Hierarchy of Self-Consistent Approximate Methods and the Parquet ...

Hierarchy of Self-Consistent Approximate Methods and the
Parquet Formalism
Shuxiang Yang
April 29, 2011
1

Motivation
Parquet formalism
• Vertex
• Scattering channels
• Reducibility
• Equations
Outline
Hierarchy of self-consistent approximate methods
• Hartree-Fock approximation F = 0
• Second-Order perturbation theory (SOPT)
F = U
• Random Phase approximation (RPA), T-matrix approximation and Fluctuation Exchange
(FLEX) approximation Γ = U
• Parquet approximation (PA)
Λ = U
• Multi-Scale Many-Body (MSMB) approach
Λ II = Λ I,QMC
Numerical results for a toy model and Hubbard model
Conclusion
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 2

Motivation
For the strongly-correlated electron systems, the Hubbard model is a minimum
model
H = −t ∑ 〈i,j〉,σ (c† i,σ c j,σ + h.c.) + U ∑ i n i,↑n i,↓
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 3

Motivation
For the strongly-correlated electron systems, the Hubbard model is a minimum
model
H = −t ∑ 〈i,j〉,σ (c† i,σ c j,σ + h.c.) + U ∑ i n i,↑n i,↓
Diagrammatic method is a popular approach to analyze this model
Central question:
How to sum up as many as possible physically relevant Feynman diagrams
=⇒ Self-consistent diagrammatic approximate methods
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 4

Green function and self-energy
(x 2 , t 2 )
(x 1 , t 1 )
• 1-particle Green function G
ψ(x 2 , t 2 ) = ∫ G(x 2 , t 2 ; x 1 , t 1 )ψ(x 1 , t 1 ) dx 1 dt 1
⊲ amplitude (phase) accumulated when a particle moving around some path
⊲ long-ranged
• self-energy Σ
⊲ renormalization due to the interaction with other particles
⊲ short-ranged
• Similarly, we can define 2-partilce Green function and 2-particle “self-energy” (vertex).
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 5

Self-Consistent Diagrammatic Calculation
• Conventional perturbation calculation (Green function G based)
⊲ G = G[G 0 , U], e.g., 1st-order (Hartree)
• Conventional perturbation calculation (self-energy Σ based)
⊲ Σ = Σ[G 0 , U]
⊲ G = G 0
1−ΣG 0
• Self-Consistent Diagrammatic Calculation
⊲ Σ = Σ[G, U, F ]
⊲ G = G 0
+ + +
G
Σ =
1−ΣG 0
Σ
1st iter
+ +
+
2nd iter
+ + +
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 6

Schwinger-Dyson equation
Schwinger-Dyson equation: Σ = Σ[G, U,F ]
• relating Σ to F , a bridge between one-particle and two-particle quantities
• exact, derived from the Equation of Motion
Σ
=
+
F
• Two questions about F :
⊲ What is F – Vertex describing the 2-particle scattering process.
⊲ How to determine F — Using parquet formalism.
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 7

Vertex
v
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 8

Vertex
v
+
initial
3 1
+ + 4
H
2
P
F ph
h
final
|3, 4 >→ |1, 2 >
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 9

Vertex
3
P
1
4
initial
F ph
h
H
final
2
|3, 4 >→ |1, 2 >
• vertex: matrix used to describe scattering process
• F ph
h
(12; 34): the amplitude of a particle-hole pair scattered from its initial state |3, 4 > to
the final state |1, 2 >.
• i ≡ P i ≡ (k i , iω ni ), or i ≡ (P i , σ i , m i ) if the internal degrees of freedom are included, say
the spin σ i or band index m i .
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 10

scattering channels
p-h horizontal channel
3
P
1
4
initial
F ph
h
H
final
2
|3, 4 >→ |1, 2 >
3 1
P
4
F ph v
H
2
|3, 1 >→ |4, 2 >
p-h vertical channel
initial
final
3
P
1
4
F pp
P
p-p channel
2
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 11

scattering channels
p-h horizontal channel
3
P
1
4
initial
F ph
h
H
final
2
|3, 4 >→ |1, 2 >
analogous to the way to
arrange small pieces of parquet
3 1
P
4
F ph v
H
2
|3, 1 >→ |4, 2 >
p-h vertical channel
initial
final
3
P
1
4
F pp
P
p-p channel
2
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 12

1-particle reducibility
reducible
+
⇒T
irreducible
+
⇒
Σ
cutting one line !
One-particle formalism
1-particle
re
irre
G T Σ
Dyson equations
G = G 0 + G 0 ∗ T ∗ G 0
G = G 0 + G 0 ∗ Σ ∗ G
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 13

2-particle reducibility
reducible
+
cutting two lines !
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 14

2-particle reducibility
reducible
+
irreducible
+
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 15

2-particle reducibility
reducible
+
irreducible
+
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 16

2-particle reducibility
reducible
+
irreducible
+
+
fully irreducible
+
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 17

2-particle reducibility
reducible
+
irreducible
+
+
⇒
⇒
F
Γ
fully irreducible
+
v
+
⇒
Λ
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 18

2-particle reducibility
reducible
+
irreducible
+
+
⇒
⇒
F
Γ
fully irreducible
+
v
+
⇒
Λ
How are F , Γ and Λ related –> Bethe-Salpeter equation and Parquet equation.
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 19

Bethe-Salpeter equation and Parquet equation
• Bethe-Salpeter equation
⊲ describing how a reducible vertex is decomposed in its own scattering channel
⊲ exact, derived from the categorization of the Feynman diagrams
F
= Γ + Γ F
• Parquet equation
⊲ describing how a irreducible vertex is still decomposable in the other two scattering
channels
⊲ exact, derived from the categorization of the Feynman diagrams
Γ
Γ v
= Λ + + Γ p F p
F v
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 20

Bethe-Salpeter equation and Parquet equation
• Bethe-Salpeter equation
⊲ describing how a reducible vertex is decomposed in its own scattering channel
⊲ exact, derived from the categorization of the Feynman diagrams
F
= Γ + Γ F
• Parquet equation
⊲ describing how a irreducible vertex is still decomposable in the other two scattering
channels
⊲ exact, derived from the categorization of the Feynman diagrams
Γ
Γ v
= Λ + + Γ p F p
F v
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 21

Parquet formalism
1-particle
2-particle
re irre fully irre
G T Σ
S.D.
χ F Γ Λ
B.S.
more localized
Parquet
Bethe-Salpeter equation and Parquet equation
• relating F , Γ and Λ
2-particle Dyson equations (recall Dyson e.q. for 1-particle formalism)
• χ = χ 0 + χ 0 ∗ F ∗ χ 0
• χ = χ 0 + χ 0 ∗ Γ ∗ χ
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 22

Hierarchy of self-consistent approximate methods
1-particle
2-particle
F = 0
G T Σ
S.D.
χ F Γ Λ
H-F
re irre fully irre
B.S.
more localized
Parquet
Better approximation
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 23

Hartree-Fock approximation: F = 0
• No vertex correction to the self-energy
• 〈n i↑ n i↓ 〉 = 〈n i↑ 〉〈n i↓ 〉 (χ = G ∗ G)
• Lowest-order correction: O(U 2 )
Σ
=
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 24

Hierarchy of self-consistent approximate methods
1-particle
2-particle
F = 0
F ≈ v
G T Σ
S.D.
χ F Γ Λ
H-F
SOPT
re irre fully irre
B.S.
more localized
Parquet
Better approximation
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 25

Self-consistent Second-Order Perturbation Theory (SOPT): F ≈ v
• using Schwinger-Dyson equation only
• Lowest-order correction: O(U 3 )
Σ =
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 26

Hierarchy of self-consistent approximate methods
1-particle
2-particle
F = 0
F ≈ v
Γ ≈ v
re irre fully irre
G T Σ
S.D.
χ F Γ Λ
B.S.
more localized
H-F
SOPT
RPA, TMA, FLEX
Parquet
Better approximation
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 27

RPA, TMA and FLEX: Γ ≈ v
• using both Schwinger-Dyson and Bethe-Salpeter equations
• more diagrams are included
⊲ ring-type: Random Phase Approximation (RPA)
⊲ ladder-type: T-matrix approximation (TMA)
⊲ ring-type and ladder-type: Fluctuation Exchange approximation (FLEX)
• Lowest-order correction: O(U 3 )
+
+
+
+
+
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 28

Hierarchy of self-consistent approximate methods
1-particle
2-particle
F = 0
F ≈ v
Γ ≈ v
Λ ≈ v
re irre fully irre
G T Σ
S.D.
χ F Γ Λ
B.S.
more localized
Parquet
H-F
SOPT
RPA, TMA, FLEX
Parquet Approximation
Better approximation
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 29

Parquet approximation: Λ ≈ v
• using Schwinger-Dyson, Bethe-Salpeter and Parquet equations
• more diagrams: ring+ladder-type and simply-crossed, are included
• Lowest-order correction: O(U 5 )
+ + +
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 30

Parquet approximation: Λ ≈ v
• using Schwinger-Dyson, Bethe-Salpeter and Parquet equations
• more diagrams: simply-crossed and ring+ladder-type, are included
• Lowest-order correction: O(U 5 )
+ + +
• still some contributions are missing: fully irreducible (complicatedly crossed) → QMC
+
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 31

Hierarchy of self-consistent approximate methods
1-particle
2-particle
F = 0
F ≈ v
Γ ≈ v
Λ ≈ v
re irre fully irre
G T Σ
S.D.
χ F Γ Λ
B.S.
more localized
Parquet
H-F
SOPT
RPA, TMA, FLEX
Parquet Approximation
Λ II ≈ Λ I,DCA/QMC
Better approximation
DCA/QMC+Parquet
MSMB
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 32

DCA/QMC + parquet:
Λ II ≈ Λ I,DCA/QMC
• Multi-Scale Many-Body (MSMB) approach: SciDAC project
⊲ short length scales treated explicitly with QMC methods
⊲ intermediate length scales treated diagrammatically using fully irreducible vertices obtained
from QMC
⊲ long length scales treated in the mean field.
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 33

summary of the different approximate methods
G
Σ
⇐ F = 0
HF
G
Σ
⇐ F ≈ v
SOP T
G
⇐ Γ ≈ v
RP A, T MA, F LEX
Σ
F
G
Γ
P A
⇐ Λ ≈ v
MSMB
Λ II ≈ Λ I,DCA/QMC
Σ
F
• iterate until self-consistency is satisfied
• It is also a hierarchy of methods with different computational costs
• there are two levels of self-consistency for the parquet calculation
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 34

application I: toy model
The Hamiltonian for the classical anharmonic oscillator is
Its partition function is
H = p2
2m + 1 2 mω2 0x 2 + ux 4 (1)
Z c
= 1 ∫ ∞
Z c,0 Z c,0
−∞
dp
∫ ∞
−∞
dxe −βH =
1
√
2πG
0
∫ ∞
−∞
dψe − ψ2
2G 0 e − g 4! ψ4 (2)
where G 0 = 1/(βmω 2 0), βu = g/4! and ψ ≡ x. From this, we can construct the
conventional perturbation or the self-consistent field formalism.
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 35

esult
3
−Σ
2
1
Hartree
2nd-order
self-consistent 2nd-order
FLEX
parquet approximation
exact
0
0 5 10 15 20 25 30
g
• from 2nd-order approximation, FLEX, to parquet approximation, the results become better.
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 36

application II: Hubbard model (N c = 1)
Hubbard model: H = −t ∑ 〈i,j〉,σ (c† i,σ c j,σ + h.c.) + U ∑ i n i,↑n i,↓
-0.46
-0.47
Exact
SC 2nd-order
FLEX
PA
G(τ)
-0.48
-0.49
U/T=1.5, Nc=1, Nl=64
-0.5
0 0.2 0.4 0.6 0.8 1
τ/β
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 37

application II: Hubbard model (N c = 4 × 4)
U = 2t, T = 0.3t
S. X. Yang et al., PRE 80, 046706 (2009).
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 38

Conclusion
• Parquet formalism
• Hierarchy of different approximate methods
1-particle
2-particle
F = 0
F ≈ v
Γ ≈ v
Λ ≈ v
re irre fully irre
G T Σ
S.D.
χ F Γ Λ
B.S.
more localized
Parquet
H-F
SOPT
RPA, TMA, FLEX
Parquet Approximation
Λ II ≈ Λ I,DCA/QMC
Better approximation
DCA/QMC+Parquet
MSMB
Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 39