Quantum Dots
Model, Methods and Implementation
Results and Discussions
Many-Body Approaches to Quantum Dots
Patrick Merlot
Department of Computational Physics
November 4, 2009
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Table of contents
1
Quantum Dots
2
Model, Methods and Implementation
3
Results and Discussions
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
What is a quantum dot?
Physics of QDs
Applications of QDs
Table of contents
1
Quantum Dots
What is a quantum dot?
Physics of QDs
Applications of QDs
2
Model, Methods and Implementation
3
Results and Discussions
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
What is a quantum dot?
Physics of QDs
Applications of QDs
What is a quantum dot?
(a)
Definition
Semiconductor whose
charge-carriers are confined in
space.
(b)
6= types/shapes/fabrication
⇒ 6= QDs
(c)
Figure 1: Possible types/shapes of QDs.
(a) Various shapes of QDs pillars∼ 0.5µm,
(b) Colloidal QD (InGaP+ZnS+lipid)∼ 10nm,
(c) QD defined by 5 metallic gates on GaAs where 2-DEG is
trapped.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
What is a quantum dot?
Physics of QDs
Applications of QDs
Physics of QDs
Quantum dot properties
Semiconductor band gap
increased by size quantization
Figure 2: Electronic band structure of semiconductor and
quantum dots (Courtesy of J.Winter[4]).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
What is a quantum dot?
Physics of QDs
Applications of QDs
Physics of QDs
Quantum dot properties
Semiconductor band gap
increased by size quantization
Tunable optical/electrical
properties
Perfect system for computational
studies
Figure 3: Fluorescent emission (Courtesy of J.Winter[4]).
Figure 4: Emission spectra of various quantum dots.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
What is a quantum dot?
Physics of QDs
Applications of QDs
Applications of QDs
Possible applications
Biological nano-sensors
Qubits for QCA
LEDs
Solar cells
Figure 5: QDs imaging in live animals compared to classical
organic dyes. (Courtesy of X. Gao)
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Table of contents
1
Quantum Dots
2
Model, Methods and Implementation
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
3
Results and Discussions
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Model of a quantum dot
Simple model of a quantum dot
Atomic scale problem ⇒ Quantum mechanics for an accurate description of
the system.
⇒ at rest, solve the time-independent Schrödinger equation Ĥ|Ψi = E |Ψi.
Not modelling all nuclei/electrons ⇒ just model the few quasiparticles confined
by the semiconductor.
Figure 6: Illustration of a quantum dot model
(Courtesy of S.Kvaal[5]).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Model of a N-particle system
The Schrödinger equation
The Schrödinger equation of a N-particle system
Ĥ(r1 , r2 , . . . , rN )|Ψκ (r1 , r2 , . . . , rN )i = Eκ |Ψκ (r1 , r2 , . . . , rN )i (1)
where ri reprensents the (spatial/spin) coordinates of quasiparticle i,
κ stands for all quantum numbers needed to classify a given
N-particle state,
|Ψκ i and Eκ are the eigenstates and eigenenergies of the system.
How to define our Hamiltonian?
Ĥ =
Ne
X
pi 2
+ ...
2m∗
i=1
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Definitions of the interactions/potentials
Forces/Fields acting on the quasiparticles:
Forces confining the particles ⇒ Confining potential
Interactions between the particles ⇒ Interaction potential
The Hamiltonian of our two-electron quantum dot model
Confining potential ⇒ the harmonic oscillator (parabolic) potential
Ĥ =
NX
e =2
i=1
NX
e =2
pi 2
1 ∗ 2
e2
1
+
m ω0 kri k2 +
,
∗
2m
2
4π0 r kr1 − r2 k
i=1
Interaction potential ⇒ the two-body Coulomb interaction
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Many-Body Approaches to Quantum Dots
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Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Applying an external magnetic field
1
pi −→ pi + eA , where A is the vector potential defined by B = ∇ × A.
in coordinate space pi → −i~∇i ,
using a Coulomb gauge ∇ · A = 0 (by choosing A(ri ) =
(pi + eA)2 →
B × ri
),
2
«
„
e
e2 2
~2
∇2i − i~ ∗ A · ∇i +
A .
−
∗
∗
2m
m
2m
(3)
In terms of B, the linear and quadratic terms in A have the form
−i~e
e2 2
e
e2 2 2
A · ∇i =
B · L , and
A =
B ri . where L = −i~(ri × ∇i )
m∗
2m∗
2m∗
8m∗
is the orbital angular momentum operator of the electron i.
2
ωc ˆ
Sz , where Ŝ
2
∗
is the spin operator of the electron and gs is its effective spin gyromagnetic
ratio and ωc = eB/m∗ is known as the cyclotron frequency.
B also acts on spin with the additional energy term: Ĥs = gs∗
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Final Hamiltonian
The final Hamiltonian reads:
Coulomb
interactions
Harmonic ocscillator
potential
Ĥ =
}|
{
z
}|
{ 2
X
1
e
1
∇2 +
m∗ ω02 kri k2
+
2m∗ i
2
4π0 r
|ri − rj |
Ne X
−~2
i=1
+
z
Ne X
i=1
|
i<j
1 ∗
m
2
ω 2
c
2
1
1 ∗
(i)
(i)
kri k + ωc L̂z + gs ωc Ŝz ,
2
2
{z
}
2
single particle interactions
with the magnetic field
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Many-Body Approaches to Quantum Dots
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Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Scaling the problem: dimensionless form of Ĥ
r
New constant, the oscillator frequency ω =
ω0 +
“ ω ”2
c
2
,
New units:
the energy unit ~ω,
the length unit, the oscillator length defined by l =
p
~/(m∗ ω) .
The dimensionless Hamiltonian is now
Ĥ =
–
«
Ne »
Ne „
X
X 1
X
1
1
1 ωc (i) 1 ∗ ωc (i)
− ∇2i + ri2 + λ
+
ml + gs
ms
.
2
2
rij
2 ~ω
2 ~ω
i=1
i<j
i=1
where the new dimensionless parameter λ = l/a0∗ describes the strength of
the electron-electron interaction (a0∗ being the effective Bohr radius).
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Many-Body Approaches to Quantum Dots
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Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
The Hamiltonian solved in this project
„
4~2
2 ∗2
4ω0 m + e 2 B 2
« 41
Role of B?
Squeezing the particles should
increase the strength of the
electron-electron interaction.
λ only decreases as the magnetic field
increases in this model.
Confinement strength λ
1
λ(B) = ∗
a0
1.6
1.4
1.2
1.0
0.8
0
5
10
B (T)
15
20
Figure 7: Dimensionless confinement strength λ as a
function of the magnetic field strength in GaAs.
In the rest of the project, we will solve the following Hamiltonian:
Ĥ =
–
Ne »
X
X 1
1
1
− ∇2i + ri2 + λ
.
2
2
rij
i=1
i<j
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Quantum Dots
Model, Methods and Implementation
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Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
The method of variational calculus
Definition
Method to solve the Schrödinger eq. more efficiently than using numerical integration.
Based on the method of Lagrange multipliers, where the functional to minimize
(the energy functional) is an integral over the unknown wave function |Φi
E [Φ] =
R ∗
Φ HΦdτ
hΦ|H|Φi
= R ∗
,
hΦ|Φi
Φ Φdτ
(7)
while subject to a normalization constraint hΦ|Φi = 1.
The method introduces new variables for each of the constraints (the Lagrange
multipliers ) and defines the Lagrangian (Λ) with respect to |Φi as
Λ(Φ, ) = E [Φ] − (hΦ|Φi − 1) ,
Find stationnary solutions by solving the set of equations by writing
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(8)
∂Λ
= 0.
∂Φ
Many-Body Approaches to Quantum Dots
Quantum Dots
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Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
The Hartree-Fock method
Definition
The HF method is a particular case of variational method in accordance with
the independent particle approximation,
the Pauli exclusion principle.
The approximated wave function
To fullfil these criteria, the wave-function must be antisymmetric with respect
to an interchange of any two particles:
Φ(r1 , r2 , . . . , ri , . . . , rj , . . . , rN ) = −Φ(r1 , r2 , . . . , rj , . . . , ri , . . . , rN ).
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Quantum Dots
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Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
The Hartree-Fock wave function
The Slater determinant is an antisymmetric product of the single particle orbitals:
˛
˛ ψα (r1 )
˛
˛
˛ ψ (r1 )
1 ˛˛ β
Φ(r1 , r2 , . . . , rN , α, β, . . . , σ) = √
..
˛
N! ˛
.
˛
˛
˛ ψσ (r1 )
ψα (r2 )
...
ψβ (r2 )
...
..
.
..
ψσ (r2 )
...
.
˛
ψα (rN ) ˛˛
˛
ψβ (rN ) ˛
˛
˛,
..
˛
˛
.
˛
˛
ψσ (rN ) ˛
(10)
It can be rewritten as
1 X
ΦT (r1 , r2 , . . . , rN , α, β, . . . , σ) = √
(−)p PΨα (r1 )Ψβ (r2 ) . . . Ψσ (rN )
N! p
√
`
´
= N! A Ψα (r1 )Ψβ (r2 ) . . . Ψσ (rN ) ,
by introducing the antisymmetrization operator A =
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1 X
(−)p P̂.
N! P
Many-Body Approaches to Quantum Dots
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Quantum Dots
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Matrix elements calculations
Definition
We write the Hamiltonian for N electrons as Ĥ = Ĥ0 + Ĥ1 =
N
X
i=1
ĥi +
N
X
v (ri , rj ),
i<j
where rij = k~
ri − r~j k, ĥi and v (ri , rj ) are respectively the one-body and the two-body
Hamiltonian.
Using properties of A and commutation rule with Ĥ0 and Ĥ1 , one can write:
Z
Φ∗T Ĥ0 ΦT dτ =
N Z
X
Ψ∗µ (r)ĥΨµ (r)dr =
µ=1
Z
Φ∗T Ĥ1 ΦT dτ =
N
X
hµ|h|µi.
N
N
1 XX
hµν|V |µνiAS .
2 µ=1 ν=1
where we define the antisymmetrized matrix element as
hµν|V |µνiAS =
Z hµν|V |µνi − hµν|V |νµi, with the following shorthands
hµν|V |µνi =
Ψ∗µ (ri )Ψ∗ν (rj )V (rij )Ψµ (ri )Ψν (rj )dri drj .
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µ=1
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Model of a quantum dot
The method of Variational calculus
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The many-body perturbation theory
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Quantum Dots
Model, Methods and Implementation
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The Hartree-Fock energy in the harmonic oscillator basis
The energy functional
The energy functional is our starting point for the Hartree-Fock calculations.
E [ΦT ] = hΦT |Ĥ0 |ΦT i + hΦT |Ĥ1 |ΦT i
=
N
X
hµ|h|µi +
µ=1
1
2
N X
N
X
hµν|V |µνiAS .
(15)
(16)
µ=1 ν=1
We expand each single-particle eigenvector Ψi in terms of a convenient complete set
of single-particle states |αi (the harmonic oscillator eigenstates in our case),
X
Ψi = |ii =
ciα |αi.
(17)
α
The energy functional now reads
E [Φ] =
N X
X
i=1 αγ
Ciα∗ Ciγ hα|h|γi +
N
1 X X α∗ β∗ γ δ
C Cj Ci Cj hαβ|V |γδiAS .
2 i,j=1 αβγδ i
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The Hartree-Fock equations (1)
Remember the method of the Lagrange multipliers
1
Define a functional E [ΦT ],
2
Identify the constraints: hΨi |Ψj i = δij which implies hΦT |ΦT i = 1,
X
X
with hΨi |Ψj i =
Ciα∗ Cjβ hα|βi =
Ciα∗ Cjα
|
{z
}
α
αβ
3
Define the Lagrangian Λ
δαβ
Λ(C1α , C2α , . . . , CNα , 1 , 2 , . . . , N ) = E [ΦT ] −
N
X
i=1
!
i
X
Ciα∗ Ciα − δij
. (19)
α
where i are the Lagrange multipliers for each of the normalization constraints.
4
Get the system of equations to solve by setting Λ
dΛ
d
≡
[Λ(C1α , C2α , . . . , CNα , 1 , 2 , . . . , N )] = 0,
dΦT
dCiα∗
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∀ i ∈ N∗ .
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The Hartree-Fock equations (2)
Treating Ciα and Ciα∗ as independent, we arrive at the Hartree-Fock equations
(one equation for each basis state |αi)
Effective potential
hα|U|γi
z
"
X
γ
hα|h|γi
Ciγ
+
N X
X X
γ
which we can rewrite as
}|
Cjβ∗
j=1 βδ
X
{
#
hαβ|V |γδiAS
|
{z
}
Cjδ Ciγ = i Ciα ,
Two-body interaction
matrix elementVαβγδ
Oαγ Ciγ = i Ciα ,
∀ α ∈ H.
γ
⇒ System of non-linear equations in the Ciα∗ , since Oαγ depends itself on the
unknowns.
⇒ To be solved by an iterative procedure.
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Model of a quantum dot
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Implementation
The Hartree-Fock (self-consistent) iterative procedure
1
Compute the effective Coulomb interaction potential
α(0)
hα|U (0) |γi with an initial guess of the Ci
.
2
Build the resulting Fock matrix O.
3
Solve the linearised system given by the equations
(Fock matrix diagonalization)
X
Initial guess of
HF orbitals
Fock matrix
computation
Fock matrix
diagonalization
[hα|h|γi + hα|U|γi] Ciγ = i Ciα .
γ
no
Selfconsistency
achieved?
(k)
at iteration (k), store the output eigenenergies i
and the coefficients of the new eigenvectors
4
α(k)
Ci
.
Substitute back the new coefficients to compute a new
Coulomb interaction potential.
5
...
6
Continue the process until self-consistency is reached.
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yes
Compute the
total energy
End
Figure 8: Flowchart of
Hartree-Fock algorithm.
Many-Body Approaches to Quantum Dots
Quantum Dots
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Model of a quantum dot
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The many-body perturbation theory
Implementation
Many-body perturbation theory
Take the Hamiltonian Ĥ = Ĥ0 + Ĥ 0 , and treat Ĥ 0 as a perturbation, such as the
Coulomb interaction.
Suppose Φn eigenfunctions of Ĥ0 corresponding to the eigenvalues En : Ĥ0 Φn = En Φn .
Consider the effect of the perturbation on a particular state Φ0 .
We denote by Ψ0 the state into which Φ0 changes under the action of the
perturbation, so that Ψ0 is an eigenfunction of Ĥ, corresponding to the eigenvalue E .
Ĥ0 Φ0 = E0 Φ0 .
(22)
ĤΨ0 = E Ψ0 .
(23)
Therefore Φ0 and Ψ0 denote the ground states of the unperturbed and perturbed
systems respectively.
Since Ĥ0 is Hermitian, one can show that:
E − E0 =
hΦ0 |Ĥ 0 |Ψ0 i
.
hΦ0 |Ψ0 i
(24)
which is an exact expression and independent of any particular perturbation method.
Since Ψ0 is unknown, using a projection operator R for the state Φ0 defined by the
equation
RΨ = Ψ − Φ0 hΦ0 |Ψi,
(25)
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Many-Body Approaches to Quantum Dots
Model of a quantum dot
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Quantum Dots
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The perturbed energy
The perturbed energy can be derived from the iterated Ψ0 which gives
E − E0 =
∞
X
˙
n=0
Φ0 |Ĥ 0
„
R
E0 − Ĥ0
«n
¸
(E0 − E + Ĥ 0 ) |Φ0 .
We shall write
∆E = E − E0 = ∆E (1) + ∆E (2) + ∆E (3) + . . .
where the mth -order energy correction ∆E (m) contains the mth -order power of the
perturbation Ĥ 0 .
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The many-body perturbation corrections
The 1st -order correction is
∆E (1) = hΦ0 |Ĥ 0 |Φ0 i.
(27)
The 2nd -order correction is
∆E (2) = hΦ0 |Ĥ 0
R
E0 − Ĥ0
(E0 − E + Ĥ 0 )|Φ0 i.
(28)
The 3rd -order energy correction reads
∆E (3) =
∞ X
∞
X
hΦ0 |Ĥ 0 |Φm ihΦm |Ĥ 0 |Φn ihΦn |Ĥ 0 |Φ0 i
(E0 − Em )(E0 − En )
n=0 n=0
− hΦ0 |Ĥ 0 |Φ0 i
∞
X
hΦ0 |Ĥ 0 |Φn ihΦn |Ĥ 0 |Φ0 i
.
(E0 − En )2
n=0
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The MBPT corrections expanded in a basis set
It is possible to rewrite the many-body energy corrections in particle and hole state
formalism by using the expression of Ĥ 0 as expressed in terms of anihilation (ck ) and
creation (ck† ) operators
1X
Ĥ 0 =
hij|v |klici† cj† cl ck ,
2 ijkl
The previous many-body perturbation corrections now read
∆E (1) = hΦ0 |Ĥ 0 |Φ0 i. =
1X
hh1 h2 |v |h1 h2 ias ,
2hh
(30)
1 2
∆E (2) =
∞
X
hΦ0 |Ĥ 0 |Φn ihΦn |Ĥ 0 |Φ0 i
1
=
E
−
E
4
n
0
n=0
X
h1 h2 p1 p2
|hh1 h2 |v |p1 p2 i|2as
,
h1 + h2 − p1 − p2
where hi and pi are respectively hole states and particles states, and i are the single
particle energies of the basis set.
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Quantum Dots
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The MBPT corrections expanded in a basis set
The 3rd -order many-body perturbation correction reads
∆E (3) =
∞
∞ X
X
hΦ0 |Ĥ 0 |Φm ihΦm |Ĥ 0 |Φn ihΦn |Ĥ 0 |Φ0 i
(E0 − Em )(E0 − En )
n=0 n=0
− hΦ0 |Ĥ 0 |Φ0 i
∞
X
hΦ0 |Ĥ 0 |Φn ihΦn |Ĥ 0 |Φ0 i
(E0 − En )2
n=0
(3)
(3)
(3)
= ∆E4p−2h + ∆E2p−4h + ∆E3p−3h ,
where
(3)
∆E4p−2h is the contribution to the third-order energy correction due to the
4-particle/2-hole excitations,
(3)
∆E2p−4h is the contribution to the third-order energy correction due to the
2-particle/4-hole excitations,
(3)
∆E3p−3h is the contribution to the third-order energy correction due to the
3-particle/3-hole excitations.
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Model of a quantum dot
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The MBPT corrections expanded in a basis set
The contributions to the third-order energy correction can be written as
0
(3)
∆E4p−2h
(3)
∆E2p−4h
(3)
∆E3p−3h
1
X hp1 p2 |v |p3 p4 ias hp3 p4 |v |h1 h2 ias
hh
h
|v
|p
p
i
as
1
2
1
2
@
A,
h1 + h2 − p1 − p2 p p
h1 + h2 − p3 − p4
h1 h2 p1 p2
3 4
1
0
X hh1 h2 |v |h3 h4 ias hh3 h4 |v |h1 h2 ias
1 X @ hh1 h2 |v |p1 p2 ias
A,
=
8hhpp
h1 + h2 − p1 − p2 h h
h3 + h4 − p1 − p2
1 2 1 2
3 4
0
0
11
X
X X hh1 h3 |v |p1 p3 ias hp3 h2 |v |h3 h2 ias
hh1 h2 |v |p1 p2 ias
@
@
AA ,
=
h1 + h2 − p1 − p2
h1 + h3 − p1 − p3
p
h h p p
h
1
=
8
X
1 2 1 2
3
3
where the pi denote the particle states, hi the hole states, and i the single particle
energies of the corresponfing state.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Code implementation
Tools
C++ language: for flexibility using classes and efficiency.
Blitz++ library: managing dense arrays.
Lpp / Lapack library: (Fortran) routines for solving linear algebra problems.
Message-Passing Interface: for parallel computing
Functionality
Read parameters from a unique textual input file or command line.
The simulator class performs the initialization and calls other objects.
The orbitalsQuantumNumbers class: generates the harmonic oscillator states.
The CoulombMatrix class generates the Coulomb interaction matrix outside
Hartree-Fock.
The HartreeFock class computes the HF energy and generates the interaction
matrix in the HF basis.
The PerturbationTheory class computes many-body perturbation corrections
from 1st - to 3rd -order, either in the harmonic oscillator or in the HF basis set.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Model of a quantum dot
The method of Variational calculus
The Hartree-Fock method
The many-body perturbation theory
Implementation
Implementation issues
Difficulties encountered if not making use of symmetries
Huge Fock matrix to diagonalize (grows exponentially with R b ).
Huge Coulomb interaction matrix to store
Vαβγδ = h(n1 , ml1 )(n2 , ml2 )|V |(n3 , ml3 )(n4 , ml4 )i is an 8-dimensional array.
Solutions implemented
By using the symmetries and properties of the Coulomb interaction:
Vαβγδ does not act on spin: ms1 = ms3 & ms2 = ms4 .
Vαβγδ conserves the total spin and angular momentum: ml1 + ml2 = ml3 + ml4 .
By sorting the states of the basis by blocks of identical angular (ml ) and spin (ms )
quantum numbers:
It allows to reduce the storage of the Coulomb interactions per blocks of couple
of states by avoiding to store zeros’s elements.
The Fock matrix appears as block diagonal, allowing much smaller eigenvalue
problems to solve.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Table of contents
1
Quantum Dots
2
Model, Methods and Implementation
3
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Validation of the simulator
Simple checks
Reproduce the non-interacting ground
state energies.
Reproduce the two-body interaction
matrix elements of OpenFCI almost with
machine precision
Comparison of MBPT results with similar
experiments
0
2nd order MBPT Correction [meV ]
∆E (max(|ml |))
∆E (max(|n|))
with
with
max(n) = 5
max(|ml |) = 11
-0.5
Waltersson computed the open-shell
2nd -order MBPT correction.
-1
Our closed-shell 2nd -order MBPT
correction shows close agreement.
-1.5
-2
0
5
10
15
20
25
30
35
max(n), max(|ml |)
nd
Figure 9: 2 -order perturbation theory correction
−
for the 2e QD. Comparison between results of
Waltersson (top) [3] and our results (down).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Level crossing as a function of B without interactions (1/3)
Fock-Darwin orbitals
When neglecting the repulsions between the particles, the eigenenergies n ml as a
function of the magnetic field B can be solved analytically for a parabolic confining
potential V (r ) = 1/(2m∗ ω02 r 2 ) leading to a spectrum known as the Fock-Darwin
states
Rewriting the eigenenergies in units of ~ω0 , n ml becomes dimensionless and we obtain
s
n ml = (2n + |ml | + 1)
1+
(ωc /ω0 )2
1
− (ωc /ω0 ) ml
4
2
(32)
eB
eB
)2 −
ml .
2m∗ ω0
2m∗ ω0
(33)
s
= (2n + |ml | + 1)
1+(
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Level crossing as a function of B without interactions (2/3)
Twelve-electrons dot
4.5
4.0
4.0
3.5
3.5
3.0
3.0
Energy n ml (~ω0 )
Energy n ml (~ω0 )
Six-electrons dot
4.5
2.5
2.0
2.5
2.0
1.5
1.5
1.0
1.0
0.5
6th shell electron states
5th shell electron states
4th shell electron states
3rd shell electron states
|0, 1i 2nd shell electron state
|0, 0i 1st shell electron state
0.5
3rd shell electron states
|0, 1i 2nd shell electron state
|0, 0i 1st shell electron state
0.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Magnetic field (T)
Figure 10: Spectrum of Fock-Darwin orbitals for 6
non-interacting particles (GaAs:~ω0 = 5meV ,r = 12).
MSc. in Computational Physics
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Magnetic field (T)
Figure 11: Spectrum of Fock-Darwin orbitals for 12
non-interacting particles (GaAs:~ω0 = 5meV ,r = 12).
Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 4
Rb = 3
Rb = 2
Rb = 1
12 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 4
Rb = 3
Rb = 2
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 4
Rb = 3
Rb = 2
Rb = 1
12 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 4
Rb = 3
Rb = 2
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
12 non-interacting particles
Rb = 4
|1, 1i
|0, 3i
b
R =3
|1, 0i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 2
Rb = 1
|0, −3i
|1, −1i
|0, −2i
Rb = 4
|1, 1i
Rb = 3
|1, 0i
|0, −1i
|0, 3i
|0, 2i
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 2
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
|1, −1i
|0, −2i
12 non-interacting particles
Rb = 4
|1, 1i
|0, 3i
b
R =3
|1, 0i
|0, 2i
|0, −3i
|1, −1i
|0, −2i
Rb = 4
|1, 1i
|1, 0i
|0, 3i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
R =2
Rb = 1
|0, −1i
Rb = 3
|0, 2i
b
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 2
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|0, −2i
12 non-interacting particles
|1, 1i
|0, 3i
b
R =3
|1, 0i
R =2
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 1
Rb = 4
|1, −1i
|0, −2i
|0, 2i
b
|0, −1i
|0, −3i
|1, 1i
|1, 0i
|0, 3i
Rb = 3
|0, 2i
Rb = 2
|0, −1i
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|0, −2i
12 non-interacting particles
|1, 1i
|1, 0i
|0, 3i
Rb = 3
R =2
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 1
Rb = 4
|1, −1i
|0, −2i
|0, 2i
b
|0, −1i
|0, −3i
|1, 1i
|1, 0i
|0, 3i
Rb = 3
|0, 2i
Rb = 2
|0, −1i
|0, 1i
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
12 non-interacting particles
6 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
b
R =3
Rb = 2
|0, 1i
|0, −3i
Rb = 4
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
MSc. in Computational Physics
R =1
Rb = 2
|0, 1i
b
|0, 0i
Rb = 3
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|0, −2i
12 non-interacting particles
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
Rb = 3
Rb = 2
|0, 1i
|0, −3i
Rb = 4
|1, −1i
|0, −2i
|1, 1i
|1, 0i
|0, −1i
|0, 3i
|0, 2i
MSc. in Computational Physics
R =1
Rb = 2
|0, 1i
b
|0, 0i
Rb = 3
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
Rb = 4
|1, −1i
|0, −3i
Rb = 4
|1, −1i
|1, 1i
|1, 1i
|0, 3i
|0, −2i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
|0, 3i
|0, −2i
|1, 0i
|0, 2i
|0, −1i
|0, 1i
MSc. in Computational Physics
Rb = 2
|0, 1i
b
|0, 0i
Rb = 3
R =1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
12 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|1, 1i
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
Rb = 4
|1, −1i
|1, 1i
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
Rb = 3
Rb = 2
Rb = 4
|1, −1i
|1, 1i
|0, −2i
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
12 non-interacting particles
|0, −3i
|0, −3i
|1, −1i
b
R =4
|1, 1i
|0, −2i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
12 non-interacting particles
|0, −3i
|0, −3i
|1, −1i
b
R =4
|1, 1i
|0, −2i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|1, 1i
|0, −2i
Rb = 4
|1, 1i
|0, −2i
|0, 3i
|1, 0i
|0, 2i
|0, −1i
|1, −1i
Rb = 3
Rb = 2
|0, 2i
|0, −1i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
|1, 0i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
Rb = 4
|0, −2i
|1, 1i
|1, 0i
|0, 3i
|0, −1i
|1, −1i
|0, 2i
Rb = 3
Rb = 2
|1, 1i
|1, 0i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
Rb = 4
|0, −2i
|1, 1i
|1, 0i
|0, 3i
|0, −1i
|1, −1i
|0, 2i
Rb = 3
Rb = 2
|1, 1i
|1, 0i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
|0, 3i
Rb = 3
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
|1, −1i
|1, 1i
Rb = 3
|1, 0i
|0, 2i
Rb = 2
|1, 1i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
Rb = 4
|0, −2i
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
|1, −1i
Rb = 3
|1, 0i
|0, 2i
Rb = 2
|1, 1i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
Rb = 4
|0, −2i
|1, 1i
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
|1, −1i
|1, 1i
Rb = 3
|1, 0i
|0, 2i
Rb = 2
|1, 1i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
Rb = 4
|0, −2i
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
12 non-interacting particles
|0, −3i
|0, −3i
|1, −1i
Rb = 4
|0, −2i
|1, −1i
|1, 1i
Rb = 3
|1, 0i
|0, 2i
Rb = 2
|1, 1i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
Rb = 2
|0, 1i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
Rb = 4
|0, −2i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
Rb = 4
|0, −2i
|1, −1i
|1, 1i
Rb = 3
|1, 0i
|0, 2i
Rb = 2
|1, 1i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
Rb = 4
|0, −2i
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
|1, −1i
Rb = 4
|0, −2i
Rb = 4
|0, −2i
|1, 1i
Rb = 3
|1, 0i
|0, 2i
MSc. in Computational Physics
|0, 3i
|0, −1i
|0, 2i
Rb = 2
Rb = 2
|0, 1i
|0, 1i
|0, 0i
Rb = 3
|1, 0i
|0, 3i
|0, −1i
|1, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
|1, −1i
Rb = 4
|0, −2i
Rb = 4
|0, −2i
|1, 1i
|1, 1i
Rb = 3
|1, 0i
Rb = 3
|1, 0i
|0, 3i
|0, 3i
|0, −1i
|0, −1i
|0, 2i
|0, 2i
Rb = 2
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Level crossing as a function of B without interactions (1/3)
6 non-interacting particles
|0, −3i
12 non-interacting particles
|0, −3i
|1, −1i
|1, −1i
Rb = 4
|0, −2i
Rb = 4
|0, −2i
|1, 1i
|1, 1i
Rb = 3
|1, 0i
Rb = 3
|1, 0i
|0, 3i
|0, 3i
|0, −1i
|0, −1i
|0, 2i
Rb = 2
|0, 2i
|0, 1i
|0, 0i
MSc. in Computational Physics
Rb = 2
|0, 1i
Rb = 1
|0, 0i
Many-Body Approaches to Quantum Dots
Rb = 1
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Level crossing as a function of λ with interactions (1/2)
Ground state energy for a six-particle QD with λ = 0.1, R = 4
Ground state energy for a six-particle QD with λ = 1, R = 5
19
28
S =0
S =2
S =4
S =6
18
26
16
25
(~ ω)
17
E0 (M, S)
(~ ω)
E0 (M, S)
S =0
S =2
S =4
S =6
27
15
24
14
23
13
22
12
11
21
0
1
2
3
4
5
6
7
8
20
9
0
1
2
3
4
Total angular momentum M
5
6
7
8
9
10
11
12
10
11
12
Total angular momentum M
Ground state energy for a six-particle QD with λ = 15, R = 5
Ground state energy for a six-particle QD with λ = 50, R = 5
107
290
S =0
S =2
S =4
S =6
106
S =0
S =2
S =4
S =6
285
105
(~ ω)
103
E0 (M, S)
E0 (M, S)
(~ ω)
104
102
280
275
101
100
270
99
98
0
1
2
3
4
5
6
7
8
9
10
11
12
Total angular momentum M
265
0
1
2
3
4
5
6
7
8
9
Total angular momentum M
Figure 12: FCI ground state energies for a 6-electron QD with λ = {0.1, 1} (top) and λ = {15, 50} (bottom).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Level crossing as a function of λ with interactions (2/2)
Summary of FCI results using OpenFCI
λ
0.1
0.5
1
2
5
10
11
12
13
15
20
50
FCI ground
state energy
(~ω) R=5
11.197
15.561
20.257
28.032
46.482
73.067
78.143
83.168
88.152
98.027
122.325
266.157
(M,S)
For a 6-particle QD, break of the
model of a single Slater determinant
from λ ' 13.
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(1,2)
(1,2)
(1,4)
A similar study performed on a
2-particle QD indicates a break of the
closed-shell model from λ ' 150.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Exponential convergence of HF as a function of R b (1/3)
Definition
The size of the basis set characterized by R b (R b ∈ N R b ≥ R f )
It defines the maximum shell number in the model space for our Hartree-Fock
computation. It implies the number of orbitals in which each single particle
wavefunction will be expanded, with spin degeneracy the number of states NS is
NS = (R b + 1)(R b + 2)
(34)
The bigger the basis set, the more accurate the single particle wavefunction is
expected. In mathematical notation, R b and the size of the basis set B are defined by
B = B(R b ) =
n
|φnml (r)i
o
: 2n + |ml | ≤ R b ,
(35)
where |φnml (r)i are the single orbital in the Harmonic oscillator basis with quantum
numbers n, ml such that the single orbital energy reads: nml = 2n + |ml | + 1 in
two-dimensions.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Exponential convergence of HF as a function of R b (2/3)
Ground state of a 2-particles QD
Ground state of a 6-particles QD
5
45
λ = 0.5
λ=1
λ=2
λ=0
40
4
35
30
E0HF [~ω]
E0HF [~ω]
3
25
20
2
15
10
1
λ = 0.5
λ=1
λ=2
λ=0
5
0
0
0
2
4
6
8
10
0
2
4
6
8
10
8
10
Ground state of a 20-particles QD
Ground state of a 12-particles QD
350
λ = 0.5
λ=1
λ=2
λ=0
140
120
300
λ=2
250
E0HF [~ω]
E0HF [~ω]
100
80
200
λ=1
150
λ = 0.5
60
100
λ=0
40
50
20
0
0
0
0
2
4
6
Figure 13: Hartree-Fock ground state (E
(bottom).
8
HF
2
4
6
Max. shell number in the basis R b
10
b
as a function of R for 2-,6-electron QD (top) and 12-,20-electron QD
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Exponential convergence of HF as a function of R b (3/3)
Relative error wrt best HF energy for 2-particles QD
Relative error wrt best HF energy for 6-particles QD
100
100
λ = 0.5
λ=1
λ=2
λ = 0.5
λ=1
λ=2
Relative error
10−1
Relative error
10−1
10−2
10−2
10−3
10−3
0
2
4
6
8
10
0
2
4
6
8
10
Relative error wrt best HF energy for 20-particles QD
Relative error wrt best HF energy for 12-particles QD
100
100
λ = 0.5
λ=1
λ=2
λ = 0.5
λ=1
λ=2
10−1
Relative error
Relative error
10−1
10−2
10−2
10−3
0
10−3
0
2
4
6
Figure 14: Hartree-Fock relative error (E
12-,20-electron QD (bottom).
8
HF
b
(R ) −
HF
HF
Emin )/Emin
MSc. in Computational Physics
2
4
6
8
10
Max. shell number in the basis R b
10
b
as a function of R for 2-,6-electron QD (top) and
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
“Convergence history” as a function of λ (1/3)
Definition
The convergence history of a simulation shows how the convergence “improve” over
iterations. We could plot
the energy difference from one iteration to the next
δ(iter ) = |E HF (iter ) − E HF (iter − 1)|.
more intuitive on the form: δ(iter ) ' 10−β iter .
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
“Convergence history” as a function of λ (2/3)
First iterations of the history for a 2-particles QD
λ = 2,
100
slower convergence as
λ increases.
First iterations of the history for a 6-particles QD
λ = 1, R b = 4, β = −1.1
Rb
λ = 1, R b = 4, β = −0.75
= 4, β = −0.8
λ = 2,
100
λ = 5, R b = 4, β = −0.57
Rb
= 4, β = −0.66
λ = 5, R b = 4, β = −0.4
λ = 1, R b = 8, β = −1.1
λ = 1, R b = 8, β = −0.7
λ = 2, R b = 8, β = −0.8
λ = 2, R b = 8, β = −0.48
λ = 5, R b = 8, β = −0.51
λ = 5, R b = 8, β = −0.32
much less impact due
to R b or to the nb.of
particles, except when
leading to unstability.
10−5
δ(iter )
δ(iter )
10−5
10−10
10−10
10−12
10−15
10−15
0
5
10
15
20
25
30
35
40
0
45
5
10
15
20
25
30
iterations
iterations
First iterations of the history for a 12-particles QD
First iterations of the history for a 20-particles QD
λ = 1, R b = 4, β = −0.7
40
45
λ = 1, R b = 4, β = −1.2
λ = 2, R b = 4, β = −0.63
100
35
λ = 2, R b = 4, β = −0.86
100
λ = 5, R b = 4, β = −0.5
λ = 5, R b = 4, β = −0.47
λ = 1, R b = 8, β = −0.43
λ = 1, R b = 8, β = −0.31
λ = 2, R b = 8, β = −0.14
λ = 2, R b = 8, β = −1.5e − 05
λ = 5, R b = 8, β = −3.6e − 06
λ = 5, R b = 8, β = −0.31
10−5
δ(iter )
δ(iter )
10−5
10−10
10−10
10−15
10−15
0
5
10
15
20
25
30
35
40
45
0
5
10
iterations
15
20
25
30
35
40
45
iterations
Figure 15: Convergence history of the Hartree-Fock iterative process for
2-,6-electron QD (top) and 12-,20-electron QD (bottom).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
“Convergence history” as a function of λ (3/3)
expected “plateau” when
reaching machine precision.
Zoom over the convergence plateau for a 20-particles QD
λ = 1, R b = 4, β = −1.2
However it seems that increasing the
interaction strength:
λ = 2, R b = 4, β = −0.86
λ = 5, R b = 4, β = −0.47
10−12
λ = 1, R b = 8, β = −0.31
slows down the convergence
process, as adding error at each
iteration,
δ(iter )
10−13
10−14
10−15
0
20
40
60
80
100
120
140
160
180
iterations
200
induces a lower accuracy, as if it
could “decrease the machine
precision”,
⇒ Phenomena maybe due to
round-off error, proportional to λ,
and entering the eigenvalue solver
in a non-trivial way.
Figure 16: Zoom over the limit of convergence of the
Hartree-Fock iterative process for a 20-particle QD.
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Quadratic error growth of HF/MBPT wrt FCI ground state
Relative error wrt CI energy for a 2-particles QD (R b = 4, R = 4)
100
Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8)
100
10−2
10−2
10−4
10−6
10−8
Relative error
Relative error
10−4
β = 2.97
β = 2.00
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−10
10−12
10−14
10−7 10−6 10−5 10−4 10−3 10−2 10−1
Confinement strength λ
100
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−14
10−7
Relative error
Relative error
β = 2.00
β = 2.00
10−12
10−6
10−5
10−4
10−3
10−2
10−1
100
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−10
10−7 10−6 10−5 10−4 10−3 10−2 10−1
Confinement strength λ
102
10−4
10−10
β = 2.96
β = 1.98
10−14
101
10−2
10−8
10−8
10−12
Relative error wrt CI energy for a 6-particles QD (R b = 4, R = 4)
100
10−6
10−6
100
101
102
Relative error wrt CI energy for a 12-particles QD (R b = 3, R = 3)
100
HF
−2
MBPT(HF) 2nd order
10
MBPT(HF) 3rd order
st order
−4
MBPT(HO)
1
10
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−6
β = 2.96
10−8
β = 1.99
10−10
10−12
10−14
101
102
Confinement strength λ
10−7 10−6 10−5 10−4 10−3 10−2 10−1
Confinement strength λ
100
101
102
rd
Figure 17: Comparison of HF/MBPT and HF corrected by MBPT up to 3 -order wrt to the FCI ground state
taken as reference for 2-electron QD (top) and 6-,12-electron QD (bottom).
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Respective accuracy of HF and MBPT (1/2)
10−12
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−5
Relative error
Relative error
10−11
Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8)
10−11
10−12
10−4
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−5
10−4
Confinement strength λ
Confinement strength λ
Relative error wrt CI energy for a 6-particles QD (R b = 4, R = 4)
Relative error wrt CI energy for a 12-particles QD (R b = 3, R = 3)
10−11
10−12
Relative error
Relative error
Relative error wrt CI energy for a 2-particles QD (R b = 4, R = 4)
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−5
10−11
10−12
10−4
Confinement strength λ
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−5
10−4
Confinement strength λ
Figure 18: Zoom on the quadratic growth of the error when λ is relatively small (λ < 0.05), showing different
accuracies with respect to the method, the number of particles and the size of the basis. for 2-electron QD (top)
and 6-,12-electron QD (bottom).
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] e−
R FCI
R HF
4
4
8
8
4
4
4
8
3
3
2
6
12
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Relative error shift between each method for λ = 10−3
MBPT(HF)-2nd order
→ min
MBPT(HF)-3rd order
→ 3 min
MBPT(H0)-2nd order and MBPT(H0)-3rd order
→ 2.1 × 103 min
HF
→ 4.6 × 103 min
MBPT(HO)-1st order
→ 7.5 × 103 min
MBPT(HF)-2nd order and MBPT(HF)-3rd order
→ min
MBPT(H0)-2nd order and MBPT(H0)-3rd order
→ 2.7 × 103 min
HF
→ 7 × 103 min
MBPT(HO)-1st order
→ 10.7 × 103 min
MBPT(HF)-2nd order and MBPT(HF)-3rd order
→ min
HF
→ 9.65 min
MBPT(H0)-2nd order and MBPT(H0)-3rd order
→ 40.3 min
st
MBPT(HO)-1 order
→ 52.55 min
MBPT(HF)-2nd order and MBPT(HF)-3rd order
→ min
HF
→ 1.31 min
nd
rd
MBPT(H0)-2 order and MBPT(H0)-3 order
→ 5.47 min
st
MBPT(HO)-1 order
→ 8 min
nd
MBPT(HF)-2 order
→ min
MBPT(HF)-3rd order
→ 5.98 min
HF
→ 5.6 × 103 min
nd
rd
MBPT(H0)-2 order and MBPT(H0)-3 order
→ 51.3 × 103 min
MBPT(HO)-1st order
→ 58.4 × 103 min
Table 1: Classification of the methods with respect to their relative accuracy in the range of λ that exhibits a
quadractic error growth.
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Many-Body Approaches to Quantum Dots
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Break of the methods
Relative error wrt CI energy for a 2-particles QD (R b = 4, R = 4)
102
Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8)
102
101
101
Relative error
100
Relative error
100
10−1
10−1
10−2
10−4
10−5
10−2
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−3
0.1
1
2
10
Confinement strength λ
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−3
10−4
10−5
50
0.1
1
2
10
Confinement strength λ
50
Relative error wrt CI energy for a 12-particles QD (R b = 3, , R = 3)
100
Relative error wrt CI energy for a 6-particles QD (R b = 4, R = 4)
101
100
10−1
Relative error
Relative error
10−1
10−2
10−2
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−3
10−4
10−5
0.1
1
2
10
Confinement strength λ
HF
MBPT(HF) 2nd order
MBPT(HF) 3rd order
MBPT(HO) 1st order
MBPT(HO) 2nd order
MBPT(HO) 3rd order
10−3
50
10−4
0.1
1
2
10
Confinement strength λ
50
Figure 19: The plots display a zoom for λ approaching the limit of the closed-shell model for 2-electron QD (top)
and 6-,12-electron QD (bottom).
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Many-Body Approaches to Quantum Dots
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Model, Methods and Implementation
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Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Summary of the results
Exponential convergence of HF as a function of R b .
Increasing λ slows down the convergence of HF, and decreases its accuracy.
Compared to FCI, HF and MBPT have a quadratic error growth wrt λ.
Unstability of the 2nd - 3rd -order MBPT corrections before 1st -order MBPT and
HF.
Break of the method before the closed-shell model
# particles
2
6
12
Break of the
method
λ '5
λ '2
λ '1
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Break of the
model
λ '150
λ '14
λ '???
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
Thank you for your attention ;)
MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
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MSc. in Computational Physics
Many-Body Approaches to Quantum Dots
Quantum Dots
Model, Methods and Implementation
Results and Discussions
Validation of the simulator
Limit of the closed-shell model as a function of λ
Convergence/Stability/Accuracy of HF
Comparison on HF/MBPT/FCI calculations
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MSc. in Computational Physics
Many-Body Approaches to Quantum Dots