlated systems
physicsworld.com
Itinerant ferromagnetism
John Hubbard: the man behind the model
Those who knew John Hubbard describe him as
a very shy man – to the point that others, who
did not know him so well, may have perceived
him as somewhat aloof. Born on 27 October
1931, Hubbard was educated first at Hampton
Grammar school and then at Imperial College,
London, where he obtained his PhD in 1958
under Stanley Raimes. Unusually for his time
and social context, he lived with his parents in
Teddington throughout his university education.
At the end of his PhD, Hubbard was recruited
to the Atomic Energy Research Establishment in
Harwell, Oxfordshire, by Brian Flowers, who was
then heading the theory division. An anecdote
from this period of Hubbard’s career illustrates
his retiring personality. While at Imperial,
Hubbard had dealt with the project assigned to
him for his PhD fairly quickly, and had then
looked for a more challenging problem. At the
time, quantum-field-theory methods, particularly
Feynman diagrams, were being applied to
problems in many-body theory. However, it was
difficult to bring the same methods to bear on
the many-electron problem – relevant to
solid-state systems – because the Coulomb
interaction between electrons made quantities
like the total energy diverge.
Hubbard realized that these divergences
could be controlled: the trick was to sum up an
infinite series of a particular class of Feynman
Stoner
T=0
T<Tc
T>Tc
he saw an article by other researchers who
Heisenbergithadupintroduced
aHubbard
different method to solve the
same problem. Hubbard had found their
method physically appealing, checked privately
that their results coincided with his, and
concluded there was no need for an additional
publication on the topic. Flowers then issued an
explicit order that Hubbard should publish his
groundbreaking work.
Hubbard’s most famous papers are the series
he wrote on his eponymous model, starting in
1963. He was not the only one working on the
strong-correlations problem: some months
earlier, Takeo Izuyama, working at Nagoya
University, and Duk-Joo Kim and Ryogo Kubo,
at the University of Tokyo, both in Japan, had
argued that a proper description of correlations
in metals with strong electron–electron
k
k
k
From Stoner to Hubbard
k
o discuss
the
different
models
ofband
magnetic
materials
(Fig.
1) awith
theasimplest
where
"
is
the
energy
band
spectrum
and
I
is
Stone
k
where
"
is
the
energy
spectrum
and
I
is
Stoner
interat
k
is the energy band spectrum and I is a Stoner interatomic exchange parameter.
ange
interaction
in athe
two-site
quantumdependent
model. magnetic properties
er Hamiltonian
this
case
temperature
this
the
temperature
magnetic
properties
areSton
rel
the temperature
dependent
magneticdependent
properties and
are
related
the
so-called
are
thecase
hopping
parameters
U with
the
characteri
X
+
s from
occupied
”spin-up”
band
to
unoccupied
”spin-down”
band,
reduce
th
excitations
from
occupied
”spin-up”
band
towhich
unoccupied
H
=
("
+
I
<
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>)c
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unoccupied
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s
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Stoner mode
easily
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Stoner
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just
E
k that
ean
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finally
atthe
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the
itinerant
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becom
F Cu
magnitude
the
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so
that
finally
at
the
e themagnitude
hopping parameters
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Coulomb
of the magnetization,
so that finallyHubbard
at the Curie
poin
netic
Mean Field
is
themetal.
energy
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model.
In
the
correlated
the
only
po
a
nonmagnetic
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n easily
realize
thatweakly
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a mean-field
approxim
a
nonmagnetic
metal.
I
m Hami
mpare
the
a standardproperties
Hubbard approach
withwith
the the
following
ΔhσStoner
= U < nmodel
he temperature
dependent
are related
so-called
S
−σ > nσwith magnetic
Ifcompare
we compare
the Stoner
model
with
a standard
Hubbar
del. IfInand
the
weakly
correlated
case,
the
onlya possible
magnetic
excitat
we
the
Stoner
model
with
standard
Hubbard
appro
nes,
the
corresponding
energy
is
of
the
order
s from occupied ”spin-up”X
band to unoccupied
”spin-down”
band,
which
reducI
X
+ is of the order I · M with M =< n
Hubbard
mode Hh = energy
tonian:
s,
and
the
corresponding
t
c
+the Curie
U ni" npoint
"bec(
tonian:
ij
i#
i cj at
e of the magnetization, so that finally
the itinerant
systemX
X
X
X
+For the o
arger
then
realistic
Curie
temperatures.
ij
i
+ t limit
ger
then
realistic
Curie
temperatures.
For
the
opposite
ofU+
strongl
H
=
c
c
U
h
ij
j
netic metal.
H
=
t
c
c
+
n
n
i
h
ij
j
i"
i#
i
U >> t
ij with the
i H
2
del
theat
half-filling
casea[1],
onecase
can
derive
anone
effective
m
pareatthe
Stoner
with
standard
Hubbard
approach
ij
ifollowing
model
the
half-filling
[1],
canHeisenberg
derive
an
tmodel
J = −2
U
Hh =
Heisenberg exchange
X
Hije =
exchange interactions Jij =
X
X+
U ni" ni#
~
~j .
JijiSi · S
tij c+
i cj
ij
2tij tji
U
He =
X
ij
~i · S
~j .
Jij S
are of the order of magnetic (Nee
Magnetism of H2: Heisenberg vs. Slater
Exchange interaction:
F
i
j
F
U
t
U
AF
i
AF
e2
J =< ij | | ji >
r12
j
Exchange interactions: Anderson
U
t
U
ED in subspace
N- =1,N+ =1
U>>t
Anderson kinetic exchange
Exchange: Local force approach
θ
U
t
Spectrum:
Exchange energy:
U
Spin-polarized LSDA
From Atom to Solids
Electrons in solids:
-Effective potential
-Bloch states
-Pauli principle
Density Functional Theory (DFT)
Effective one-particle states
Local Density Approximation (LDA)
⇥r )]
=>
n0 (⌅
<=
⇥
E[n(⌅r)] = ⇥ |H| ⇤ =Vextn(⌅
r)Vext
(⌅rr))d⌅
r 0+ VFext
[n(⌅
1 ⇥1
2
0
0
2 r
<E⇥1 <2 |H
| |H
|(H
H12|)|H⇤+⇥
=2E
+E2n
r)|n
)[V
(⌅rext
)E 0(⌅
V+
(⌅
rext
)]d⌅
⇥ 1 |2 |H
=
⇥
|H
|
⇤+⇥
)|
⇤
=
+
(⌅
r
)[V
r
)
V
(⌅
r
0 (⌅
2⇤
2 |H
22
2 ⇤+⇥
2=
1 |(H
22⇤
2
ext
ext
0
1 |=
2⇥0⇤ <
2
2
2
E
⇥
|
⇤
⇥
|H
|(H
H
⇤
=
n
(⌅
r
)
0
2
1
2
2
2
2
2
1
2
2
1
2
0
0
1
T [n]
+ U [n] )| 2 ⇤ = E2 + n0 (⌅r)[Vext (⌅r
E1 < ⇥ 2 |H1 | 2 ⇤ = ⇥ 2 |HF2 |[n]2=
⇤+⇥
2 |(H1 H2 ⇥
⇥
⇥
F [n] =1 T [n] + U [n]
0
02
2 ⇥2
1
0
0
1 r
0
0
E2 <E⇥2 <1 |H
| |H
|(H
H21n|)|0H
=1E
+E1n
r)|n
)[V
(⌅rext
)E (⌅
V+
(⌅
rext
)]d⌅
V
(⌅
r
<=
V
⇥ 2 |1 |H
=
⇥
|H
|
⇤+⇥
|(H
)|
⇤
=
+
(⌅
r
)[V
r
)
V
1⇤
1 |H
11
1 ⇤+⇥
1=
2=>
11)⇤
0 (⌅
1
ext
ext
2 |=
1⇥⇤ <
1
1
1
0
ext
ext
E
⇥
|
⇤
⇥
|H
⇤+⇥
|(H
H
⇤
=
n
r(⌅
)r
1
2 11
1
1
1 2
1
2
1
10
02(⌅
2
1
0
Vext
r1 ⇤+⇥
) <= 1V|(H
E2 < ⇥ 1 |H2 | 1 ⇤ =
⇥ =>
r)[Vext (⌅r
1 |H1n|0 (⌅
ext 2 H1 )| 1 ⇤ = E⇥
1 + n0 (⌅
0
0
0
0
0
0
1
0 <
0⇤+⇥
0 |(H
⇥ E0 + n (⌅
0 E1 0 H
02 )| 20⇤ =
E1⇥ +E
E
+
E
E
<
⇥
|H
|
r
)[V
(⌅
r)
Effective
one-electron
equation:
+
E
<
E
+
0
2 |H1 | Schrödinger-like
2⇤ =
2E
2
2
2
2
2
1
1
2
ext
1
2 0 2 E01 +1 E20 < E020 + E1
0
1
2
E1 < ⇥ 2 |H1 | 2 ⇤ = ⇥ 2 2 |H2 | 2 ⇤+⇥ 2 E
|(H
H22<
)| E22⇤ +
=E
E12 + n0 (⌅r⇥)[Vext (⌅r) Vext (⌅r
1 E
1 +
2
2
h̄ h̄2
0
2
h̄
2
⇥ E 0 + n (⌅
2
2r⇤)i 2
⌅
V
(⌅
r
))⇥
(⌅
r
)⌅
⇥
ri)⇥
(
E
<
⇥
|
|(H
H
)|
⇤
=
r
)[V
(⌅
r)
V
(⌅
r
(⌅
=
⇤
(⌅
r
)
ef f1h̄
i ))⇥
i (⌅
1 |H2 | (1 ⇤ = ⇥ ⌅1 |H
1f⇤+⇥
1=
1
1
0
ef
i
i
2
1
ext
V
(⌅
r
))⇥
(⌅
r
)
=
⇤
⇥
(⌅
r
)
(
ef f
i i 2
0
0i
1
2
2m
2m
E2 < ⇥ 1 |H2 | 1 ⇤ = ⇥ 1 |H1 | (1 ⇤+⇥ ⌅
Hf1(⌅
)|r))⇥
=r)E=1 +
⇤i ⇥in(⌅r0)(⌅r)[Vext (⌅r) Vext (⌅r
2m2 Vef
1 |(H
1 ⇤i (⌅
2m
0
0
0
N 0 N
N1
E
+
E
<
E
+
E
1
2
2
2
20
0=
0|⇥ (⌅
0
N
n(⌅rE
) n(⌅
r
)|
r
)
=
|⇥
(⌅
r
)|
i
Charge density:
+
E
<
E
+
E
i
n(⌅
r
|⇥
r)|2
1 2
2
2
1) =
2 i (⌅
h̄ i 2 n(⌅
|⇥i (⌅ri )|
i r) =
2(
⌅⇥ ⇥Vef f (⌅r))⇥
r) =
r)
h̄
i i (⌅
⇥ ⇤i ⇥i (⌅
22m
Vef f+
(⌅rr))⇥
(⌅r)V
)r)d⌅
=
⇤r+
⇥iE
(⌅rxc
)[n]
( T+[n]
⇥ r(⌅
ir
i)d⌅
E[n]E[n]
= Ts=
[n]
VH⌅
n(⌅
)V
(⌅
+[n]
VH+[n]
n(⌅
r
+
Er)V
ext
s2m
ext
xc [n]
E[n]
=
T
[n]
+
V
[n]
+
n(⌅
r)d⌅r + Exc [n]
s
ext (⌅
N H
Energy Functional:
E[n] = Ts [n] + VH [n] + n(⌅r2)Vext (⌅r)d⌅r + Exc [n]
N
|⇥
(⌅
e2 ⇥ e2 ⇥⇥n(⌅r)⇥=
n(⌅r)n(⌅
r2i)n(⌅
)⇥r)|r )⇥
n(⌅
2
e
n(⌅r)n(⌅r )
n(⌅rd⌅
) r= d⌅
i )|
VH [n]
d⌅
r|⇥
⇥[n]
i (⌅
2 d⌅
N
VH=[n] =
rH
d⌅
rr=
V
d⌅
r
r
2 2
|⌅r ⇥ |⌅
⌅r2r | ⌅r | h̄
2 |⌅
i
T
[n]
=
d⌅
r
⇥
(⌅
r
)(
⌅
)⇥ri (⌅r)⌅r |
s
i
KS-kinetic energy: E[n] = Ts [n] + VH⇥[n] + n(⌅r)Vext2m
(⌅
r
)d⌅
r
+ Exc [n]
i
E[n] = Ts [n] + VH [n]
r⇥)Vext⇥(⌅r)d⌅r⇥ + Exc [n]
⇥ + ⇥ n(⌅
n(⌅
[n]
2
⇥
Erxc
[n]
⇥
⇥ r ) Excn(⌅
2
2 e r )n(⌅
n(⌅
r) )
Exc [n]
)n(⌅
r
2
2
⇥
e
n(⌅
r
)n(⌅
r
)
V
(⌅
r
)
=
V
(⌅
r
)
+
e
d⌅
r
+
V
(⌅
r
)
=
V
(⌅
r
)
+
e
d⌅
r
+
ef
f
ext
ef
f
ext
⇥
⇥
V
(⌅
r
)
=
V
(⌅
r
)
+
e
d⌅
r
+
V
[n]
=
d⌅
r
d⌅
r
Hartree potential:
2Hf =
ef
VH e[n]
d⌅
r⌅rn(⌅
|⌅r2ext
| rd⌅
)rn(⌅
|⌅
⌅rr| rn(⌅
rr⌅)r⇥ | ⌅r |
)n(⌅
)r|⌅
|⌅
n(⌅r)
2
|⌅
r
⌅
r
|
VH [n] =
d⌅r d⌅r
E10
DFT: KS-equation (1965)
Effective potential:
2
|⌅r
⇥ ⇥
⌅r |
n(⌅
r⇥ ) ExcE[n]
xc [n]
⇥
n(⌅
r
)
2
⇥
V
(⌅
r
)
=
V
(⌅
r
)
+
e
d⌅
r
+
efrf) = Vext (⌅
ext
Vef f (⌅
r) + e n(⌅
d⌅
+⇥[n]
rr) |⌅r |⌅r⌅rE| xc
⌅
r
| n(⌅rn(⌅
2
) r)
Vef f (⌅r) = Vext (⌅r) + e
d⌅r
+
|⌅r ⌅r |
n(⌅r)
2
Computational Material Science: DFT
CERAN-plate
DFT-theory:
LiAlSiO4
A.L, R. O. Jones,
H. Xu,P. J. Heaney,
Phys. Rev. B 58, 6219 (1998
Correlation driven MIT
photoemission
spectra (DOS)
A. Fujimori et al.
U/W
Spectral function: Correlations effects
ARPES
Free electrons
Correlated electrons
Strongly Correlated Electron Systems
3d - 4f
Charge fluct.
open shells
materials
I
H
Li
Na
K
Rb
Cs
Fr
II
Be
Mg
Ca
Sr
Ba
Ra
•  Kondo
•  Mott-Hubbard
•  Heavy Fermions
•  High-Tc SC
•  Spin-charge order
•  Colossal MR
U<<W
U>>W
Spin fluct.
IIIb IVb Vb
Sc
Y
La *
Ac**
VIb VIIb
IIb
III
Cu
Ag
Au
Zn
Cd
Hg
B
C
Al
Si
Ga Ge
In Sn
Tl
Pb
Pm Sm Eu Gd Tb
Np Pu Am Cm Bk
Dy
Cf
Ho
Es
Ti
Zr
Hf
Rf
V
Nb
Ta
Db
Cr Mn
Mo Tc
W Re
Sg Bh
Lanthanides * Ce
Th
Actinides **
Pr
Pa
Nd
U
VIIIb
Fe
Ru
Os
Hs
Co
Rh
Ir
Mt
Ib
Ni
Pd
Pt
Control parameters
VII
N
P
As
Sb
Bi
O
S
Se
Te
Po
F
Cl
Br
I
At
Er Tm Yb
Fm Md No
Lu
Lr
•  Orbital
•  Lattice
La1-xCaxMnO3
Nd2-xCexCuO4
La2-xSrxCuO4
'Normal'
Metal
200
eu
Ps
100
AF
p
AF
ga
do
Temperature (K)
300
0
0.3
Dopant Concentration x
VI
•  Charge / Spin
•  Band filling
•  Dimensionality
FM
V
Degrees of freedom
•  Bandwidth (U/W)
CMR
IV
SC
0.2
0.1
0.0
0.1
SC
0.2
Dopant Concentration x
0.3
0
He
Ne
Ar
Kr
Xe
Rn
Hubbard model for correlated electrons
H = ∑ t ijci+σc jσ + U∑ n i↑ n i↓
ij
i
U
t
The Theory of Everything
Hamiltonian for multi-fermionic system in field-operators:
Atomic Units:
Coilomb interaction:
Second quantisation operators in orthonormal basis:
Wannier Basis:
with site, orbital andspins quantum numbers
where'0
e
Hoehnberg
and
~~i*(r)~~i
(r)dr, Kohn: DFT(2)
PHYSICAL REVIEW
NOVEMEBR 1964
I N HOMOGENEOUS
136,
VOLUM E
2
NUM B ER 3 8
EL
EleCtrOn Gaa*
InhOmOgeIIeouS
P.
9
HOHENBERGt
ELECTRON
HOMOGENEOUS
whe
and GAS
potential
an external
of
v(r)
C
under the
V= influence
v(r)i(*(r)P(r)dr,
W. KonNt
num
hasis a universa
st
where Pfn]
Coulomb
potential Hamiltonian
an external The
the mutual Ecole
of
the influencerepulsion.
v(r) and
under
I'aris,
France
number of particles" and
the mutual Coulomb repulsion. The Hamiltonian has
func
the form
at
La
San
Jolla,
of
functional plays a central
the form
18 June 1964)
H=
With its aid we define, W
Hamiltonian in
T+V+U,
H=
v
T+V+U,
where'0
state of P*(r)P*(r')
It energy functional
This
v(r).
drdr'
(r')
P (r)
that
that
of v(r),
a
F I (r)
where'0
enea
= fs(r)n (r)dr+Ft I (r) j
E—
its
correct
Field Operators:
I NFrance
Ecole Xornzale Superzeure, I'aris,
AND
Prance and I'aculte des Sciences, Orsay,
Xonnale Superieure,
and
University
Calzfo&nia
Diego,
(Received
Calzfornia
f
an interacting electron gas in an external potential
is
paper deals with the ground
the exthere exists universal functional of the density,
such
t
g, independent
has as
minimum value the
ground-state energy associated with
s(r). The functional FLn(r)j is then discussed for two situations: (1) n(r) @san(r), 8/ao((1, and
~~i*(r)~~i
~. In both cases F can be expressed entirely in terms (2)
of the cor(2) a(r) = q (r/ra) with p arbitrary and 1'p
2 order electronic polarizabilities of a uniform electron gas. This approach
relation energy and linear and higher
also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of
these methods are presented.
proved
pression
~
(r)dr,
E„gn]=—
ELECTRON GAS
MOGENEOUS
that
assume for simplicity
follows
We shall in all that
~~i*(r)~~i
for
the correct
(r)dr,
(2)
Clearly,
V= v(r)i(*(r)P(r)dr,
E.
state
in which
situations
ground
INTRODUCTION
universal
valid
forthe
functional,
2dealing
only
ntial v(r)weandare where
any
is a description
theoretical
considerations
ofenergy
this
Pfn] is awith
'
v
(
is(
We shall now show that
functional. Once known, it is relatively easy to deterthe last decade there has been considerable
&~ IJRINGnumber
particles"
This
and
of
potential.
miltonian state
has
anythe external
electronic
the
density
denote
in a given
ground-state
external
energy
is nondegenerate.
value
for the correct n(r)
the properties We
in understanding
of a mine
progress
drdr'
(r')
P*(r)P*(r')
(r)
P
'
f
potential.
The
of
homogeneous functional
interacting electron
gas.
point
are restricted by the condi
central role
in the present paper.
plays aelectrons
In Part II, we obtain an expression for FLnj when tr
in general, to state
as
has been,
regard the 0'
inviewthe
ground
by
Cle
i.e. , the
deviates
from uniformity,
n(r)=1'cp
slightly
for
a only
aid we define,
collection
potential
similar to a V=
noninteracting
particles
With ofits
given
v(r)i(*(r)P(r)dr,
v(r),
—0; In this that
case FLej is entirely
assume
with for
follows+ts(r),
ts/tss simplicity
in allofthat
We shall
collective
with the important additional
concept
—
=en
stat
Ãfm]
functional
the ground
exact ground-state energy
expressible in
excitations. energy we are only dealing with situations
in terms
whichof the
'
&
the exact electronic polarizability n(g) of a uniform
On the other hand, there has been in existence sinceWeand
electronic
the This
density
denote gas.
state
is nondegenerate.
electron
procedure will describe correctly
7920's
the
the
a different approach, represented by
0'
ell known
Thomas-Fermi method' in
in whichby the long-range Friedel charge oscillations' set isupv by
its ground
and the
re6nements,state
All
refinements
localized
a
perturbation.
previous
of
the
and in
the electronic density n(r) plays a central role=—
energy functional
v
which the system of electrons is pictured more like a Thomas-Fermi method have failed to include these.
In Part III we consider the case of a slowly varying,
classical liquid. This approach has been useful, up to
functionalbut of+ofv(r).
which is clearly ofa inhomonecessarily almost constant density, tr (r)
W
that fo
It
of
functional
a
is
which
clearly E„gn]
v(r).
4'
(2)
of valu
F
dr+
I
LN].
(10)
(r)
(r)
drdr'
P*(r)P*(r') (r') P (r) is a unique are
We shall now show that conversely v(r)
f
Path Integrals for Fermions
Short introduction from Alexei Kamenev
“Field Theory of Non-Equilibrium Systems” (Cambridge, 2011)
Fermions second-quantization operators (Pauli principle)
Algebra of Grassmann anti-commuting numbers:
Grassmann numbers anticommute with fermionic operators
Grassmann calculus
Differentiation:
N.B. order:
Example:
Integration:
(equivalent to differetition)
Coherent State
Eigenstate of annihilation operator
Diefinition of coherent states
Proof
Left Coherent State:
is just another Grassman number
(NOT a complex conjugate)
Unity operator in coherent states
Overlap of Coherent States (non-orthogonal)
Resolution of Unity
Proof
Trace of Fermionic Operators
Matrix elements of normally ordered operators
Trace-formula
''Minus'' due to commutation Left and Right coherent state
Gaussian Path Integrals
Only one analytical path integral:
Short notation
Proof - '‘det‘‘: expand the exponent only N-th oder is non-zero
Examples:
N=1
N=2
Correlation Function: U=0
Change of variables
Using:
Single-particle correlation function:
Two-particle correlation function:
Path Integral for Everything
Euclidean action
One- and two-electron matrix elements:
Shot notation:
One- and Two-particle Green Functions
One-particle Green function
1
2
Two-particle Green function (generalized susceptibilities)
Vertex function:
Baym-Kadanoff functional
Source term
Partition function and Free-energy:
Legendre transforming from J to G:
Decomposition into the single particle part and correlated part
=
Functional Family
Exact representation of Φ: Vαee=α Vee
Different Functionals and constrained field J:
G=ρ
G=G(iω)
G=G(k,iω)
G. Kotliar et. al. RMP (2006)
J=V=Vh+Vxc
J=Σloc(iω)
J=Σ(k,iω)
DFT
LDA+DMFT
GW++
Dynamical Mean Field Theory
BZ

1
ˆ k , iω
Gˆ (iωn ) = ∑
G
n
Ω k
(
G0 (τ − τ ʹ′)
Σ
Σ
Σ
U
Σ
Σ
Gˆ0−1 (iωn ) = Gˆ −1 (iωn ) + Σˆ (iωn )
G0 (τ − τ ʹ′)
Σ
QMC
Σ
Σ
)
Σ
W. Metzner and D. Vollhardt (1987)
A. Georges and G. Kotliar (1992)
ED
Single Impurity Solver
DMRG
IPT
FLEX
Σˆ new (iωn ) = Gˆ0−1 (iωn ) − Gˆ −1 (iωn )
DMFT: Charge+Spin+Orbital Fluctuations
DMFT
self-consistensy
DMFT
Impurity solver
Analogy with conventional MFT
Dieter Vollhardt
Metal-Insulator Transition
Dynamical Mean-Field Approach for Strongly Correlated Materials
1
1.18
Fig. 9: Mott-Hubbard MIT phase diagram showing the metallic phase and the insulating pha
respectively, at temperatures below the critical end point, as well as a coexistence region; fr
Ref. [54].
than linearly with the temperature, the difference ∆S = Smet −Sins eventually becomes posit
whereby the slope also becomes positive at lower temperatures;8 this is indeed observed
cluster DMFT calculations [60]. Since ∆S = 0 at T = 0 the phase boundary must terminat
T = 0 with infinite slope.
At half filling and for bipartite lattices in dimensions d > 2 (in d = 2 only at T = 0),
paramagnetic phase is unstable against antiferromagnetic long-range order. The metal-insula
transition is then completely hidden by the antiferromagnetic insulating phase, as shown
Fig. 10.
G. Kotliar and D. Vollhardt,
correlations
in materials
Physics
Today
3, 53 (2004)
spectral
function
(“density
of states”)6 ofElectronic
the Hubbard
model
in the
Strong correlation limit and Magnetism
PM
AFM
V. Janis, et al, EPL 24, 287 (1993)
U>W/2=4
P. Werner, et al, PRB 86, 205101 (2012)
 
Jττ ' Sτ ⋅ Sτ'
Formation of Local Moments and AFM correlations
What is the Mott transition?
a correlation driven metal-insulator transition
Mott ’49
σ
a
n ~ 1/a3 ~ t
cannot be obtained in band theory:
t
not due to AF (weak coupling effect):
2a
n
Comparison of LDA and realistic DMFT
arison of LDA and realistic DMFT schemes
LDA
LDA+DMFT
Density functional
Baym-Kadanoff functional
Density ⇢(r)
Green-Function G(r, r0 , !)
Potential Vxc (r)
Self-energy ⌃i (!)
Etot = Esp Edc
P
Esp = k<kF "k
R
Edc = EH + ⇢Vxc dr
⌦ = ⌦sp
⌦sp =
⌦dc
T r ln[ G 1 ]
Exc ⌦dc = T r⌃G
LW
DMFT model of ferromagnetism
DOS-peaks
D. Vollhardt, et. al.,
In:Bandferromagnetism,
Springer, 2000
Band degeneracy
W=4
U =6
V0=4
F0=2
Orbital degrees of freedom
•  e
g
orbitals
•  t
2g
orbitals
Mn (3+) = 3d4
3d-ion in cubic crystal field
5x
d
2x
eg
3x
t2g
J. van der Brink, D. Khomskii
Charge transfer TMO insulators
Zaanen-Sawatzky-Allen
(ZSA) phase diagram
Phys. Rev. Lett. 55, 418 (1985)
LDA+U: static mean-filed approximation
LDA+U functional:
U
U
E = E LDA + ∑ n i n j - n d (n d -1)
2 ij
2
∂E
1
εi =
=ε
+ U( − ni )
One-electron energies:
LDA
∂n i
2
U
n i = 1 ⇒ ε i = ε LDA −
Occupied states:
Mott2
Hubbard
U
gap
Empty states:
n i = 0 ⇒ ε i = ε LDA +
2
LDA
U≡
V. Anisimov et al, PRB, 44, 943 (1991)
review: J. Phys. Cond. Mat. 9, 767 (1997)
∂ε LDA
∂n d
Rotationally invariant LDA+U
LDA+U functional
Local screened Coulomb correlations (Orbital Polarization!)
LDA-double counting term (nσ =Tr(nmm0 σ ) and n=n- +n+ ):
Occupation matrix for correlated electrons:
Spin and Orbital moments in CoO
—  LDA+U+SO+non-collinear
S
L
h
h
L
I. Solovyev, A. L., and K. Terakura, PRL 80, 5758 (1998)
Electronic structure of TMO: LDA+U
DOS
MnO
MnO
12
8
NiO
NiO
LSDA
LSDA
U= 5e V
U= 5e V
Spin-wave
Spectrum
NiO
I. Solovyev
0
8
4
8
U= 9e V
U= 9e V
4
0
8
U= 1 3e V
U= 1 3e V
3d
0
4
0
-­‐12
-­‐8
-­‐4
0
Energ y (eV)
O2p
4
-­‐12
3d
-­‐8
-­‐4
0
Energ y (eV)
U=
1
3
LDA
5
400
0
w(q), meV
100 200 300
Density of Sta tes (sta tes/eV formula unit)
4
4
8
G
7
9
11
13
exp
Z
F
G
L
Slater: Magnetic Transition State
DFT: Janak theorem
DFT: Transition State
∂E[n]
εi =
∂ni
∂E[n]
ΔE = E[ni = 1]− E[ni = 0] = Δni
= ε (ni = 1 / 2)
∂ni n =1/2
i
Exchange interaction:
J = E[AF]− E[F] = Δε (n↑ = n↓ = 1 / 2)
ε↓ − ε↑
Local Force Theorem: Functionals
here
= 2 ⌃i ± ⌃i , ⌃i = ⌃i ei ,and
with
ei being the unit vector in0the dire
hence
i
1 is⌃an
ch
analog
of
the
”local
force
theorem”
inis an
the
density
fu
= 2 T r (Gij œ). We assume that the⇥ bare Green
function
G
does
which
analog
of
the
”loc
c
⇤
⇤
⇤
⌦
=
⌦
=
.ctive
Spin
excitations
with
low
energies
are
connected
with
the
rT
spin-dependent
potential
on
site
i,
=
(
,
,
)
are
Pauli
matrices,
G
=
sp
x
y
z
s with low energies
are
connected
with
⇤
⇤
1 the rotations ofijve
⌦ = scheme,
⌦sp = the
T rself
ln including
⌃energy
G0 the
the
LDA+DMFT
t
and
terms
Hartree-Fock
term
s all 1the spin-dependent
0
e
LDA+DMFT
isIn
local,
i.e.
isscheme,
diago
Magnetic
Torque
nd Gij = 2 T r (Gij œ).
We assume that the
bare
Green
function
G
does
not
depe
which is an analogthe
of spin-matrix
the ”local force
theorem”
structure
of thin
energy.
Spin
excitations
low
are
connected
with
the
rections
and
all structure
the
spin-dependent
including
the
Hartree-Fock
terms
ini
the
LDA+DMFT
scheme,
the self
energyare
is loc
pin-matrix
ofwith
theinterms
self
energy
and
Green
function
log
of the
”local
force
theorem”
theInenergies
density
functional
theory
[57].
'
e
=
eare
the
self
energy.
Spin
excitations
with
low
energies
connected
with
rotation
i
i structure
⌃G
the
spin-matrix
of the
self the
energy
iis⇥diagonal
i
'
ei the
= self ienergy
⇥ ei is local, i.e.
MFT scheme,
in site
indices.
Letandus
:
⇣ s
⌘
s
c
c
c
s
(c,s)
"
#
structure of the self energy and
Green
function
in
the
following
form
1
'
e⌃
e,i where
⌃
==⌃i ++
⌃i iG
,⌃Gi ij
i =
⌃=
⌃
±of
i ⇥
⌃
+
G
i i G
”local force theorem” ⌃
(47)
corresponding
variation
2
i =the
ij
i
i
ij
ij th
ei = ' i ⇥ ei
orem” (47) the corresponding variation
of
the
⇣fective
⌘ thermodyn
spin-dependent
potenti
(c,s)
"
#
1
s
s
s
c
c
s
as⌃i ”local
where
⌃G
= 2 and
⌃i G
±s⌃=
⌃
⌃iœ).
ei , with
gwritten
to the ”local
the
corresponding
of
i =
i
i ,1variation
=force
⌃i force
+theorem”
⌃ itheorem”
, G
=
G
+corresponding
⇣
⌘ij (47)
T
r
(G
We t
ij
ccording
to the
(47)
the
variation
of
the
therm
ij
ij
ij
2
(c,s)
"
#
1
spin-dependent
potential
on
=( x
s
⇤ sfective
directions
and
allsite
thei,the
spin-dep
re
⌃
=
⌃
±
⌃
,
⌃
=
⌃
e
,
with
e
being
un
an⇣ bei can
written
as
otential
be⌘written
as
i
⌦
=
⌦
=
1i i ' i
i
i
s V
i
i
sp
2
and Gij = 2 T r (G
œ).self
Weenergy.
assumeSpin
that excitat
the bar
inijthe
"
# ⇤ s
1
s
⇤
⌦ = ,⌃
⌦isp
'⌦i⌦e=i=being
Theorem:
⇤⌦spthe
⌃Magnetic
==
⌃i V
ei ,iwith
unit
vector
in the direction
o
==
V
'i ithe
i ± ⌃i Force
i all
directions
and
spin-dependent
terms
inclu
2
⌦
V
'
e
:
sp
ve spin-dependent potential on site i, =i (i x , y , c z ) are
P
1
in, the
self
energy.
Spin
excitations
with=lowT
energ
indextorque
is
equal
to
endent
potential
on
site
i,
=
(
,
)
are
Pauli
matrices,
G
r
x
y
z
ij
2
1
s
here the
torque
indextorque
is We
equalassume
to e : that the bare Green functi
G
=
T
r
(G
œ).
i
ij
equal
to
ijijMagnetic
torque
is equal
to Green
2We Torque:
(G
œ).indextorque
assume that
the bare
function According
G0 doestonot
depend on
the ”localeiforce
= '
s
s
s
ctions
and all the spin-dependent
Hartr
ll the spin-dependent
terms
Hartree-Fock
terms
are
incorpo
⌃⇥
V
2T rthe
[⌃
⇥G
Gincluding
]]
potential
can bethe
written
as
!L
⌃terms
Vi =including
2Ti =r!L
[⌃
i
ii
i
ii
s
s force theorem” (47) the c
According
to the
”local
slow energies
s= 2T
⌃
V
r
[⌃
⇥
G
y.
Spin
excitations
with
are
connected
with
of ve
i
!L
i energies
ii ] the rotations
e self
Spin
with can
low
are
connec
⌃ iexcitations
Vi energy.
= 2T r!L
[⌃
⇥ Gii ] potential
be written as
where the torque indextorque
⌦ = ⇤ ⌦sp
ei = ' i ⇥ ei
where the torque indextorque is equal to
center.
near
magnetic structures in the following form
In the nonrelativistic case one can rewrite
the spin Hamiltonian
for small spin de
⇣
⌘
# Functional
s "from
s X
Exchange
interactions
collinear magneticJstructures
in
the
following
form
T r!L ⌦
⌃i Gij=
⌃j Gji J e · e
ij =
spin
ij i
j
X
⌦spin = generalize
Jijijei · ethe
Heisenberg
exchagne:
j
e exchange
parameters.
This formula
LDA expression
ij
related
re wheresystems.
⇣
⌘
⇣
⌘
"
#both direc
exspinExchange
wave spectrum in ferromagnets
can
be considered
s "⌃ ssG# ⌃ s G
J
=
T
r
Jijij= T r!L !L
⌃i Gij ⌃ij Gjiij j ji
interactions:
meters or by the consideration of the energy of corresponding spiral s
are
the effective
exchangeparameters.
parameters. This
formula
generalize
the LDA expressio
he
effective
exchange
This
formula
generalize
the LD
onrelativistic
case
when
the
anisotropy
is
absent
one
has
theSpin
case ofwave
correlated
systems.
spectrum:
caseSpin
of wave
correlated
systems.
indexspin
wave spectrum in ferromagnets
X
4
4 can be considered both dire
!indexspin
Jor0jbyspectrum
1the consideration
cos qR
⌘the energy
[J(0)of can
J(q)]
n wave
wave
inj ferromagnets
be consider
exchange
of
corresponding
spiral
q =parameters
M j
M
Ref.parameters
[57]). In nonrelativistic
when the anisotropy
is absent
oneof
hascorrespo
hange
or by thecase
consideration
of the
energy
Non-collinear excitation:
X
4
4 is ion.
[57]).
In
nonrelativistic
case
when
the
anisotropy
e magnetic moment (in
Bohr
magnetons)
per
magnetic
!q =
J0j 1 cos qRj ⌘
[J(0) absent
J(q)] one ha
M j
M
ted that the expression for4 spin
Xstiffness tensor D↵ defined
4 by the re
!q = moment
cos
qR
⌘as the[J(0)
J
M is of
the exchange
magnetic
(inJ0j
Bohr1has
magnetons)
per
ion.
0)where
in terms
parameters
to
be
exact
conseque
j magnetic
M j for spin stiffness tensor D Mdefined by the r
It should
be
noted
that
the
expression
↵
M. Katsnelson
and A. L. , which
Phys. Rev.
8906 (2000)
LandauLifshitz equations
are61,
definitely
correct
in the long-
Exchange interactions and Band structure
3d in Ni
Fe, Co, Ni
Ni
J0
Co
Fe
Green Function calculations : J0=Σ Jij
V. Antropov, et. Al. PRB (1988)
LDA+Exchange Interactions
Spin-waves T<Tc
εF
J ij = π1
i
i
ij
j
j
ji
(
V
−
V
)
G
(
V
−
V
)
G
∫ ↑ ↓ ↑ ↑ ↓ ↑ dε
−∞
Curie temperature
Tc =
2
3
∑J
0j
j
S. Halilov, et. al.,
PRB 58, 293 (1998)
Ni
Fe
Co
LDA+Disordered Local Moments
DLM
The best first-principle
Spin-fluctuation model
with classical moments
EXP
DLM
J. Staunton and B. Gyorffy
PRL69, 371 (1992)
EXP
Orbital order: KCuF3
In KCuF3 Cu+2 ion has
d9 configuration
with a single hole in eg doubly degenerate subshell.
Experimental crystal structure
LDA+U calculations for undistorted
perovskite structure
hole density of the same symmetry
antiferro-orbital order
A.L. V. Anisimov and J. Zaanen, Phys. Rev.B 52, R5467 (1995)
1d-AFM in KCuF3
J(K)
Jc Jab
Theory -240 +6
Exp.
-202 +3
KCuF3
Superexchange interaction
LSDA gave cubic perovskite crystal
structure stable in respect to JahnTeller distortion of CuF6 octahedra
LDA+U produces
total energy minimum
for distorted structure
Quadrupolar distortion in KCuF3
Magnetism
5.25
Exchange in Iron: LSDA++
500
Fe spin-wave
Energy, meV
400
300
LSDA
LDA+ (E)
200
100
Exp
0
0,0
0,5
Wave vector (001)
1,0
H
Quantum Impurity Solver
Σ
Σ
For a general N × N super-site impurity model (simp) the partition function can b
functional integral over the 2N-component spin and site-dependent spinor Grassm
and c :
'
Z = D[c∗, c]e−Ssimp ,
Σ
where
Σ
Σ
U
Σ
Σ
Σ
Ssimp = −
N '
(
I,J=0 0
+
β
dτ
'
β
dτ # c∗Iσ (τ)
0
N '
(
) −1
*
#
Gσ (τ − τ ) IJ cJσ (τ # )
β
dτUnI,↑ (τ)nI,↓(τ),
I=1 0
τ’
U
where G is the N × N matrix of effective bath Green’s function for a spin-colline
The main problem of all cluster extension of DMFT is to find an optimal self-cons
best scheme?
obtain the bath Green’s functionWhat
matrixisinaimaginary
time GIJ (τ − τ # ) or in Ma
Quantum Monte Carlo !
GIJ (iω). In the free-cluster version of the CDMFT scheme [6] which is equivalent
DMFT method [8] or to the molecular CPA scheme in alloy theory [9] we can use
prescription. First, we need to integrate out the superlattice degrees of freedom, si
G(τ−τ’)
τ
Continuous Time Quantum Monte Carlo
Partition function:
Continuous Time Quantum Monte Carlo (CT-QMC)
E. Gull, A. Millis, A.L., A. Rubtsov, M. Troyer, Ph. Werner, Rev. Mod. Phys. 83, 349 (2011)
Weak coupling QMC: CT-INT
A. Rubtsov, 2004
Random walks in the k-space
Z=… Zk-1 + Zk + Zk+1+ ….
k-1
k+1
Acceptance ratio
increase
decrease
k D k −1
k
w D
Step k+1
D is tribution
Step k-1
0
0
20
Maximum at βUN
2
40
k
60
w D k +1
k + 1 Dk
Convergence with Temperature: CT-INT
Maximum:
βUN
2
Strong-Coupling Expansion CT-HYB
P. Werner, 2006
Strong-Coupling Expansion CT-HYB
P. Werner, 2006
Comparison of different CT-QMC
Σ
Σ
Σ
Σ
U
Σ
Σ
Σ
Σ
U
τ
τ’
G(τ−τ’)
Ch. Jung, unpublished
CT-QMC review:E. Gull et al. RMP (2011)
Magnetism vs. Kondo resonance
#1
Three impurity atoms with Hubbard
#2
repulsion and exchange interaction
U=2.4, J=-0.2 and J=0, β=64
0.6
DOS
0.4
0.2
0.0
-4
-2
0
2
4
Energy
M. Crommie, PRL(2001)
CT-QMC: single vs. trimer
V. Savkin, et al, PRL 94, 026402 (2005)
Equilateral and Isosceles Trimers
Density of states at geometry modification of the trimer
Equilateral (ET) and isosceles (IT) trimers
J23=J, J12=J13=J/3
AFM
V. Savkin et al, PRL 94, 026402 (2005)
0,7
0,7
0,6
0,6
ET 1-2-3
IT 1
IT 2-3
0,4
0,3
1
0,2
2
0,1
ET 1-2-3
IT 1
IT 2-3
0,5
DOS
DOS
0,5
FM
1
ET
0,4
0,3
1
1
ET
IT
0,2
3
2
2
0,1
3
3
2
3
0,0
0,0
-4
-3
-2
-1
0
Energy
1
2
3
4
-4
-3
-2
-1
0
1
Energy
One can see a reconstruction of the Kondo resonance for
isosceles trimer at antiferromagnetic exchange interaction
2
3
4
Hybridization function Co on/in Cu(111)
•  Hybridization of Co in bulk twice
stronger than on surface
•  Hybridization in energy range of Cu-d
orbitals more anisotropic on surface
•  Co-d occupancy: n= 7-8
B. Surer, et al PRB (2012)
Orbitally resolved Co DOS from QMC
Orbitally resolved DOS of the Co impurities in bulk Cu and on Co (111) obtained from QMC simulations at
temperature. T = 0.025 eV and chemical potential μ = 27 eV and μ = 28 eV, respectively.
All Co d-orbitals contribute to LDOS peak near EF=0
B. Surer, et al, PRB (2012).
Magnetic susceptibility: nanosystems
Bethe-Salpeter
Equation:
Susceptibility:
Local correlated nano-system:
U
Spin and Charge susceptibility near impurity
K. Patton, H. Hafermann,et.al
PRB (2009)
From Atom to Solid
Atomic physics
Bands effects (LDA)
N (E)
N (E)
E
n
d | SL> F
d
n+ 1 E
EF
N(E)
LHB
QP
EF
UHB
E
LDA+DMFT
E
Spectral Function Fe: ARPES vs. DMFT
SP-ARPES (BESY)
J. Sánchez-Barriga,
et al, PRL (2010)
Magnetism of metals: LDA+DMFT
l  Exchange
l  Finite
interactions in metals
temperature 3d-metal magnetism
1,2
1,2
M(T) and χ(T): LDA+DMFT
1,0
M(T)
0,8
χ(T)
Fe
0,6
0,8
0,6
Ni
0,4
0,4
0,2
0,2
0,0
0,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
T/Tc
A. L., M. Katsnelson and G. Kotliar, PRL87, 067205 (2001)
2,2
χ-1Meff2/3Tc
Global spin flip
M(T)/M(0)
1,0
Interaction of electrons
with collective excitations
Magnon
Plasmon
Orbiton
Non-local Coulomb interactions
General non-local action for solids:
Atomic action with local Hubbard-like interaction
Bosonic charge and spin variables:
A. Rubtsov et al, Annals of Physics 327, 1320 (2012)
Efficient DB-perturbation theory
Separate local and non-local effective actions:
Imuprity action with fermionic and bosonic bathes (CT-QMC)
F
B
Dual boson-fermion transformation:
EDMFT
Diagrams:
Dual Boson: General Idea
2
+
1
⌫
HTSC
+ ⇤!
tk ]
(14)
~
Vk ]
 
Jττ ' Sτ ⋅ Sτ'
g⌫
1
1
(15)
X⇣
⌫ ⇤ †
! ⌘! f⌫+! f⌫
!
!0 ⌦
⌫
!
1 X
+
4 0
†
†
!
f
f
0
⌫⌫ ⌫+! ⌫ 0 ! f⌫ f⌫ 0
DB-diagrammatic
scheme
⌫⌫ ⌦
+
⌫⇤
†
⌦ ⌘! f⌫
f⌫+!
Bosonic Selfenergy
=
⌘
< c!+⌦ c!0
† †
c
⌦ ! c! 0 >imp
g!+⌦ g!0
< c⌫+! c⌫ a† ⇢! >imp
= Renormalized vertex:
g⌫ g⌫+!
g! g! 0 (
⌦ g! g! 0
+ ...
Fermionic Selfenergy
⌦)
⌦+! ! 0
< ⇢ >imp g⌫
!
!
˜ k⌫ g⌫ ) 1 + ⌫ tk ] 1
Gk⌫ = [(g⌫ + g⌫ ⌃
˜ k⌫ g⌫ ) 1 + ⌫ tk ] 1
Gk⌫ =![(g⌫ + g⌫ ⌃
˜ q! ! ) 1 + ⇤! Vk ] 1
X
=
[(
+
⇧
X
q!
!
!
0
0
⌫
1
0
1
gX
! = !
⌫⌫ 0 ! g⌫ 0 s
⌫ 0q!!= [( ! + ! ⇧
˜ q! ! ) 1 + ⇤! Vk ] 1
0
0
X
Fermionic and Bosonic⌫ Green Functions
SCF-condition
Gk⌫ = g⌫
X
k
1
1
1
˜ k⌫ g⌫ ) + ⌫ tG
= )gg⌫⌫ )1 +
˜gk⌫
˜gG⌫k⌫
Gk⌫ = [(g⌫ + g⌫ ⌃
]
= [(g+⌫ g+ ⌃
⌃
+ ⌫
k G=
k⌫ [(g
k⌫
Xq! = [(
!
+
˜
! ⇧q! ! )
1
⌫
⌫
k⌫ ⌫
X
Xq! =
k
⌫
1
1
X˜q! ⇧
˜=q!!)! )1 +
+ ⇤! XVq!
]=
kX
= [(
+
+
+
⇧
⇤!⇤
q! [(
!
!
! X
! q! !
q
q
A. Rubtsov, M.I. Katsnelson, A. L., Annals of Phys. 327, 1320 (2012)
XX
!
X̃
⇥
G̃ G̃
⌃k ⇤+⌃,K+k
† ⇤K
− (0)
< cω+Ω cω ρΩ >imp
−
<
ρ>
g
δ
imp
ω
Ω
⌃k⇤
λωΩ = X̃⇤K
g g
χ lattice
,
Simple
Test:
Hubbard
(0)
usceptibility” of the dual
1 system.
X̃
⇤
.
ω ω+Ω Ω
Next, we use t
m
in the
typically
equals
zero
as of
< the
ρ> imp
vanishes.
ding
the nominator
triangle vertices
to
the
both
ends
ladder
and
usin
"
...
†
U>>t
#
#
#
d λ are related, since
(0) ρΩ =
ω # sσσ cω⇤# σ cω −Ω,σ ,
⌅ 1
⇥
X̃⇤K ⇥
G̃⌃k G̃⇤+⌃,K+k
1
(0)
⌥⌥⌥↵↵
%
'
X̃
⌃k⇤
⇤K
⌥⌥⌥↵↵↵⇥A triangle represents the ⇥ vertex⇥and square represents
Figure A.7. Bosoinc dual self-energy in the⌅ladder approximation.
the verte
&
⇥
=
+
⇧
Note
that
t
⇤in
−1 ⇤K
⌥
⌦
(0)
⇧
⌃
#
#
#
#
λ
=
χ
1
−
γ
g
g
s
Ωω
σσ
Ω
1ω,ω ,Ω⇤ X̃ω⇤Kω −Ω
e dual system.
Let
us cons
ω#
ertices to the both ends of the ladder and using the exact
relationof(1t
behaviour
Fermionic Selfenergy
substitute
formulas
(26,28)
for
⇧
and
⇥.
in
cancella
13 This results
in
this
casebu
sh
rtant note that for⇤ the Gaussian impurity
problem
γ
vanishes,
⌅
1
⇥
1
(0)
⌥
pression
⌥⌥⌥↵↵↵ fermionic
X̃⇤K and bosonic Green’s functions of ori
t relations between
⌅
⇤K
⇥⇤K = ⌥⌥⌥⇧↵↵⇥
1
Gωk
⇥⌦
+ ⇧⇥⇤⌅⌃ =
. ⇥⇤ X̃ (0) ⇥⇤ .
⇤K
⇤K
(0)
X̃
= (∆ω − εk )⇤−1⇤K
gω−1 G̃ωk gω−1 (∆ω − $k )−1 + (∆ω − $k )
om Eq.(28) has a meaning of an e⌅ective local Stoner parameter
2
s (26,28) for ⇧ and ⇥. This results
of −1out⌅ of the tex
−1 in cancellation
−1
−1
JijΩ
=−
Generalization
of =
Anderson
case
Xcontribution
(ΛΩ −superexchange
Vto
χΩnon-local
X̃toΩkfrequency-dependent
χΩ (Λ
+ (Λ
Vk
Ωk
k ) the
Ω − Vk ) with
fluctuation
self-energy
⇥
is
desc
U
⇤K
not reproduce
(0)
8
⌅
A. Rubtsov, et al, Annals
⇥ Phys.
= 327,
⇥ X̃1320⇥(2012)
.
⇤K
⇤ ⇤K ⇤
from a single
Introduction
Plasmon mode in ladder DB
Beyond DMFT
Dual Bosons
How Mott transition affect plasmon mode?
Dual fermion susceptibility: Plasmon mode
U=1.5 V=0.4 t=0.25 Beta=10
Square 64x64 lattice
6
5
E
4
3
2
Γ1
X
M
Γ
0 Γ
X
M
Susceptibility using Dual Fermion: h⇢⇢iK !0 6= 0
Γ
0
0
0
0
0
0
0
0
0
0
0
Summary
—  Magnetism of correlation systems can be well
described in the LDA+DMFT scheme
—  Local correlations efficiently included in
CT-QMC impurity solver
Imaginary Time and Matsubara space
Constrain GW calculations of U
Polarisation
F. Aryasetiawanan et al
PRB(2004)
Double-Bethe Lattice: exact C-DMFT
A.  Ruckenstein
PRB (1999)
Self-consistent condition: CDMFT
AF-between plane
AF-plane
Finite temperature phase
diagram
•  order-disorder transition at t? / t=p2 for large U
•  MIT for intermediate U
H. Hafermann, et al. EPL, 85, 37006 (2009)
Density of States: large U
Slater parametrization of U
Multipole expansion:
Coulomb matrix elements in Ylm basis:
Angular part – 3j symbols
Slater integrals:
CT-HYB: General Interaction
CT-HYB: Krylov code
CT-QMC-Krylov: performance
Satellite structure in Ni
Ni: LDA+DMFT (T=0.9 Tc)
2.5
<S(τ)S(0)>
-1
2.0
Density of states, eV
nd in final states
d-orbital spectral function
n
9
full U
1.5
EF
LDA
0.5
1.5
0.0
0
2
4
6
-1
band
τ, eV
1.0
PES
E
=that6(?)
eV
satfind
We
one-particle
0.5
DMFT
0.0
8
(LDA)
Eex =PES
0.3(0.6)
eV
WE
= 0.3(0.6)
3(4) eV eV
band =
ex
E
6(?)
eV
Eex=
==
0.3(0.6)
eV
B. Valence-ba
Wsat
3(4)
eV
1.0
-8
-6
-4
-2
Energy, eV
full U
1
G1
3
1
S
9
d →d
8
D
P
3
F
0
2
s
double-counting potential UH i
LDA+DMFT+QMC
only barely distinguishable.
Fig
! PRL (2001)
A. L., M. Katsnelson and G.P
Kotliar,
tral function Im m Gm! .E !
displayed result is relatively di
main band (" 4 eV) as well as th
identical to those obtained with
thus share the same poor agree
symmetry-resolved exchange s
given directly by the selfenergy
†eg " .EF / ! †eg
T-Lanczos (5d+10k)
† t2g " .EF / ! † t2g
J. Kolorenc et al PRB (2012)