Quantum physics

S. Johri and R. Bhatt developed a real-space renormalization group approach aimed at extracting the localized single-particle eigenstates of the Anderson model from a large system by identifying clusters of resonant site potentials. E. Campbell generalized this real-space renormalization group approach using standard perturbation theory. Both approaches were intended to approximate the single-particle density of states of the Anderson model. In this thesis, we aimed to test the potential of applying a similar real-space renormalization group approach to calculate the density of states of the interacting Anderson-Hubbard model. Our interest in the density of states of this model is due to a V-shaped zero-bias anomaly in two-dimensional systems. A real-space renormalization group approach is best applied to a one-dimensional system. We found that the zero-bias anomaly is not V-shaped in one-dimension. To test the potential of a real-space renormalization group approach, we used the cluster approach which is the same as the non-interacting renormalization group approach but without the perturbation theory and found that for strong disorder this technique could accurately calculate the density of states over a wide range of energies but deviated from exact results at the band edge, at $\omega=\pm U$ and near $\omega=0$. The first two inaccuracies will be reduced with a proper real-space renormalization group approach. We suspect that the last inaccuracy is associated with long range physics and may be difficult to recover. We also developed a technique that adjusts the identification of clusters in the cluster approach to improve the computation time of the density of states with minimal loss of accuracy in a tunable range around the Fermi level. We found that this technique significantly reduced the computation time and was able to preserve the density of states near the Fermi level, except at the smallest energies near $\omega=0$.

Author Keywords: Anderson-Hubbard model, renormalization group, Strong electron correlations, Zero-bias anomaly

The time-dependent Gross-Pitaevskii Equation, describing the movement of parti-

cles in quantum mechanics, may not be solved analytically due to its inherent non-

linearity. Hence numerical methods are of importance to approximate the solution.

This study develops a discrete scheme in time and space to simulate the solution

defined in a finite domain by using the Crank-Nicolson difference method and Sinc

Collocation Methods (SCM), respectively. In theory and practice, the time discretiz-

ing system decays errors in the second-order of accuracy, and SCMs are decaying

errors exponentially. A new SCM with a unique boundary treatment is proposed

and compared with the original SCM and other similar numerical techniques in time

costs and numerical errors. As a result, the new SCM decays errors faster than the

original one. Also, to attain the same accuracy, the new SCM interpolates fewer

nodes than the original SCM, which saves computational costs. The new SCM is

capable of approximating partial differential equations under different boundary con-

ditions, which can be extensively applied in fitting theory.

Author Keywords: Crank-Nicolson difference method, Gross-Pitaevskii Equation, Sinc-Collocation methods