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Quantum Monte Carlo Methods for Studying Quantum Dots

Alexander Fleischer

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

This thesis aims to give an understanding of quantum dots trapped in different potentials by computationally finding the ground state energies of the system.The potentials studied are the single and double harmonic oscillator wells and the finite square well.To achieve this, we have implemented a robust Variational Monte Carlo framework that allows for different single-particle wavefunctions.This allowed us to study the use of a linear expansion of the single harmonic oscillator basisfunctions as an approximation to the single-particle wavefunctions of the other potential wells.We investigated how many of these basisfunctions needed to achieve sufficiently good single-particle wavefunctions.The framework has several optimizations that drastically reduce the computational time of the Monte Carlo method.The framework has a rigorous testing regime that ensures the authenticity of the results.To verify the results, we have compared them to the results of other Variatonal Monte Carlo computations, as well as Diffusion Monte Carlo, the Coupled Cluster method, the Similarity Renormalization Group method and the Full Configuration Interaction method.In addition, we have verified the results of the finite square well by comparing with a general series solution.

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This thesis aims to give an understanding of quantum dots trapped in different potentials by computationally finding the ground state energies of the system.The potentials studied are the single and double harmonic oscillator wells and the finite square well.To achieve this, we have implemented a robust Variational Monte Carlo framework that allows for different single-particle wavefunctions.This allowed us to study the use of a linear expansion of the single harmonic oscillator basisfunctions as an approximation to the single-particle wavefunctions of the other potential wells.We investigated how many of these basisfunctions needed to achieve sufficiently good single-particle wavefunctions.The framework has several optimizations that drastically reduce the computational time of the Monte Carlo method.The framework has a rigorous testing regime that ensures the authenticity of the results.To verify the results, we have compared them to the results of other Variatonal Monte Carlo computations, as well as Diffusion Monte Carlo, the Coupled Cluster method, the Similarity Renormalization Group method and the Full Configuration Interaction method.In addition, we have verified the results of the finite square well by comparing with a general series solution.
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.i Preface I first discovered physics during high school, thanks to my brother Alexander Fleischer.He knew how much I liked mathematics, so he recommended a course in physics to me.I ended up liking it even more than mathematics, and as a result, after high school I applied to the physics, meteorology and astronomy Bachelor program at the University of Oslo.The very first semester I was introduced to a field I liked just as much as physics, namely programming.Unfortunately, my Bachelor did not involve much programming after the first year.During the third year I had the option of taking an introductory course in computational physics, but I had several other physics courses I wanted to take, and I prioritized these over computational physics.When it was time to apply for a Masters program, I was unsure of what to pick.I then remembered how fun all the programming had been during the first year, and decided to give the combination of physics and programming a try, so I applied to the computational physics Masters program.The first semester I took the introductory computational physics course, and knew immediately that I had picked the right Masters program for me.I would like to thank my supervisor Morten Hjorth-Jensen for all the help and motivation he gave me during the past two years.I would also like to thank my brother Alexander for introducing me to physics in the first place, and for the helpful discussions we had about this thesis.Additionally, I would like to thank Håkon V. Treider for the fun times we have had sharing an office for the better part of two years, and for the help he has given me with general programming related issues.Finally, I would like to thank the various other people I shared an office with over the past two years, and the rest of the people at the computational physics research group, for making it a great place to be. Oslo,
This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all. In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
This thesis investigates the use of wave-function methods for the study of quantum-dot systems. It investigates single dots, using quantum Monte Carlo techniques in a wide range of magnetic field values, and a double-dot system, using the exact-diagonalization method.\n\nThe thesis proposes simple yet accurate many-particle wave functions for various angular-momentum and spin states, for both weak and strong magnetic fields. Using these trial wave functions, it evaluates various properties of dots and studies Wigner crystallization and spin polarization for the weak-field limit. For strong magnetic fields, the thesis investigates ground states of different spin polarizations as a function of the magnetic field and tests the commonly used lowest-Landau-level approximation. The results are compared to calculations from the density-functional theory.\n\nFinally, the thesis presents a method that combines the accuracy of the exact-diagonalization method and the scalability of Monte Carlo methods. The Monte Carlo-based diagonalization is a promising tool for use in situations that cannot be handled with a simple trial wave function, and have too many particles for an exact-diagonalization treatment. For quantum dots, methods for efficiently evaluating the high-magnetic-field basis functions and their gradients are demonstrated.
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 different 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 effective but has the advantage that no truncation occurs. We analyse the ground state using two different generators, Wegner's canonical generator and a modified version of that one, and meet numerical problems by replacing the standard Coulomb by an effective 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 different generators, Wegner's canonical generator and White's generator. We demonstrate that Wegner's generator leads to numerical instabilities and stiff equation systems, especially for systems with comparatively high correlations. Computations with White's generator, on the other hand, are shown to be much more efficient 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 IM-SRG(2), we compare our results for the ground state energy with other ab-initio many-body methods, including Hartree-Fock, Coupled Cluster, Diffusion Monte Carlo and Full Configuration Interaction. Finally, we use the IM-SRG(2) results to study the role of correlations in two-dimensional quantum dots.