We present and compare several many-body methods as applied to two-dimensional quantum dots with circular symmetry. We calculate the approximate ground state energy using a harmonic oscillator basis optimized by Hartree-Fock (HF) theory and further improve the ground state energy using two post-HF methods: in-medium similarity renormalization group and coupled cluster with singles and doubles. With the application of quasidegenerate perturbation theory or the equations-of-motion method to the results of the previous two methods, we obtain addition and removal energies as well. Our results are benchmarked against full configuration interaction and diffusion Monte Carlo where available. We examine the rate of convergence and perform extrapolations to the infinite basis limit using a power-law model.
We present and compare several many-body methods as applied to two-dimensional quantum dots with circular symmetry. We calculate the approximate ground state energy using a harmonic oscillator basis optimized by Hartree-Fock (HF) theory and further improve the ground state energy using two post-HF methods: in-medium similarity renormalization group and coupled cluster with singles and doubles. With the application of quasidegenerate perturbation theory or the equations-of-motion method to the results of the previous two methods, we obtain addition and removal energies as well. Our results are benchmarked against full configuration interaction and diffusion Monte Carlo where available. We examine the rate of convergence and perform extrapolations to the infinite basis limit using a power-law model.
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 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.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
We introduce a generic approach to study interaction effects in diffusive or chaotic quantum dots in the Coulomb blockade regime. The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. We classify the possible induced two-body ensembles, both in the presence and absence of spin degrees of freedom. The ensembles depend on the underlying space-time symmetries as well as on features of the two-body interaction. Confining ourselves to spinless electrons, we then use the Hartree-Fock (HF) approximation to calculate HF single-particle energies and HF wave functions for many realizations of the ensemble. We study the statistical properties of the resulting one-body HF ensemble for a fixed number of electrons. In particular, we determine the statistics of the interaction matrix elements in the HF basis, of the HF single-particle energies (including the HF gap between the last occupied and the first empty HF level), and of the HF single-particle wave functions. We also study the addition of electrons, and in particular the distribution of the distance between successive conductance peaks and of the conductance peak heights.
Quantum many-body systems can be studied theoretically from first principles using microscopic models, but the computational cost of numerically exact calculations increases exponentially with the number of particles and the complexity of the interaction, limiting the systems that can be analysed. The Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) methods, heavily used in nuclear physics, rely on mean-field approximations but scale more favourably with increasing system size and complexity. In this work we apply these methods to the problem of a two-dimensional harmonically trapped Fermi gas with attractive interactions, with the goal of investigating the validity of the approximations in the few-body limit. We implement the approximate methods in code and then compare them with exact calculations. We find that the HFB method can account for pairing correlations that arise due to the attractive interaction, predicting well the ground state energy even at high interaction strengths. Also, the excitation spectra predicted by the QRPA method qualitatively match the true spectra, reproducing the expected subset of excitations. Finally, we see that perturbative corrections can be used to improve the ground state energy estimate of the HFB.