Studies of quantum dots: Ab initio coupled-cluster analysis using OpenCL and GPU programming

We have implemented two different quantum many-body method: Hartree-Fock and Variational Monte Carlo. The system studies were closed-shell quantum dot systems with the single-well harmonic oscillator potential and the double-well potential. The basis used consisted of harmonic oscillator functions and calculations were performed in cartesian coordinates. The theory is presented in detail including the calculation of the two-body elements in the Hartree-Fock method. The resulting basis from the Hartree-Fock method was then used as function in the Slater determinant for the variational Monte Carlo method. The Hartree-Fock limit was reached for 30 particles in two dimensions and 20 in three for the single-well. Said limit was reached for 2 up to 12 particles for the two dimensional double-well and for 2 up to 14 for the three dimensional double-well.

Quantum many-Body Simulations of Double Dot System

We approximate single particle wave functions for some given potential by expanding the solutions from diagonalizing the single particle problem in a harmonic oscillator basis. These single particle wave function approximations allow us to perform variational Monte Carlo (VMC) simulations without having explicit expressions for the single particle wave functions. We use the VMC simulations to analyze the ground states of quantum dots trapped in various confinement potentials, namely single and double harmonic oscillator potentials and single finite square well potentials. A point of interest is to see how many harmonic oscillator basis functions are needed for high quality approximations to the single particle wave functions for the different potentials. This is of interest because the number of basis functions needed is an important constraint of the method when it comes to computational efficiency. To verify the method used, we compare our ground state energy results with other ab initio many-body methods, such as the Similarity Renormalization Group method, the Coupled Cluster method, and the Full Configuration Interaction method, as well as with other VMC and DMC (Diffusion Monte Carlo) results.

Monte Carlo Studies of Confined Fermions

We apply the Similarity Renormalization Group (SRG) method to quantum dots, using the same methodology that has recently had great success in the nuclear physics community. The SRG method can be realized in two di erent ways: In free space with respect to the physical vacuum state, or in-medium using the particle-hole formalism of second quantization. We start with the free-space approach, which is computationally less e ective but has the advantage that no truncation occurs. We analyse the ground state using two di erent generators, Wegner's canonical generator and a modi ed version of that one, and meet numerical problems by replacing the standard Coulomb by an e ective interaction and the harmonic oscillator by a Hartree-Fock basis. Afterwards, we apply the recently evolved in-medium SRG approach to our electronic systems. Here we choose the IM-SRG(2) method, meaning that all operators are truncated on a two-body level. Again, we apply two di erent generators, Wegner's canonical generator and White's generator. We demonstrate that Wegner's generator leads to numerical instabilities and sti equation systems, especially for systems with comparatively high correlations. Computations with White's generator, on the other hand, are shown to be much more e cient and require less CPU time, especially as the size of the basis is increased. This enables us to look at systems up to N = 42 particles. To analyse the capabilities of IMSRG(2), we compare our results for the ground state energy with other ab initio many-body methods, including Hartree-Fock, Coupled Cluster, Di usion Monte Carlo and Full Con guration Interaction. Finally, we use the IM-SRG(2) results to study the role of correlations in two-dimensional quantum dots.

/pdf/quantum-mechanical-systems-in-traps-and-similarity-40tk6n3ko1.pdf

Quantum-mechanical systems in traps and Similarity Renormalization Group theory

Full Configuration Interaction Simulation of Quantum Dots

/pdf/coupled-cluster-studies-of-double-quantum-dots-mp65knddxo.pdf

Coupled-Cluster Studies of Double Quantum Dots

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Coupled-Cluster Studies of Double Quantum Dots