Doped asymmetric two-band Hubbard models Peter van Dongen 25 August 2009
Doped asymmetric two-band Hubbard models
Doped asymmetric two-band Hubbard models
Peter van Dongen
25 August 2009
EPSRC Symposium Workshop on Quantum Simulations
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Overview
Introduction: experimental motivation and Hamiltonian
Part 1: Results at half-filling ( n = 2)
Part 2: results away from half-filling ( n = 2)
Part 3: n = 2 , including crystal field splitting
Phase diagram and spectral functions
Additional results at and away from half-filling
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Orbital-selective Mott transitions in Ca
2 − x
Sr x
RuO
4
Experimental phase diagram of Ca
2 − x
Sr x
RuO
4
:
[S. Nakatsuji, Y. Maeno,
PRL 84 , 2666 (2000)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Orbital-selective Mott transitions in Ca
2 − x
Sr x
RuO
4
Experimental phase diagram of Ca
2 − x
Sr x
RuO
4
:
[S. Nakatsuji, Y. Maeno,
PRL 84 , 2666 (2000)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Orbital-selective Mott transitions in Ca
2 − x
Sr x
RuO
4
Experimental phase diagram of Ca
2 − x
Sr x
RuO
4
:
[S. Nakatsuji, Y. Maeno,
PRL 84 , 2666 (2000)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ 0
Rotational invariance:
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0 X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
J
3
= J
⊥
= J
0 ≡ J i m
U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian?
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ 0
Rotational invariance:
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0 X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
J
3
= J
⊥
= J
0 ≡ J i m
U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0
X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ i m
Rotational invariance: J
3
= J
⊥
= J
0 ≡ J U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0
X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
Rotational invariance: J
3
= J
⊥
= J
0
≡ J i m
U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0
X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
Rotational invariance: J
3
= J
⊥
= J
0
≡ J i m
U
0
= U − 2 J
In QMC -calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0
X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
Rotational invariance: J
3
= J
⊥
= J
0
≡ J i m
U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Introduction: experimental motivation and Hamiltonian
Phase diagram of Ca
2 − x
Sr x
RuO
4
, two-band Hubbard model
Two-band Hubbard model for Ca
2 − x
Sr x
RuO
4
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
+ U
X n i m ↑ n i m ↓
( ij ) m σ i m
+
X
( U
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0 i σσ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0
X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ
Rotational invariance: J
3
= J
⊥
= J
0
≡ J i m
U
0
= U − 2 J
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Expansion of self-energy for multi-band Hubbard models:
Σ
γ
( ω ) =
X
U
αγ h n
α i +
α = γ
1
X X
U
αγ
U
βγ
ω
α = γ β = γ
( h n
α n
β i − h n
α ih n
β i ) + O ( ω
− 2
)
[Also by V.S. Oudovenko and G. Kotliar, PRB 65 , 75102 (2002)]
[See also J.K. Freericks and V. Turkowski, PRB (to be published)]
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
The spectral function
Part 1: n = 2 , spectral functions
(narrow/wide band)
0.2
0.1
0.0
0.1
0.2
0.3
0.6
0.5
0.4
0.3
0
T=1/40, "# =0.40
narrow band
U=0.0
U=1.8
U=2.05
U=2.2
U=2.4
U=2.6
U=2.8
1 wide band (inverted scale)
2
!
3 4
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
The spectral function
Part 1: n = 2 , spectral functions
(narrow/wide band)
0.2
0.1
0.0
0.1
0.2
0.3
0.6
0.5
0.4
0.3
0
T=1/40, "# =0.40
narrow band
U=0.0
U=1.8
U=2.05
U=2.2
U=2.4
U=2.6
U=2.8
1 wide band (inverted scale)
2
!
3 4
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
The spectral function
Spectral weight at Fermi level
(narrow/wide band) narrow band
T=1/40
T=1/32
T=1/25
0.6
0.5
0.4
0.3
0.2
0.1
0
1.8
2 wide band
2.2
U
2.4
2.6
2.8
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
The spectral function
Self energy at low frequencies
(narrow/wide band) a)
2
1 b)
0
2
2
1
0
T=1/32 narrow band
U=2.8
U=2.6
U=2.4
U=2.2
U=2.0
U=1.8
2
U
2.5
3 wide band
1
0
0 0.2
0.4
0.6
!
0.8
1 1.2
n = 2
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
The spectral function
Self energy at low frequencies
(narrow/wide band) a)
2
1 b)
0
2
2
1
0
T=1/32 narrow band
U=2.8
U=2.6
U=2.4
U=2.2
U=2.0
U=1.8
2
U
2.5
3 wide band
1
0
0 0.2
0.4
0.6
!
0.8
1 1.2
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
Magnetic phase diagram
Results at half-filling: magnetic phase diagram
0.25
0.2
1 band
2 bands
Strong-coupling limit (1OPT)
Weak-coupling limit (2OPT)
0.15
PM
0.1
MIT of 1-band HM
AF
MIT of narrow band
0.05
0
0 1 2 3
U
4 5 6 7 appendix
Doped asymmetric two-band Hubbard models
Part 1: Results at half-filling ( n = 2)
Magnetic phase diagram
Results at half-filling: magnetic phase diagram
0.25
0.2
1 band
2 bands
Strong-coupling limit (1OPT)
Weak-coupling limit (2OPT)
0.15
PM
0.1
MIT of 1-band HM
AF
MIT of narrow band
0.05
0
0 1 2 3
U
4 5 6 7
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Static properties
Part 2: n = 2 , U -dependence of orbital occupancy
1.3
U=2.80
U=2.60
U=2.40
U=2.20
U=2.00
U=1.80
1.2
wide band
1.1
narrow band
1
2 2.1
2.2
2.3
n (total filling)
2.4
2.5
2.6
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Static properties
Part 2: n = 2 , U -dependence of orbital occupancy
1.3
U=2.80
U=2.60
U=2.40
U=2.20
U=2.00
U=1.80
1.2
wide band
1.1
narrow band
1
2 2.1
2.2
2.3
n (total filling)
2.4
2.5
2.6
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Static properties
Rigid-band model ↔ QMC results a) wide band narrow band b)
1.3
wide band (RB) narrow band (RB) wide band (QMC) narrow band (QMC)
1.2
1.1
-4 -3 -2 -1 0 1 2 3 4
!
1.0
2 2.1
2.2
n (total filling)
2.3
Phase diagram
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Static properties
Rigid-band model ↔ QMC results a) wide band narrow band b)
1.3
wide band (RB) narrow band (RB) wide band (QMC) narrow band (QMC)
1.2
1.1
-4 -3 -2 -1 0 1 2 3 4
!
1.0
2 2.1
2.2
n (total filling)
2.3
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Static properties
U -dependence of intraorbital double occupancy b)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
U=2.80
U=2.40
U=2.00
U=2.40 (HF)
U=0.00
2.5
narrow band
3 n (total filling) wide band
3.5
4
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Phase diagram
Phase diagram of the doped two-band Hubbard model fully metallic phase orbital-selective Mott phase
U c1
at half-filling
U c2
at half-filling
Mott insulating phase (black line)
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Phase diagram
Phase diagram of the doped two-band Hubbard model
Narrow band provides major contribution
Wide band provides major contribution
2
1.6
U c1
at half-filling orbital-selective Mott phase
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
DOS at Fermi-level
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2 2.1
narrow band wide band
2.2
2.3
n (total filling)
2.4
U=2.80
U=2.60
U=2.40
U=2.20
U=2.00
2.5
2.6
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
Imaginary part of the self-energy a) 0.0
b)
T=1/40
-0.2
n=2.20
-0.4
-0.6
-0.8
-1.0
0 n=2.10
1 n=2.25
n=2.20
n=2.15
n=2.10
n=2.05
2
" n
3 4 0 1
T=1/20
T=1/30
T=1/40
2
" n
3 4
Part 3
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
Imaginary part of the self-energy a) 0.0
b)
T=1/40
-0.2
n=2.20
-0.4
-0.6
-0.8
-1.0
0 n=2.10
1 n=2.25
n=2.20
n=2.15
n=2.10
n=2.05
2
" n
3 4 0 1
T=1/20
T=1/30
T=1/40
2
" n
3 4
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
DOS for wide band ↔ DOS for single-band model:
˜
= 0 .
855
Two-band model narrow band
Single-band model n = 2.20
n = 1.09
-3 -2 -1 0
E
1 n = 2.00
2 3 -3 -2 -1 0
E
1 n = 1.00
2 3
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
DOS for wide band ↔ DOS for single-band model:
˜
= 0 .
945
Two-band model narrow band
Single-band model n = 2.20
n = 1.08
-3 -2 -1 0
E
1 n = 2.00
2 3 -3 -2 -1 0
E
1 n = 1.00
2 3
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
DOS for wide band ↔ DOS for single-band model:
˜
= 1 .
035
Two-band model narrow band
Single-band model n = 2.20
n = 1.06
-3 -2 -1 0
E
1 n = 2.00
2 3 -3 -2 -1 0
E
1 n = 1.00
2 3
Doped asymmetric two-band Hubbard models
Part 2: results away from half-filling ( n = 2)
Dynamical properties
DOS for wide band ↔ DOS for single-band model:
˜
= 1 .
125
Two-band model narrow band
Single-band model n = 2.20
n = 1.05
-3 -2 -1 0
E
1 n = 2.00
2 3 -3 -2 -1 0
E
1 n = 1.00
2 3
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Part 3: n = 2 , including crystal field splitting
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
( ij ) m σ
+ U
X n i m ↑ n i m ↓
+
X
( U
0 i m i σσ
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0 X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓
+
1
2
∆
X
( n i 1
− n i 2
) i σ i m i
In QMC-calculations: J
⊥
= J
0
= 0 U
0
=
1
2
U J
3
=
1
4
U
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Part 3: n = 2 , including crystal field splitting
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
( ij ) m σ
+ U
X n i m ↑ n i m ↓
+
X
( U
0 i m i σσ
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0 X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ i m
+
1
2
∆
X
( n i 1 i
− n i 2
)
In QMC-calculations: J
⊥
= J
0
= 0 U
0
=
1
2
U J
3
=
1
4
U
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Part 3: n = 2 , including crystal field splitting
Hamiltonian:
H = −
X t m c
† i m σ c j m σ
( ij ) m σ
+ U
X n i m ↑ n i m ↓
+
X
( U
0 i m i σσ
0
− J
3
δ
σσ
0
) n i 1 σ n i 2 σ
0
− J
⊥
X c
† i 1 σ c i 1 ¯ c
† i 2 ¯ c i 2 σ
− J
0 X c
† i m ↑ c
† i m ↓ c i ¯ ↑ c i ¯ ↓ i σ i m
+
1
2
∆
X
( n i 1 i
− n i 2
)
In QMC-calculations: J
⊥
= J
0
= 0 U
0
= 1
2
U J
3
= 1
4
U
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting static properties
Orbital-dependent filling
2 n=n w
+n n
=2.00
Wide Band
1.5
2 n=n w
+n n
=2.20
1.5
1 1
0.5
Narrow Band
0
-4 -3 -2 -1 0 1 2 3 4
Crystal field splitting
∆
0.5
0
U = 4.00
U = 3.60
U = 3.20
U = 2.80
U = 2.40
U = 2.00
-4 -3 -2 -1 0 1 2 3 4
Crystal field splitting
∆
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting static properties
Intraorbital double occupancy
1
Wide Band
0.8
Narrow Band
0.6
1
0.8
0.6
U = 4.00
U = 3.60
U = 3.20
U = 2.80
U = 2.40
U = 2.00
0.4
0.4
0.2
0 n=n w
+n n
=2.10
-4 -3 -2 -1 0 1 2 3 4
Crystal field splitting
∆
0.2
n=n w
+n n
=2.20
0
-4 -3 -2 -1 0 1 2 3 4
Crystal field splitting
∆
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Phase diagram and spectral functions
The phase diagram
5
4.5
Band insulator narrow band n=2.10
4
3.5
3
2.5
2
1.5
-4
OSMP wide band n=2.10
increase filling
OSMP narrow band n=2.10
Band insulator
-3
U c
for wide B, n=2.10
U c
for narrow B, n=2.10
U c
for wide B, n=2.20
U c
for narrow B, n=2.20
-2 -1 0
Crystal field splitting
∆
1
Metallic regime
Non-Fermi liquid
2 3 wide band n=2.10
increase filling
4
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Phase diagram and spectral functions
DOS at selected points in the phase diagram
5
4.5
Band insulator narrow band n=2.10
4
3.5
3
2.5
2
1.5
-4
OSMP wide band n=2.10
U=3.00,
∆
=-2.20
Wide band
Narrow band
-3
U c
for wide B, n=2.10
U c
for narrow B, n=2.10
U c
for wide B, n=2.20
U c
for narrow B, n=2.20
-2 -1 0
Crystal field splitting
∆
1
-6 -4 -2 0 2 4 6
E
2 3 4
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Phase diagram and spectral functions
DOS at selected points in the phase diagram
5
4.5
Band insulator narrow band n=2.10
4
3.5
3
2.5
2
1.5
-4
OSMP wide band n=2.10
U=4.00,
Wide band
Narrow band
∆
=-1.60
-3
U c
for wide B, n=2.10
U c
for narrow B, n=2.10
U c
for wide B, n=2.20
U c
for narrow B, n=2.20
-2 -1 0
Crystal field splitting
∆
1
-6 -4 -2 0 2 4 6
E
2 3 4
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Phase diagram and spectral functions
DOS at selected points in the phase diagram
5
4.5
4
3.5
3
2.5
2
1.5
-4
U=3.00,
∆
=2.60
Wide band
Narrow band
OSMP narrow band n=2.10
Band insulator wide band n=2.10
U c
for wide B, n=2.10
U c
for narrow B, n=2.10
-6 -4 U c
for wide B, n=2.20
U c
-3 -2 -1 0
Crystal field splitting
∆
1
Metallic regime
Non-Fermi liquid
2 3 4
Doped asymmetric two-band Hubbard models
Part 3: n = 2, including crystal field splitting
Phase diagram and spectral functions
DOS at selected points in the phase diagram
5
4.5
4
3.5
3
2.5
2
1.5
-4
U=3.40,
∆
=0.60
Wide band
Narrow band
OSMP narrow band n=2.10
Band insulator wide band n=2.10
U c
for wide B, n=2.10
U c
for narrow B, n=2.10
-6 -4 U c
for wide B, n=2.20
U c
-3 -2 -1 0
Crystal field splitting
∆
1
Metallic regime
Non-Fermi liquid
2 3 4
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models
:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling
I
I
I
:
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping
I
I
I
:
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting
I
I
:
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Summary
Summary
Doped asymmetric two-band Hubbard models:
1.
at half-filling:
I
I
I
. . .
as a model for OSMTs (Ca
2 − x
Sr x
RuO
4
)
Results: orbital selective Mott phase , non-Fermi-liquid magnetic phase diagram: weak coupling AFM
2.
for general doping:
I
I
I
OSMTs even away from half-filling n ' 2: pinning of particle density of narrow band
Phase diagram and DOS as a function of interaction/filling
3.
including crystal field splitting:
I
I
OSMTs also in presence of crystal field splitting
Phase diagram and DOS including crystal field splitting
Thanks for your attention!
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
= J
3
≡ J :
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
:
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
= J
3
≡ J :
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
:
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
= J
3
≡ J :
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
:
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
= J
3
≡ J :
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
:
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
Perturbation theory at strong coupling
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
I
:
Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
= J
3
≡ J :
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling:
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
= J
3
≡ J :
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
∗
2
)
2
U + J
3
= ZJ
Is
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
Perturbation theory at weak and strong coupling
Self-consistent perturbation theory at weak coupling:
I Expand free energy: f ( U , ∆) = f
0
(∆) + Uf
1
(∆) + U
2 f
2
(∆) + · · ·
I
I
Keep order parameter fixed: h ( U , ∆) = h
0
(∆) + Uh
1
(∆) + U
2 h
2
(∆) + · · ·
Optimize in each order: df d ∆
= 0 k
B
T
HF c
∼ t
1
∗ exp I d
− t
1
∗ b
1
U ν (0)
T c
∼ T
HF c exp − t
∗
1 b
2
ν (0)
Perturbation theory at strong coupling:
I Effective Hamiltonians for J
⊥
< J
3
H t
0 and J
⊥
= J
3
≡ J :
=
X
J
Heis
( S i
· S j
− n i n j
) ↔ H t
0
=
X
J
Is
( S i 3
( T = 0)
· S j 3
− n i n j
) h ij i h ij i
I Critical temperature: k
B
T c
∼
( t
1
∗
)
2
+ (
U + J t
∗
2
)
2
(2nd order)
= ZJ
Heis
↔ k
B
T c
∼
( t
1
∗
)
2
+ ( t
2
∗
)
2
U + J
3
= ZJ
Is
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
DOS of two-band Hubbard model at strong coupling
Sketch of density of states (non-interacting):
ν m σ
( ω ) phase diagram m = 1
ω m = 2
0
Density of states for
ν
LHB m σ
ν
LHB m σ
( ω ) =
( ω ) =
√
2
√
3
J
⊥
ν
0 m
(
<
»
J
4
3
3 and
ν
0 m
(
√
2 [ ω +
[ ω +
1
2
1
2
J
⊥
= J
3
( U + J
3
)])
( U + J )])
≡ J : ( T = 0)
( J
⊥
< J
3
)
( J
⊥
= J
3
)
ω
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
DOS of two-band Hubbard model at strong coupling
Sketch of density of states ( non-interacting ):
ν m σ
( ω ) m = 1
ω m = 2
0
ω phase diagram
Density of states for
ν
LHB m σ
ν
LHB m σ
( ω ) =
( ω ) =
√
2
√
3
J
⊥
ν
0 m
(
<
»
J
4
3
3 and
ν
0 m
(
√
2 [ ω +
[ ω +
1
2
1
2
J
⊥
= J
3
( U + J
3
)])
( U + J )])
≡ J : ( T = 0)
( J
⊥
< J
3
)
( J
⊥
= J
3
)
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
DOS of two-band Hubbard model at strong coupling
Sketch of density of states ( at strong coupling ):
ν m σ
( ω ) m = 1
ω m = 2
0
ω phase diagram
Density of states for
ν
LHB m σ
ν
LHB m σ
( ω ) =
( ω ) =
√
2
√
3
J
⊥
ν
0 m
(
<
»
J
4
3
3 and
ν
0 m
(
√
2 [ ω +
[ ω +
1
2
1
2
J
⊥
= J
3
( U + J
3
)])
( U + J )])
≡ J : ( T = 0)
( J
⊥
< J
3
)
( J
⊥
= J
3
)
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
DOS of two-band Hubbard model at strong coupling
Sketch of density of states (at strong coupling):
ν m σ
( ω ) m = 1
ω m = 2
0
Density of states for
ν
LHB m σ
ν
LHB m σ
( ω ) =
( ω ) =
√
2
√
3
J
⊥
ν
0 m
(
<
»
J
4
3
3 and
ν
0 m
(
√
2 [ ω +
[ ω +
1
2
1
2
J
⊥
= J
3
( U + J
3
)])
( U + J )])
≡ J : ( T = 0)
( J
⊥
< J
3
)
( J
⊥
= J
3
)
ω phase diagram
Doped asymmetric two-band Hubbard models
Appendix
Perturbation theory results
DOS of two-band Hubbard model at strong coupling
Sketch of density of states (at strong coupling):
ν m σ
( ω ) m = 1
ω m = 2
0
Density of states for
ν
LHB m σ
ν
LHB m σ
( ω ) =
( ω ) =
√
2
√
3
J
⊥
ν
0 m
(
<
»
J
4
3
3 and
ν
0 m
(
√
2 [ ω +
[ ω +
1
2
1
2
J
⊥
= J
3
( U + J
3
)])
( U + J )])
≡ J : ( T = 0)
( J
⊥
< J
3
)
( J
⊥
= J
3
)
ω
Doped asymmetric two-band Hubbard models
Appendix
Introduction to DMFT
Dynamical Mean Field Theory (DMFT)
DMFT maps lattice problem onto SIAM
⇔ neglects non-local correlations
⇔ assumes local self-energy Σ( k , ω )
!
= Σ( ω )
Several impurity solvers for SIAM available (QMC, NCA, NRG, ED, IPT, · · · )
Doped asymmetric two-band Hubbard models
Appendix
Introduction to DMFT
Dynamical Mean Field Theory (DMFT)
DMFT maps lattice problem onto SIAM
⇔ neglects non-local correlations
⇔ assumes local self-energy Σ( k , ω )
!
= Σ( ω )
Several impurity solvers for SIAM available (QMC, NCA, NRG, ED, IPT, · · · )
Doped asymmetric two-band Hubbard models
Appendix
Introduction to DMFT
Dynamical Mean Field Theory (DMFT)
DMFT maps lattice problem onto SIAM
⇔ neglects non-local correlations
⇔ assumes local self-energy Σ( k , ω )
!
= Σ( ω )
Several impurity solvers for SIAM available (QMC, NCA, NRG, ED, IPT, · · · )
Doped asymmetric two-band Hubbard models
Appendix
Introduction to DMFT
Dynamical Mean Field Theory (DMFT)
DMFT maps lattice problem onto SIAM
⇔ neglects non-local correlations
⇔ assumes local self-energy Σ( k , ω )
!
= Σ( ω )
Several impurity solvers for SIAM available (QMC, NCA, NRG, ED, IPT, · · · )
Doped asymmetric two-band Hubbard models
Appendix
Introduction to DMFT
Numerical implementation of DMFT
DMFT iteration scheme:
Doped asymmetric two-band Hubbard models
Appendix
Additional results at and away from half-filling
Result:
ratio of QP-renormalization factors:
r
≡
Z narrow
/ Z wide
-0.7
a)
-0.8
T=1/25
T=1/32
T=1/40
0.7
0.6
0.5
b)
0.4
0.3
0.2
0.1
0
1.8
I
2
0.4
0.2
0
2
II
U
2.5
2.2
U
2.4
2.6
III
2.8
Doped asymmetric two-band Hubbard models
Appendix
Additional results at and away from half-filling
Result: intra-orbital double occupancies
( m =narrow,wide)
0.38
a) wide band
0.37
b) T=1/25
T=1/40
0.15
0.1
wide band
0.05
narrow band
0
1.8
2 2.2
U
2.4
2.6
2.8
Doped asymmetric two-band Hubbard models
Appendix
Additional results at and away from half-filling
Quasi-particle weight for narrow band a) 0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7
U=3.00
U=2.80
U=2.60
U=2.40
U=2.20
U=2.00
0.1
0
2 2.1
n (total filling)
2.2
2.3
Doped asymmetric two-band Hubbard models
Appendix
Additional results at and away from half-filling
Real part of the self-energy b)
3
2.5
2
1.5
wide band n=2.60
n=2.40
n=2.20
n=2.15
n=2.10
n=2.05
n=2.00
1
0.5
0
0 0.5
1 1.5
2 2.5
" n narrow band
0 0.5 1 1.5 2 2.5
" n
