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Recent developments in Bogoliubov Many-Body
Perturbation Theory
Pierre Arthuis
IRFU, CEA, Université Paris - Saclay
with T. Duguet (CEA Saclay), J.-P. Ebran (CEA DAM), H. Hergert (MSU),
R. Roth & A. Tichai (TU Darmstadt)

Bridging nuclear ab-initio and energy-density-functional theories
IPN, Orsay - October 6th 2017

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

1/24

The BMBPT project
Particle-number-restored BMBPT formalism
Exact diagrammatic expansion with symmetry breaking and restoration
[Duguet and Signoracci, J. Phys. G 44, 2017]

Formalism actualization
Diagonal version
hΨ|H|Φi
hΨ|Φi

Off-diagonal version
hΨ|H|Φ(φ)i
hΨ|Φ(φ)i

Ab initio (This talk)

EDF (Thomas’ talk)

Realist H
High order

Effective H
Low order

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

2/24

The BMBPT project
Manual derivation of order 3

Diagonal implementation
hΨ|H|Φi
hΨ|Φi

Check against MBPT limit

Automatic derivation
[PA, Duguet, Ebran]

Numerical implementation
Ab initio

ary
n
i
lim

- 60

Egs [MeV]

- 80

Realist H
High order

- 100
- 120





- 140
- 160
- 180

16



Pre



18






20






22








HFB



MCPT(2)



IM- SRG



Gorkov SCGF



BMBPT(2)

24

A

[PA, Tichai, Hergert, Roth, Duguet, in prep.]
P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

3/24

Outline

1

From BMBPT equations to diagrams...
• Definition
• Manual derivation up to third order

2

...and from BMBPT diagrams back to equations
• Automatic generation of connected diagrams
• Automatic derivation of analytical formulas

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

4/24

Outline

1

From BMBPT equations to diagrams...
• Definition
• Manual derivation up to third order

2

...and from BMBPT diagrams back to equations
• Automatic generation of connected diagrams
• Automatic derivation of analytical formulas

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

5/24

Bogoliubov Many-Body Perturbation Theory
• Bogoliubov vacuum |Φi, βk |Φi = 0 ∀k
• Grand potential operator Ω ≡ H − λA in quasiparticle basis
Ω = Ω00 +

o
1 X 11 †
1 X n 20 † †
Ωk1 k2 βk1 βk2 +
Ωk1 k2 βk1 βk2 +Ω02
β
β
k
k
k1 k2 2 1 +. . .
1!
2!
k1 k2

k1 k2

• Perturbative expansion of ground-state energy (Ω = Ω0 + Ω1 )
Z ∞
n
dτ1 T [Ω1 (τ1 )Ω(0)]
E0 = hΦ| Ω(0) −
Z ∞0
o
1
+
dτ1 dτ2 T [Ω1 (τ1 ) Ω1 (τ2 )Ω(0)] + ... |Φic
2! 0
• Propagators (also anomalous G −−(0) for off-diagonal theory)
+−(0)
Gk1 k2 (τ1 , τ2 )

P. Arthuis - IRFU, CEA, UPSy

hΦ|T[βk†1 (τ1 )βk2 (τ2 )]|Φi
−+(0)

= −Gk2 k1 (τ2 , τ1 )
hΦ|Φi
Recent developments in BMBPT

6/24

Building blocks of the diagrammatic
• Normal-ordered form of Ω with respect to Φ
Ω=

+

+

+

+ ...

Ω00
Ω11

• Propagators

Ω20

Ω02

k2 τ2

k2 τ2
−+(0)

+−(0)

Gk1 k2 (τ1 , τ2 )

Gk1 k2 (τ1 , τ2 )
k1 τ1

k1 τ1

• Main diagrammatic rules








Wick theorem (off-diagonal Wick theorem for off-diagonal theory)
No external legs
No oriented loop between vertices
No self-contraction (anomalous one for off-diagonal theory)
Propagators go out of the Ω vertex at time 0
Equivalent lines
Discard topologically equivalent diagrams

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

7/24

Outline

1

From BMBPT equations to diagrams...
• Definition
• Manual derivation up to third order

2

...and from BMBPT diagrams back to equations
• Automatic generation of connected diagrams
• Automatic derivation of analytical formulas

P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

8/24

Low-order diagrams
• First- and second-order diagrams

PE0.1

[Duguet and Signoracci, J. Phys. G 44, 2017]

PE1.1

PE1.2

• Third-order diagrams

PE2.1

PE2.2

PE2.3

PE2.4

PE2.5

PE2.6

PE2.7

PE2.8

Validation of the manual derivation by checking the MBPT limit
P. Arthuis - IRFU, CEA, UPSy

Recent developments in BMBPT

9/24


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