1. No category

How to accurately get the polarisability
from an impurity model?
Sophie Chauvin(1,2,3), Lucia Reining (1,3), Silke Biermann (2,3)
(1)Laboratoire
des Solides Irradiés, Ecole polytechnique, France
(2)Centre de Physique Théorique, Ecole polytechnique, France
(3)European Theoretical Spectroscopy Facility (ETSF)
vendredi 30 mai 2014
Dynamical mean
field theory
Many-body
perturbation
theory
vendredi 30 mai 2014
Overview
1. Introduction to correlated materials: the Mott
transition
2. Dynamical Mean Field theory (DMFT)
3. Extended DMFT (EDMFT)
4. EDMFT and the Hubbard dimer
vendredi 30 mai 2014
Correlated materials
•
•
Transition metals, transition metal
oxides, f-electrons (rare earths,
actinides, their compounds)
Localised orbitals: 3d, 4f, ...
Hubbard model
H = −t
�
<i,j>,σ
(c†i,σ cj,σ
+
c†j,σ ci,σ )
+U
�
ni↑ ni↓
i
1 electron per site
t � U : kinetic energy
dominates
U � t : repulsion dominates
vendredi 30 mai 2014
Correlated materials
•
The Mott transition: a metalinsulator transition
band theory
Metal
•
Band theory not suited for
these spectra
quasi-particle peak
Hubbard bands
U/t
Photoemission spectrum of
metallic V2O3 near the Mott
transition
Insulator
atomic-like
from Kotliar and Vollhardt, Physics Today (2004).
vendredi 30 mai 2014
from Poteryaev et al., cond-mat/0701263,
adapted from Mo et al., Phys. Rev. Lett. 90, 186403 (2003).
Our motto
•
The many-body Hamiltonian:
H=
N
�
i=1
�
�
1
�2 ∇2i
2
−
− Ze
2m
|ri − R|
R
�
1�
e2
+
2
|ri − rj |
i�=j
1023 interacting particles!
•
The many-body wave function: ψ(r1 , r2 , . . . , rN )
we cannot even store the information!
•
Focus on reduced quantities
Physical system
Green’s function:
propagation of an
electron
Physical quantity
Experiment
vendredi 30 mai 2014
DFT
Green’s function
approaches
Impurity Green’s
function
Interacting electrons Interacting electrons Interacting electrons
Local density
n(r)
Structural
properties, phonons
Green’s function
G(r, r ; t, t )
Local Green’s
�
function Gii (t, t .)
Angle-resolved
photoemission
Direct and inverse
photoemission
�
�
Dynamical Mean-Field Theory
•
Introduce a dynamical mean field
Physical system
U
Fictitious system
t
U
t
t
Frequency-dependence
Self-consistency: Gii = −i < T c†i (t)ci (0) >= Gimp
Need approximation
vendredi 30 mai 2014
∆(ω)
frequency-dependence!
Non-locality
•
bath
see M. Gatti et al., Phys. Rev. Lett. (2007)
see A. Georges, Exact functionals, effective
actions and (Dynamical) Mean-Field Theories:
Some Remarks
or Kotliar and Vollhardt, Physics Today (2004)
Extended Dynamical Mean-Field
Theory
•
•
•
Not included in DMFT:
long-range interactions vij ,
effects from other bands
Describe dynamical screening �(ω) , could affect Mott transition
•
•
Effective, dynamical (screened) Coulomb interaction
U ni ni −→ U (τ − τ � )ni (τ )ni (τ � )
•Basic variables: electron propagation AND screening (G and W)
•Local point of view
H. Kajueter, PhD (1996). For a pedagogical introduction, see
S. Florens PhD (2003), and Th. Ayral, Phys. Rev. B (2013)
vendredi 30 mai 2014
Example: the Hubbard dimer
•
•
Can we make EDMFT construction more apparent?
Representability issue of EDMFT: can we always find the right
parameters ∆(ω) and U (ω) ?
t
U
c†1,σ , c1,σ
Hdimer = −t
vendredi 30 mai 2014
��
σ
U(ω)
?
U
bath
∆(ω)
c†2,σ , c2,σ
c†1σ c2σ
+
c†2σ c1σ
frequency-dependence!
�
+U
�
i=1,2
ni↑ ni↓
•
Integrate out degrees
of freedom on site 2
�
��
�
�
U
1
1
∗
∗
�
+
c1,σ (τ )c1,σ (τ ) −
c1,σ� (τ )c1,σ (τ ) −
2
2
2
σ,σ �
��
�
∗
∗
−t
c1,σ (τ )c2,σ (τ ) + c2,σ (τ )c1,σ (τ )
The Hubbard dimer
Intermediate results
σ
+
�
�
1
1
�
�
� �
� )φ(τ )
− iφ(τ ) +
φ(τ
2 ]U
Z∼
D[c∗1σ , c1σ
D[φ]e−S1 e−Sφ I
��
S =
�
S1 =0
��
σ
c1,σ (τ )∂τ c1,σ (τ )
σ
�
I = (det G)2 exp −t2
� ��
σ
U
c†1,σ , c1,σ
vendredi 30 mai 2014
dτ dτ � c∗1σ (τ )G(τ, τ � )c1σ (τ � )
Interpretation:
Site 1 in effective bosonic
(0.6)medium
�
���
�
�
�
�
�
UU � ∗
1
1�
∗
1
1
∗ (τ )c1,σ (τ ) −
∗c
++
cc1,σ
)−
� (τ )c� (τ �)(τ
1,σ
(τ
)c
(τ
)
−
c
−
� (τ )c1,σ1,σ
1,σ
1,σ
1,σ
22 � �
22
2 2
σ,σ
σ,σ
�
1
1
�
� β − �iφ(τ ) +
φ(τ
)φ(τ
)
11
2 U ) − iφ(τ )
Sφ =
dτ
φ(τ )φ(τ
2U
0
0
(0.5)
•
β
� �c∗ (τ )∂ c (τ )
dτ
τ 1,σ
∗
1,σ
dτ
(0.4)
•
σ
β
(0.3)
c∗2,σ (τ ) (∂τ + iφ(τ )) c2,σ (τ )
σ
�
(0.2)
�
•(0.7)Elementary process:
−1
t ∼ G−1
∼
G
12
21 ;
(0.8)
G = G22
(0.9)
Perspective: Integrate out boson...
φ
Conclusions
vendredi 30 mai 2014
•
DMFT: an effective single-site problem (or «effective atom»)
to reproduce the local part Gii (electron propagation)
•
EDMFT: transforms long-range interactions into effective
local dynamical interaction to reproduce the local
quantities Gii and Wii (electron propagation and
screening)
•
Explicit construction on Hubbard dimer: work in progress