Many-Body Approaches to Quantum Dots

Transcription

1 Many-Body Approaches to Patrick Merlot Department of Computational Physics November 4, 2009 Many-Body Approaches to

2 Table of contents Many-Body Approaches to

3 What is a quantum dot? Physics of QDs Applications of QDs Table of contents 1 What is a quantum dot? Physics of QDs Applications of QDs 2 3 Many-Body Approaches to

4 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. types/shapes/fabrication QDs (b) (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. Many-Body Approaches to

5 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]). Many-Body Approaches to

6 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. Many-Body Approaches to

7 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) Many-Body Approaches to

8 Table of contents Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 1 2 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 3 Many-Body Approaches to

9 Model of a quantum dot Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 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 Ψ = E Ψ. 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]). Many-Body Approaches to

10 Model of a N-particle system Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The Schrödinger equation The Schrödinger equation of a N-particle system Ĥ(r 1, r 2,..., r N ) Ψ κ (r 1, r 2,..., r N ) = E κ Ψ κ (r 1, r 2,..., r N ) (1) where r i reprensents the (spatial/spin) coordinates of quasiparticle i, κ stands for all quantum numbers needed to classify a given N-particle state, Ψ κ and E κ are the eigenstates and eigenenergies of the system. How to define our Hamiltonian? Ĥ = N e i=1 p i 2 2m +... Many-Body Approaches to

11 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 = N e=2 i=1 2 Ne=2 p i 2m + i=1 1 2 m ω 2 0 r i 2 + e 2 4πɛ 0 ɛ r 1 r 1 r 2, (2) Interaction potential the two-body Coulomb interaction Many-Body Approaches to

12 Applying an external magnetic field Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 1 p i p i + ea, where A is the vector potential defined by B = A. in coordinate space p i i i, using a Coulomb gauge A = 0 (by choosing A(r i ) = B r i ), 2 (p i + ea) 2 2 2m 2 i i e «m A i + e2 2m A2. (3) In terms of B, the linear and quadratic terms in A have the form i e m A i = e 2m B L, and e 2 2m A2 = e2 8m B2 ri 2. where L = i (r i i ) is the orbital angular momentum operator of the electron i. 2 B also acts on spin with the additional energy term: Ĥ s = gs ω c Ŝz, where Ŝ 2 is the spin operator of the electron and gs is its effective spin gyromagnetic ratio and ω c = eb/m is known as the cyclotron frequency. Many-Body Approaches to

13 Final Hamiltonian Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The final Hamiltonian reads: N e ( 2 Ĥ = 2m 2 i + i=1 Harmonic ocscillator potential {}}{ 1 2 m ω 2 0 r i 2 ) + Coulomb interactions {}}{ e 2 1 4πɛ 0 ɛ r r i r j N e ( 1 ( ω ) 2 c + 2 m ri ω (i) c ˆL z + 1 ) 2 g s ω c Ŝ z (i), (4) i=1 }{{} single particle interactions with the magnetic field i<j Many-Body Approaches to

14 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 New constant, the oscillator frequency ω = New units: the energy unit ω, r ω 0 + ωc 2, 2 the length unit, the oscillator length defined by l = p /(m ω). The dimensionless Hamiltonian is now XN e Ĥ = i=1» i r i 2 + λ X i<j 1 r ij + XN e i=1 1 2 ω c ω m(i) l g s «ω c ω m(i) s. (5) where the new dimensionless parameter λ = l/a0 describes the strength of the electron-electron interaction (a0 being the effective Bohr radius). Many-Body Approaches to

15 The Hamiltonian solved in this project λ(b) = a0 4ω0 2m 2 + e 2 B 2 Role of B? «1 4 Squeezing the particles should increase the strength of the electron-electron interaction. λ only decreases as the magnetic field increases in this model. Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Confinement strength λ B (T) 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: XN e Ĥ = i=1» i r i 2 + λ X i<j 1 r ij. (6) Many-Body Approaches to

16 The method of variational calculus Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 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 Φ E[Φ] = Φ H Φ Φ Φ while subject to a normalization constraint Φ Φ = 1. = R Φ HΦdτ R Φ Φdτ, (7) The method introduces new variables for each of the constraints (the Lagrange multipliers ɛ) and defines the Lagrangian (Λ) with respect to Φ as Λ(Φ, ɛ) = E[Φ] ɛ ( Φ Φ 1), (8) Find stationnary solutions by solving the set of equations by writing Λ Φ = 0. Many-Body Approaches to

17 The Hartree-Fock method Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 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(r 1, r 2,..., r i,..., r j,..., r N ) = Φ(r 1, r 2,..., r j,..., r i,..., r N ). (9) Many-Body Approaches to

18 The Hartree-Fock wave function Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The Slater determinant is an antisymmetric product of the single particle orbitals: ψ α(r 1 ) ψ α(r 2 )... ψ α(r N ) Φ(r 1, r 2,..., r N, α, β,..., σ) = 1 ψ β (r 1 ) ψ β (r 2 )... ψ β (r N ) N! , (10) ψ σ(r 1 ) ψ σ(r 2 )... ψ σ(r N ) It can be rewritten as Φ T (r 1, r 2,..., r N, α, β,..., σ) = 1 X ( ) p PΨ α(r 1 )Ψ β (r 2 )... Ψ σ(r N ) (11) N! p = N! A `Ψ α(r 1 )Ψ β (r 2 )... Ψ σ(r N ), (12) by introducing the antisymmetrization operator A = 1 X ( ) p ˆP. N! P Many-Body Approaches to

19 Matrix elements calculations Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Definition NX NX We write the Hamiltonian for N electrons as Ĥ = Ĥ0 + Ĥ1 = ĥ i + v(r i, r j ), i=1 i<j where r ij = r i r j, ĥ i and v(r i, r j ) 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 NX Z NX Φ T Ĥ0Φ T dτ = Ψ µ (r)ĥψ µ (r)dr = µ h µ. (13) µ=1 µ=1 Z Φ T Ĥ1Φ T dτ = 1 NX NX µν V µν AS. (14) 2 µ=1 ν=1 where we define the antisymmetrized matrix element as µν V µν AS = Z µν V µν µν V νµ, with the following shorthands µν V µν = Ψ µ(r i )Ψ ν(r j )V (r ij )Ψ µ(r i )Ψ ν(r j )dr i dr j. Many-Body Approaches to

20 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 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[Φ T ] = Φ T Ĥ0 Φ T + Φ T Ĥ1 Φ T (15) NX = µ h µ + 1 NX NX µν V µν AS. (16) 2 µ=1 µ=1 ν=1 We expand each single-particle eigenvector Ψ i in terms of a convenient complete set of single-particle states α (the harmonic oscillator eigenstates in our case), Ψ i = i = X α c α i α. (17) The energy functional now reads NX X E[Φ] = Ci α C γ i α h γ + 1 NX X Ci α C β j C γ i Cj δ 2 αβ V γδ AS. (18) i=1 αγ i,j=1 αβγδ Many-Body Approaches to

21 The Hartree-Fock equations (1) Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Remember the method of the Lagrange multipliers 1 Define a functional E[Φ T ], 2 Identify the constraints: Ψ i Ψ j = δ ij which implies Φ T Φ T = 1, with Ψ i Ψ j = X Ci α C β X j α β Ci α Cj α {z } αβ α 3 Define the Lagrangian Λ δ αβ = NX Λ(C1 α, C 2 α,..., C N α, ɛ 1, ɛ 2,..., ɛ N ) = E[Φ T ] ɛ i i=1 X α! Ci α Ci α δ ij. (19) where ɛ i are the Lagrange multipliers for each of the normalization constraints. 4 Get the system of equations to solve by setting Λ dλ d dφ T dci α [Λ(C1 α, C 2 α,..., C N α, ɛ 1, ɛ 2,..., ɛ N )] = 0, i N. (20) Many-Body Approaches to

22 The Hartree-Fock equations (2) Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Treating Ci α and Ci α as independent, we arrive at the Hartree-Fock equations (one equation for each basis state α ) X α h γ C γ i + X γ γ Effective potential α U γ z } { # " X N X j=1 βδ C β j αβ V γδ AS {z } Two-body interaction matrix elementv αβγδ C δ j C γ i = ɛ i C α i, (21) which we can rewrite as X O αγ C γ i = ɛ i Ci α, α H. γ System of non-linear equations in the Ci α, since O αγ depends itself on the unknowns. To be solved by an iterative procedure. Many-Body Approaches to

23 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The Hartree-Fock (self-consistent) iterative procedure 1 Compute the effective Coulomb interaction potential α U (0) γ with an initial guess of the C α(0) i. 2 Build the resulting Fock matrix O. 3 Solve the linearised system given by the equations (Fock matrix diagonalization) X γ [ α h γ + α U γ ] C γ i = ɛ i C α i. at iteration (k), store the output eigenenergies ɛ (k) i and the coefficients of the new eigenvectors C α(k) i. 4 Substitute back the new coefficients to compute a new Coulomb interaction potential Continue the process until self-consistency is reached. no Initial guess of HF orbitals Fock matrix computation Fock matrix diagonalization Selfconsistency achieved? Compute the total energy End yes Figure 8: Flowchart of Hartree-Fock algorithm. Many-Body Approaches to

24 Many-body perturbation theory Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Take the Hamiltonian Ĥ = Ĥ 0 + Ĥ, and treat Ĥ as a perturbation, such as the Coulomb interaction. Suppose Φ n eigenfunctions of Ĥ 0 corresponding to the eigenvalues E n: Ĥ 0 Φ n = E nφ n. Consider the effect of the perturbation on a particular state Φ 0. We denote by Ψ 0 the state into which Φ 0 changes under the action of the perturbation, so that Ψ 0 is an eigenfunction of Ĥ, corresponding to the eigenvalue E. Ĥ 0 Φ 0 = E 0 Φ 0. (22) ĤΨ 0 = EΨ 0. (23) Therefore Φ 0 and Ψ 0 denote the ground states of the unperturbed and perturbed systems respectively. Since Ĥ0 is Hermitian, one can show that: E E 0 = Φ 0 Ĥ Ψ 0. (24) Φ 0 Ψ 0 which is an exact expression and independent of any particular perturbation method. Since Ψ 0 is unknown, using a projection operator R for the state Φ 0 defined by the equation RΨ = Ψ Φ 0 Φ 0 Ψ, (25) Many-Body Approaches to

25 The perturbed energy Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The perturbed energy can be derived from the iterated Ψ 0 which gives X «E E 0 = Φ0 Ĥ R n (E 0 E + Ĥ ) Φ 0. (26) n=0 E 0 Ĥ 0 We shall write E = E E 0 = E (1) + E (2) + E (3) +... where the m th -order energy correction E (m) contains the m th -order power of the perturbation Ĥ. Many-Body Approaches to

26 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The many-body perturbation corrections The 1 st -order correction is E (1) = Φ 0 Ĥ Φ 0. (27) The 2 nd -order correction is E (2) R = Φ 0 Ĥ (E 0 E + E 0 Ĥ ) Φ 0. (28) Ĥ0 The 3 rd -order energy correction reads E (3) = X X n=0 n=0 X Φ 0 Ĥ Φ 0 Φ 0 Ĥ Φ m Φ m Ĥ Φ n Φ n Ĥ Φ 0 (E 0 E m)(e 0 E n) n=0 Φ 0 Ĥ Φ n Φ n Ĥ Φ 0 (E 0 E n) 2. (29) Many-Body Approaches to

27 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation 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 as expressed in terms of anihilation (c k ) and creation (c k ) operators Ĥ = 1 X ij v kl c i c j 2 c l c k, ijkl The previous many-body perturbation corrections now read E (1) = Φ 0 Ĥ Φ 0. = 1 X h 1 h 2 v h 1 h 2 as, (30) 2 h 1 h 2 X E (2) Φ 0 Ĥ Φ n Φ n Ĥ Φ 0 = = 1 X h 1 h 2 v p 1 p 2 2 as, E n=0 0 E n 4 ɛ h 1 h 2 p 1 p h1 + ɛ h2 ɛ p1 ɛ p2 2 where h i and p i are respectively hole states and particles states, and ɛ i are the single particle energies of the basis set. Many-Body Approaches to

28 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The MBPT corrections expanded in a basis set The 3 rd -order many-body perturbation correction reads where E (3) = X X n=0 n=0 X Φ 0 Ĥ Φ 0 Φ 0 Ĥ Φ m Φ m Ĥ Φ n Φ n Ĥ Φ 0 (E 0 E m)(e 0 E n) n=0 Φ 0 Ĥ Φ n Φ n Ĥ Φ 0 (E 0 E n) 2 = E (3) (3) (3) 4p 2h + E 2p 4h + E 3p 3h, (31) E (3) 4p 2h is the contribution to the third-order energy correction due to the 4-particle/2-hole excitations, E (3) 2p 4h is the contribution to the third-order energy correction due to the 2-particle/4-hole excitations, E (3) 3p 3h is the contribution to the third-order energy correction due to the 3-particle/3-hole excitations. Many-Body Approaches to

29 Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation The MBPT corrections expanded in a basis set The contributions to the third-order energy correction can be written as E (3) 4p 2h = 1 8 E (3) 2p 4h = 1 8 E (3) 3p 3h = 0 1 h 1h 2 v p 1 p 2 as X p 1 p 2 v p 3 p 4 as p 3 p 4 v h 1 h 2 as A, ɛ h 1 h 2 p 1 p h1 + ɛ h2 ɛ p1 ɛ p2 ɛ 2 p 3 p h1 + ɛ h2 ɛ p3 ɛ p h 1h 2 v p 1 p 2 as X h 1 h 2 v h 3 h 4 as h 3 h 4 v h 1 h 2 as A, ɛ h 1 h 2 p 1 p h1 + ɛ h2 ɛ p1 ɛ p2 ɛ 2 h 3 h h3 + ɛ h4 ɛ p1 ɛ p h 1h 2 v p 1 p 2 X X h 1 h 3 v p 1 p 3 as p 3 h 2 v h 3 h 2 as AA, ɛ h1 + ɛ h2 ɛ p1 ɛ p2 ɛ h3 p h1 + ɛ h3 ɛ p1 ɛ p3 3 h 1 h 2 p 1 p 2 where the p i denote the particle states, h i the hole states, and ɛ i the single particle energies of the corresponfing state. Many-Body Approaches to

30 Code implementation Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory 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 1 st - to 3 rd -order, either in the harmonic oscillator or in the HF basis set. Many-Body Approaches to

31 Implementation issues Model of a quantum dot The method of Variational calculus The Hartree-Fock method The many-body perturbation theory Implementation Difficulties encountered if not making use of symmetries Huge Fock matrix to diagonalize (grows exponentially with R b ). Huge Coulomb interaction matrix to store V αβγδ = (n 1, m l1 )(n 2, m l2 ) V (n 3, m l3 )(n 4, m l4 ) is an 8-dimensional array. Solutions implemented By using the symmetries and properties of the Coulomb interaction: V αβγδ does not act on spin: m s1 = m s3 & m s2 = m s4. V αβγδ conserves the total spin and angular momentum: m l1 + m l2 = m l3 + m l4. By sorting the states of the basis by blocks of identical angular (m l ) and spin (m s) 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. Many-Body Approaches to

32 Table of contents Many-Body Approaches to

33 Simple checks Reproduce the non-interacting ground state energies. Reproduce the two-body interaction matrix elements of OpenFCI almost with machine precision 2 nd order MBPT Correction [mev ] 0 E(max( ml )) with max(n) = 5 E(max( n )) with max( ml ) = max(n), max( ml ) Comparison of MBPT results with similar experiments Waltersson computed the open-shell 2 nd -order MBPT correction. Our closed-shell 2 nd -order MBPT correction shows close agreement. Figure 9: 2 nd -order perturbation theory correction for the 2e QD. Comparison between results of Waltersson (top) [3] and our results (down). Many-Body Approaches to

34 Level crossing as a function of B without interactions (1/3) Fock-Darwin orbitals When neglecting the repulsions between the particles, the eigenenergies ɛ n ml as a function of the magnetic field B can be solved analytically for a parabolic confining potential V (r) = 1/(2m ω0 2 r 2 ) leading to a spectrum known as the Fock-Darwin states Rewriting the eigenenergies in units of ω 0, ɛ n ml becomes dimensionless and we obtain s ɛ n ml = (2n + m l + 1) 1 + (ωc/ω 0) (ωc/ω 0) m l (32) s eb = (2n + m l + 1) 1 + ( 2m ) 2 eb ω 0 2m m l. (33) ω 0 Many-Body Approaches to

35 Level crossing as a function of B without interactions (2/3) Six-electrons dot Twelve-electrons dot Energy ɛn m l ( ω0) Energy ɛn m l ( ω0) rd shell electron states 2 nd shell electron state 1 st shell electron state Magnetic field (T) Figure 10: Spectrum of Fock-Darwin orbitals for 6 non-interacting particles (GaAs: ω 0 = 5meV,ɛ r = 12) th shell electron states 5 th shell electron states 4 th shell electron states 3 rd shell electron states 2 nd shell electron state 1 st shell electron state Magnetic field (T) Figure 11: Spectrum of Fock-Darwin orbitals for 12 non-interacting particles (GaAs: ω 0 = 5meV,ɛ r = 12). Many-Body Approaches to

36 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

37 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

38 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

39 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

40 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

41 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

42 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

43 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

44 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

45 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

46 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

47 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

48 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

49 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

50 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

51 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

52 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

53 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

54 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

55 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

56 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

57 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

58 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

59 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

60 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

61 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

62 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

63 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

64 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

65 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

66 Level crossing as a function of B without interactions (1/3) 6 non-interacting particles 12 non-interacting particles Many-Body Approaches to

67 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 = S = 0 S = 2 S = 4 S = 6 27 S = 0 S = 2 S = 4 S = E0(M, S) ( ω) E0(M, S) ( ω) Total angular momentum M 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 = S = 0 S = 2 S = 4 S = S = 0 S = 2 S = 4 S = E0(M, S) ( ω) E0(M, S) ( ω) Total angular momentum M 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). Many-Body Approaches to

68 Level crossing as a function of λ with interactions (2/2) Summary of FCI results using OpenFCI λ FCI ground (M,S) state energy ( ω) R= (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (1,2) (1,2) (1,4) For a 6-particle QD, break of the model of a single Slater determinant from λ 13. A similar study performed on a 2-particle QD indicates a break of the closed-shell model from λ 150. Many-Body Approaches to

69 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 N S is N S = (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 n B = B(R b ) = φ nml (r) : 2n + m l R bo, (35) where φ nml (r) are the single orbital in the Harmonic oscillator basis with quantum numbers n, m l such that the single orbital energy reads: ɛ nml = 2n + m l + 1 in two-dimensions. Many-Body Approaches to

70 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 λ = 0.5 λ = 1 λ = 2 λ = 0 30 E 0 HF [ ω] 3 2 E 0 HF [ ω] λ = 0.5 λ = 1 λ = 2 λ = Ground state of a 12-particles QD 350 Ground state of a 20-particles QD λ = 0.5 λ = 1 λ = 2 λ = λ = E 0 HF [ ω] E 0 HF [ ω] λ = 1 λ = λ = Max. shell number in the basis R b Figure 13: Hartree-Fock ground state (E HF as a function of R b for 2-,6-electron QD (top) and 12-,20-electron QD (bottom). Many-Body Approaches to

71 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 10 0 λ = 0.5 λ = 1 λ = λ = 0.5 λ = 1 λ = Relative error Relative error Relative error wrt best HF energy for 12-particles QD λ = 0.5 λ = 1 λ = Relative error wrt best HF energy for 20-particles QD λ = 0.5 λ = 1 λ = Relative error 10 2 Relative error Max. shell number in the basis R b Figure 14: Hartree-Fock relative error (E HF (R b ) E HF HF min )/Emin as a function of Rb for 2-,6-electron QD (top) and 12-,20-electron QD (bottom). Many-Body Approaches to

72 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. Many-Body Approaches to

73 Convergence history as a function of λ (2/3) First iterations of the history for a 2-particles QD λ = 1,, β = 1.1 First iterations of the history for a 6-particles QD λ = 1,, β = 0.75 slower convergence as λ increases. δ(iter) λ = 2,, β = 0.8 λ = 5,, β = 0.57 λ = 1, R b = 8, β = 1.1 λ = 2, R b = 8, β = 0.8 λ = 5, R b = 8, β = 0.51 δ(iter) λ = 2,, β = 0.66 λ = 5,, β = 0.4 λ = 1, R b = 8, β = 0.7 λ = 2, R b = 8, β = 0.48 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 iterations iterations First iterations of the history for a 12-particles QD First iterations of the history for a 20-particles QD λ = 1,, β = 0.7 λ = 1,, β = λ = 2,, β = 0.63 λ = 5,, β = λ = 2,, β = 0.86 λ = 5,, 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 = δ(iter) δ(iter) iterations iterations Figure 15: Convergence history of the Hartree-Fock iterative process for 2-,6-electron QD (top) and 12-,20-electron QD (bottom). Many-Body Approaches to

74 Convergence history as a function of λ (3/3) δ(iter) Zoom over the convergence plateau for a 20-particles QD iterations λ = 1,, β = 1.2 λ = 2,, β = 0.86 λ = 5,, β = 0.47 λ = 1, R b = 8, β = 0.31 Figure 16: Zoom over the limit of convergence of the Hartree-Fock iterative process for a 20-particle QD. expected plateau when reaching machine precision. However it seems that increasing the interaction strength: slows down the convergence process, as adding error at each iteration, 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. Many-Body Approaches to

75 Quadratic error growth of HF/MBPT wrt FCI ground state Relative error wrt CI energy for a 2-particles QD (, R = 4) 10 0 Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8) Relative error β = 2.00 β = 2.97 HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Relative error β = 1.98 β = 2.96 HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Confinement strength λ Relative error Relative error wrt CI energy for a 6-particles QD (, R = 4) β = β = 2.00 HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error Relative error wrt CI energy for a 12-particles QD (, R = 3) 10 0 HF 10 2 MBPT(HF) 2 nd order MBPT(HF) 3 rd order 10 4 MBPT(HO) 1 st order MBPT(HO) 2 nd order 10 6 MBPT(HO) 3 rd order β = β = Confinement strength λ Figure 17: Comparison of HF/MBPT and HF corrected by MBPT up to 3 rd -order wrt to the FCI ground state taken as reference for 2-electron QD (top) and 6-,12-electron QD (bottom). Many-Body Approaches to

76 Respective accuracy of HF and MBPT (1/2) Relative error wrt CI energy for a 2-particles QD (, R = 4) Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8) Relative error HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error wrt CI energy for a 6-particles QD (, R = 4) Relative error wrt CI energy for a 12-particles QD (, R = 3) Relative error HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error HF MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order 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). Many-Body Approaches to

77 e R FCI R HF Relative error shift between each method for λ = MBPT(HF)-2 nd order MBPT(HF)-3 rd order MBPT(H0)-2 nd order and MBPT(H0)-3 rd order HF MBPT(HO)-1 st order MBPT(HF)-2 nd order and MBPT(HF)-3 rd order MBPT(H0)-2 nd order and MBPT(H0)-3 rd order HF MBPT(HO)-1 st order MBPT(HF)-2 nd order and MBPT(HF)-3 rd order HF MBPT(H0)-2 nd order and MBPT(H0)-3 rd order MBPT(HO)-1 st order MBPT(HF)-2 nd order and MBPT(HF)-3 rd order HF MBPT(H0)-2 nd order and MBPT(H0)-3 rd order MBPT(HO)-1 st order MBPT(HF)-2 nd order MBPT(HF)-3 rd order HF MBPT(H0)-2 nd order and MBPT(H0)-3 rd order MBPT(HO)-1 st order ɛ min 3 ɛ min ɛ min ɛ min ɛ min ɛ min ɛ min ɛ min ɛ min ɛ min 9.65 ɛ min 40.3 ɛ min ɛ min ɛ min 1.31 ɛ min 5.47 ɛ min 8 ɛ min ɛ min 5.98 ɛ min ɛ min ɛ min ɛ min Table 1: Classification of the methods with respect to their relative accuracy in the range of λ that exhibits a quadractic error growth. Many-Body Approaches to

78 Break of the methods Relative error Relative error wrt CI energy for a 2-particles QD (, R = 4) HF 10 3 MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order 10 4 MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error Relative error wrt CI energy for a 2-particles QD (R b = 8, R = 8) HF 10 3 MBPT(HF) 2 nd order MBPT(HF) 3 rd order MBPT(HO) 1 st order 10 4 MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error wrt CI energy for a 6-particles QD (, R = 4) 10 1 Relative error wrt CI energy for a 12-particles QD (,, R = 3) 10 0 Relative error HF 10 3 MBPT(HF) 2 nd order MBPT(HF) 3 rd order 10 4 MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ Relative error HF MBPT(HF) 2 nd order 10 3 MBPT(HF) 3 rd order MBPT(HO) 1 st order MBPT(HO) 2 nd order MBPT(HO) 3 rd order Confinement strength λ 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). Many-Body Approaches to

79 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 2 nd - 3 rd -order MBPT corrections before 1 st -order MBPT and HF. Break of the method before the closed-shell model # particles Break of the method Break of the model 2 λ 5 λ λ 2 λ λ 1 λ??? Many-Body Approaches to

80 Thank you for your attention ;) Many-Body Approaches to

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