Quantum-cluster theoriesfrom a variational perspective
Michael Potthoff
!" #$ %'& (
self-energy SFA
!" #*) %& (
!" #+ %& (
, - . , /
Quantum-cluster theoriesfrom a variational perspective
Michael Potthoff
!" #$ %& (
self-energy SFA dynamic
!" # %'& ( Green’sfunction
LuttingerWard
dynamic
!" #*) %& (
!" #+ %& (
, - . , /
Quantum-cluster theoriesfrom a variational perspective
Michael Potthoff
!" #$ %& (
self-energy SFA dynamic
!" # %'& ( Green’sfunction
LuttingerWard
dynamic
!" #*) %& ( electrondensity
DFT static
!" #+ %& ( densitymatrix
RayleighRitz
static
, - . , /
Quantum-cluster theoriesfrom a variational perspective
Michael Potthoff
!" #$ %& (
self-energy SFA dynamic
!" # %'& ( Green’sfunction
LuttingerWard
perturbation theory dynamic
!" #*) %& ( electrondensity
DFT LDA static
!" #+ %& ( densitymatrix
RayleighRitz
Hartree-Fock,Gutzwiller, VMC, ...
static
, - . , /
Quantum-cluster theoriesfrom a variational perspective
Michael Potthoff
!" #$ %& (
self-energy SFA new approximations? dynamic
!" # %'& ( Green’sfunction
LuttingerWard
perturbation theory dynamic
!" #*) %& ( electrondensity
DFT LDA static
!" #+ %& ( densitymatrix
RayleighRitz
Hartree-Fock,Gutzwiller, VMC, ...
static
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
outline
self-energy-functional theory: variational principles, approximation strategies, reference systems construction of the self-energy functional evaluation of the self-energy functional
variational cluster approximations: cluster size, boundary conditions, causality symmetry-breaking Weiss fields, AFM and dSC
relation to dynamical mean-field theory: DMFT as an approximation within the SFA CPT, VCA, C-DMFT and DCA the role of bath degrees of freedom
summary and outlook
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
outline
self-energy-functional theory: variational principles, approximation strategies, reference systems construction of the self-energy functional evaluation of the self-energy functional
variational cluster approximations: cluster size, boundary conditions, causality symmetry-breaking Weiss fields, AFM and dSC
relation to dynamical mean-field theory: DMFT as an approximation within the SFA CPT, VCA, C-DMFT and DCA the role of bath degrees of freedom
summary and outlook
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
outline
self-energy-functional theory: variational principles, approximation strategies, reference systems construction of the self-energy functional evaluation of the self-energy functional
variational cluster approximations: cluster size, boundary conditions, causality symmetry-breaking Weiss fields, AFM and dSC
relation to dynamical mean-field theory: DMFT as an approximation within the SFA CPT, VCA, C-DMFT and DCA the role of bath degrees of freedom
summary and outlook
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
approximation strategies
Hamiltonian:
&
grand potential:
" & ! "
physical quantity:
#
functional:
" # # %
on domain
$
variational principle:
!" # # %'& (
für
#& #
Euler equation:
% # # %'&!" # # %
! #
&& (
type III
type IItype I
Isimplify Euler equation% # # % ' (% # # % general
IIsimplify functional" # # % ' (" # # % thermodynamically consistent
IIIrestict domain$ ' ($ thermodynamically consistent,
systematic, clear concept
, - . , )
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
construction of the self-energy functional
" #$ %'& ?
(one-to-one)
TrTr
Luttinger-Ward functional, universal
: Legendre transform of
Tr
SFT DFT
!" # $ %& ( !" #*) %& (
Σ space
Ω
Ω = Ω[Σ]
δ Ω[Σ] = 0
self-energy-functional theory (SFT)
stationary at physical self-energy construced formally, but unknown
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
construction of the self-energy functional
" #$ %'& ? " # % & # % /
# % &
!" # %
! (one-to-one)
# % & " # # % % Tr
# %
Tr
Luttinger-Ward functional, universal
#$ %: Legendre transform of
# %
" #$ %'& Tr
$
#$ %
!" #$ %'& (
$&
! #$ %! $
SFT DFT
!" # $ %& ( !" #*) %& (
Σ space
Ω
Ω = Ω[Σ]
δ Ω[Σ] = 0
self-energy-functional theory (SFT)
stationary at physical self-energy construced formally, but unknown
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
construction of the self-energy functional
" #$ %'& ? " # % & # % /
# % &
!" # %
! (one-to-one)
# % & " # # % % Tr
# %
Tr
Luttinger-Ward functional, universal
#$ %: Legendre transform of
# %
" #$ %'& Tr
$
#$ %
!" #$ %'& (
$&
! #$ %! $
SFT DFT
!" # $ %& ( !" #*) %& (
Σ space
Ω
Ω = Ω[Σ]
δ Ω[Σ] = 0
self-energy-functional theory (SFT)
" #$ %
stationary at physical self-energy
# $ %
construced formally, but unknown
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
reference system
Rayleigh, Ritz
Ht’,U’
Ψt’,U’Ψt’,U’t,UE [ ]
t,UH
original system reference system
# %'&
# %
&& ( Hartree-Fock approximation
SFT
Ht’,U’t,UH
t’,U’Σt,U[ ]Σ t’,U’Ω
original system reference system
?
new approximations ?
type of approximation choice of reference system
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
reference system
Rayleigh, Ritz
Ht’,U’
Ψt’,U’Ψt’,U’t,UE [ ]
t,UH
original system reference system
# %'&
# %
&& ( Hartree-Fock approximation
SFT
Ht’,U’t,UH
t’,U’Σt,U[ ]Σ t’,U’Ω
original system reference system
?
new approximations ?
type of approximation choice of reference system
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
reference system
Rayleigh, Ritz
Ht’,U’
Ψt’,U’Ψt’,U’t,UE [ ]
t,UH
original system reference system
# %'&
# %
&& ( Hartree-Fock approximation
SFT
Ht’,U’t,UH
t’,U’Σt,U[ ]Σ t’,U’Ω
original system reference system
" #$ %'& ?
" #$ %
&& (
new approximations ?
type of approximation choice of reference system
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
evaluation of the self-energy functional
t,Uδ Ω [Σ ] = 0(t’)
t,UΩ = Ω [Σ]
Ω
Σ spacet’Σ = Σ( )
# $ %
unknown but universal!
original system:
" #$ %'& Tr
$
#$ %
reference system:
" #$ %& Tr
$
#$ %
combination:
" #$ %'& " #$ %
Tr
$ Tr
$
non-perturbative, thermodynamically consistent, systematic approximations , - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
U
t
lattice model (
&
) inthe thermodynamic limit
n.n. hopping:
local interaction:
electron density : & "
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
n.n. hopping:
local interaction:
electron density : & "
reference system,
:
system of decoupled clusters
diagonalization trial self-energy:
$ & $
self-energy functional:
" #$ %
stationary point:
" # $ %'& (
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterscluster size:
: analytic
: exact diagonalization
: Lanczos method
( (
: stochastic techniques
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
example:
Hubbard model
& (
, half-filling,
&
, nearest-neighbor hopping
&
variational parameter: nearest-neighbor hopping
within the chain
-1 0 1 2-4.4
-4.2
-4.0
-3.8
-3.60.95 1.00 1.05
00.00020.00040.00060.0008Ω
t’
∆Ω4
L c=2
6810
L c=2
L c=10
L c=2
L c=10
t’
" " #$ %
stationary at
&
& (
: cluster size irrelevant
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
variational parameters:intra-cluster hoppingpartial compensation offinite-size effects
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
variational parameters:hopping between cluster boundariesboundary conditions
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
boundary conditions
-1 -0.5 0 0.5 1-4.34
-4.32
-4.30
-4.28
-4.26
-4.24
-4.22
-4.20
tr
Ω
10
L c=4
86
exact
t r
t t t
&
Hubbard model& (
, half-filling,
&
&
open or periodic b.c. ?open boundary conditions !
exact:
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
, - . , / /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
variational parameters:on-site energiesthermodynamic consistency
, - . , / /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
variational parameters:ficticious symmetry-breaking fieldsspontaneous symmetry breaking
, - . , / /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
variational parameters:ficticious symmetry-breaking fieldsdifferent order parameters
, - . , / /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
antiferromagnetism
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-4.50
-4.49
-4.48
-4.47
-4.46
-4.45Ω
h
U=8
&
Hubbard model, half-filling
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
antiferromagnetism
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-4.50
-4.49
-4.48
-4.47
-4.46
-4.45Ω
h
U=8
&
Hubbard model, half-filling
ground-state energy per site:
0 2 4 6 8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
VCA
direct ED
CPT
VMC
QMC
U
E0
QMC, VMC: extrapolated to
' ,
' (
QMC:
VMC:
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
antiferromagnetism
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-4.50
-4.49
-4.48
-4.47
-4.46
-4.45Ω
h
U=8
&
Hubbard model, half-filling
0
42
−4−6−8
ΓMXΓk
8
−2
6ω
−8−6−4
2
−20
68
4
Γ X M Γ
QMC
ω
VCA
QMC / MaxEnt:
& (
, 8 8 cluster
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
high-temperature superconductivity
hole doping | electron doping
d-wave-superconductivity
antiferromagnetism
-
-
Hubbard model&
& (
,
&
& (
Senechal, Lavertu, Marois, Tremblay (2005)
, - . , / )
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
&
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusters
&
Hubbard-I-type approximation
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
improved description of temporalcorrelations
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
improved mean-field theory
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
optimum mean-field theory, DMFTMetzner, Vollhardt (1989)Georges, Kotliar, Jarrell (1992)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
cellular DMFTKotliar et al (2001)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
variational cluster approach (VCA)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
,
&
variational cluster approach (VCA)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
original system,
:
lattice model (
&
) inthe thermodynamic limit
reference system,
:
system of decoupled clusterswith additional bath sites
&
variational cluster approach (VCA)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
classification of dynamical approximations
loca
l deg
rees
of f
reed
om
cluster size
oo
1 2
Lc
Hubbard−I
DIA cellular DIA
variational CA
oo DMFT
cellular DMFT
21
bL
dynamical mean-field theory Metzner, Vollhardt (1989), Georges, Kotliar, Jarrell (1992)cellular DMFT Kotliar, Savrasov, Palsson (2001)dynamical impurity approach (DIA) Potthoff (2003)variational cluster approach Potthoff, Aichhorn, Dahnken (2004)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
original system,
:
reference system,
:
loca
l deg
rees
of f
reed
om
cluster size
oo
Nc
1 2
DIA cellular DIA
variational CA
ns
oo DMFT
cellular DMFT
21
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
original system,
:
reference system,
:
" #$ % & (
open boundary conditions (see above)
there is no reference systemwhich generates the DCA !
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
original system,
:
reference system,
:
" #$ % & (
(
'
DCA self-consistency condition
: invariant under superlattice translations
and periodic on each cluster
systematic restores translational symmetry no implications on quality of DCA !
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
original system,
:
reference system,
:
" #$ % & (
(
" #
%
' " #
%
)
P-C-DMFT self-consistency condition
systematic restores translational symmetry
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
cluster extensions of DMFT
cellular DMFT (C-DMFT)Kotliar, Savrasov, Palsson, Biroli(2001)
dynamical cluster approximation(DCA)Hettler, Tahvildar-Zadeh, Jarrell,Pruschke, Krishnamurthy (1998)
periodized C-DMFT (P-C-DMFT)Biroli, Parcollet, Kotliar (2003)
fictive impurity modelsOkamoto, Millis, Monien, Fuhrmann(2003)
original system,
:
reference system,
:without any relation to the original system !
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
systematics of dynamical approximations
loca
l deg
rees
of f
reed
om
cluster size
oo
1 2
Lc
Hubbard−I
DIA cellular DIA
variational CA
oo DMFT
cellular DMFT
21
bL
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
dynamical impurity approximation (DIA)
4.6 4.8 5 5.2 5.4 5.6 5.8 6
0
0.01
0.02
0.03
0.04
UcUc1 Uc2
metal insulator
coex.
Tc
U
T crossover
Hubbard modelhalf-fillingsemielliptical DOS
&
DIA with
&
Potthoff (2003)
qualitative agreement wit DMFT (QMC, NRG)Georges et al (1996), Joo, Oudovenko (2000), Bulla et al (2001)
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
DIA - phase transitions
0 0.1 0.2 0.3 0.4
-2.708
-2.706
-2.704
-2.702
-2.7
-2.698
-2.696
-2.694
V
Ω
U=5.2
0.004
0.002
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
T=0
0 0.1 0.2 0.3 0.4 0.5
-0.010
-0.005
0
0.005
0.010
6.05.95.85.75.65.55.4
5.3
5.1
5.2
5.0
V
Ω (V
) −
Ω (0
)
T=0
&
, different
(
: discontinuous
& (
, different
: continuous
metastable states order of phase transitions
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
convergence with increasing
4.6 4.8 5 5.2 5.4 5.6 5.8U
0
0.01
0.02
0.03
0.04
0.05
0.06
T
Insu
lator
Meta
l
Coexis
tence
regio
n Uc1
Uc2
Uc
4.6 4.8 5 5.2 5.4 5.6 5.80
0.01
0.02
0.03
0.04
0.05
0.06
N=6N=4N=2
L =6bL =4b
L =2b
T
U
Hubbard modelhalf-fillingsemielliptical DOS
&
DIA
Pozgajcic (2004)
quantitative agreement with DMFT (QMC, NRG)Georges et al (1996), Joo, Oudovenko (2000), Bulla et al (2001)
extremely fast convergence with increasing
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
more bath sites vs. larger clusters
loca
l deg
rees
of f
reed
om
cluster size
oo
1 2
Lc
Hubbard−I
DIA cellular DIA
variational CA
oo DMFT
cellular DMFT
21
bL
, - . , /
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
: bath sites ?
0 0.2 0.4 0.6-4.34
-4.32
-4.30
-4.28
-4.26
-4.24
-4.22
-4.20
4
Ω
6
exact
L c=2
tb
t b t b
t t t
0 0.1 0.2 0.3 0.4 0.5
-0.34
-0.32
-0.30
-0.28
-0.26
-0.24
1 / L c
direct
0
# bath sites
exact
E0
2SFT
exact:
larger cluster vs. more bath sites enhanced convergence
, - . ,
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
conclusions
self-energy-functional theory: rigorous variational principle
!" # $ %'& (
trial self-energies from a reference system:
non-perturbative, thermodynamically consistent, systematic approximations
main approximations:
DIA mean-field phase diagrams, multi-band models
C-DMFT unified framework, new variants, DCA:
' VCA low-dimensional lattice models, large clusters
open problems: non-local ? bosons ? two-particle correlation functions ? upper bounds ? relation to Ritz principle ?
thanks to: M. Balzer, C. Dahnken, W. Hanke (U Würzburg)M. Aichhorn, E. Arrigoni (TU Graz)W. Nolting (HU Berlin)
, - . , )
Michael Potthoff Quantum-cluster methods for correlated materials, Sherbrooke, July 2005—————————————————————————————————————————————————————————–
conclusions
self-energy-functional theory: rigorous variational principle
!" # $ %'& (
trial self-energies from a reference system:
non-perturbative, thermodynamically consistent, systematic approximations
main approximations:
DIA mean-field phase diagrams, multi-band models
C-DMFT unified framework, new variants, DCA:
' VCA low-dimensional lattice models, large clusters
open problems: non-local
? bosons ? two-particle correlation functions ? upper bounds ? relation to Ritz principle ?
thanks to: M. Balzer, C. Dahnken, W. Hanke (U Würzburg)M. Aichhorn, E. Arrigoni (TU Graz)W. Nolting (HU Berlin)
, - . , )
&
&
density-functional theory (DFT) self-energy-functional theory (SFT)
external potential
hopping
density
self-energy
ground-state densities & # %
-representable self-energies
$ & $ # %
ground-state energy
& # % grandcanonical potential
" & " #$ %
# % & # % " # $ %'& Tr $ # $ %
: explicit Tr
$ : explicit
# % : unknown, universal ( -independent)
#$ %: unknown, universal (
-independent)
variational principle:
! # %'& (
variational principle:
!" #$ %'& (
exact but not explicit exact but not explicit
local-density approximation (LDA) different approximations
reference system: homogeneous electron gas different reference systems
approximate functional
functional
on restricted domain
, - . ,