Numerical Studies of Topological Phases

The topological phases of matter have been a major part of condensed matter physics research since the discovery of the quantum Hall effect in the 1980s. Recently, much of this research has focused on the study of systems of free fermions, such as the integer quantum Hall effect, quantum spin Hall effect, and topological insulator. Though these free fermion systems can play host to a variety of interesting phenomena, the physics of interacting topological phases is even richer. Unfortunately, there is a shortage of theoretical tools that can be used to approach interacting problems. In this thesis I will discuss progress in using two different numerical techniques to study topological phases.

Recently much research in topological phases has focused on phases made up of bosons. Unlike fermions, free bosons form a condensate and so interactions are vital if the bosons are to realize a topological phase. Since these phases are difficult to study, much of our understanding comes from exactly solvable models, such as Kitaev's toric code, as well as Levin-Wen and Walker-Wang models. We may want to study systems for which such exactly solvable models are not available. In this thesis I present a series of models which are not solvable exactly, but which can be studied in sign-free Monte Carlo simulations. The models work by binding charges to point topological defects. They can be used to realize bosonic interacting versions of the quantum Hall effect in 2D and topological insulator in 3D. Effective field theories of "integer" (non-fractionalized) versions of these phases were available in the literature, but our models also allow for the construction of fractional phases. We can measure a number of properties of the bulk and surface of these phases.

Few interacting topological phases have been realized experimentally, but there is one very important exception: the fractional quantum Hall effect (FQHE). Though the fractional quantum Hall effect we discovered over 30 years ago, it can still produce novel phenomena. Of much recent interest is the existence of non-Abelian anyons in FQHE systems. Though it is possible to construct wave functions that realize such particles, whether these wavefunctions are the ground state is a difficult quantitative question that must be answered numerically. In this thesis I describe progress using a density-matrix renormalization group algorithm to study a bilayer system thought to host non-Abelian anyons. We find phase diagrams in terms of experimentally relevant parameters, and also find evidence for a non-Abelian phase known as the "interlayer Pfaffian".

NUMERICAL STUDY OF DAMMANN ARRAY ILLUMINATORS

The numerical solutions of binary-phase (0, tau) gratings for one-dimensional array illuminators up to 32 are presented. Some fabrication errors, which are due to position-quantization errors, phase errors, dilation (or erosion) errors, and the side-slope error, are calculated and show that even-number array illuminators are superior to odd-number array illuminators when these fabrication errors are considered. One (0, tau) binary-phase, 8 x 16 array illuminator made with the wet-chemical-etching method is given in this paper.

Experimental, Numerical and Analytical Studies of the MHD-driven plasma jet, instabilities and waves

This thesis describes a series of experimental, numerical, and analytical studies involving the Caltech magnetohydrodynamically (MHD)-driven plasma jet experiment. The plasma jet is created via a capacitor discharge that powers a magnetized coaxial planar electrodes system. The jet is collimated and accelerated by the MHD forces.

We present three-dimensional ideal MHD finite-volume simulations of the plasma jet experiment using an astrophysical magnetic tower as the baseline model. A compact magnetic energy/helicity injection is exploited in the simulation analogous to both the experiment and to astrophysical situations. Detailed analysis provides a comprehensive description of the interplay of magnetic force, pressure, and flow effects. We delineate both the jet structure and the transition process that converts the injected magnetic energy to other forms.

When the experimental jet is sufficiently long, it undergoes a global kink instability and then a secondary local Rayleigh-Taylor instability caused by lateral acceleration of the kink instability. We present an MHD theory of the Rayleigh-Taylor instability on the cylindrical surface of a plasma flux rope in the presence of a lateral external gravity. The Rayleigh-Taylor instability is found to couple to the classic current-driven instability, resulting in a new type of hybrid instability. The coupled instability, produced by combination of helical magnetic field, curvature of the cylindrical geometry, and lateral gravity, is fundamentally different from the classic magnetic Rayleigh-Taylor instability occurring at a two-dimensional planar interface.

In the experiment, this instability cascade from macro-scale to micro-scale eventually leads to the failure of MHD. When the Rayleigh-Taylor instability becomes nonlinear, it compresses and pinches the plasma jet to a scale smaller than the ion skin depth and triggers a fast magnetic reconnection. We built a specially designed high-speed 3D magnetic probe and successfully detected the high frequency magnetic fluctuations of broadband whistler waves associated with the fast reconnection. The magnetic fluctuations exhibit power-law spectra. The magnetic components of single-frequency whistler waves are found to be circularly polarized regardless of the angle between the wave propagation direction and the background magnetic field.

Fast numerical methods for mixed, singular Helmholtz boundary value problems and Laplace eigenvalue problems - with applications to antenna design, sloshing, electromagnetic scattering and spectral geometry

This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

Part I

Numerical solutions to the S-limit equations for the helium ground state and excited triplet state and the hydride ion ground state are obtained with the second and fourth difference approximations. The results for the ground states are superior to previously reported values. The coupled equations resulting from the partial wave expansion of the exact helium atom wavefunction were solved giving accurate S-, P-, D-, F-, and G-limits. The G-limit is -2.90351 a.u. compared to the exact value of the energy of -2.90372 a.u.

Part II

The pair functions which determine the exact first-order wavefunction for the ground state of the three-electron atom are found with the matrix finite difference method. The second- and third-order energies for the (1s1s)¹S, (1s2s)³S, and (1s2s)¹S states of the two-electron atom are presented along with contour and perspective plots of the pair functions. The total energy for the three-electron atom with a nuclear charge Z is found to be E(Z) = -1.125•Z² +1.022805•Z-0.408138-0.025515•(1/Z)+O(1/Z²)a.u.