Topological insulators(TIs) constitute a novel state of quantum matter which possesses non-trivial topological properties. Although discovered only in the recent few years, TIs have attracted intensive interest among the community of condensed matter physics and material science. TIs are insulating in the bulk but have conductive gapless edge or surface states on the boundaries, which have their origin in the nontrivial bulk band topology that is induced by the strong spin-orbital interactions in the materials. Existing in all dimensions, TIs exhibit a variety of exotic physics such as quantum spin Hall effect, momentum-spin locked surface states, Dirac fermion transport, quantized anomalous Hall effect, Majorana fermions, etc. In this thesis, I study the transport properties of 2D and 3D TIs by numerical approaches. As an introduction, a brief review of TIs is given. A detailed description of the numerical methods is also presented. The results can be summarized in four aspects. First, disorder is found be able to induce a non-trivial TI from an originally trivial band insulator, where the conductance of a two terminal device drops to nearly zero and then rises to form an anomalous plateau as disorder strength is increased, and finally all the states become localized. The real space Chern number calculation as well as the effective medium theory suggests that disorder is fundamentally responsible for the emerging of the extended helical edge states in this system. We also present a levitation and pair annihilation picture of the extended states for this model. Second, by making the 2D TIs into singly connected quantum point contacts(QPCs), I show a coherent and fast Aharonov-Bohm oscillation of conductance caused by the quantum interference of the helical edge states. This oscillation not only happens against weak magnetic field but also against the gate voltage in the zero-field condition. This results in a giant edge magnetoresistance of the device in weak magnetic fields. The amplitude of the magnetoresistance is controllable by adjusting either the QPCs' slit width or the interference loop size in the device. The oscillation is found robust against disorder. Third, by applying a uniform spin-splitting Zeeman field in the bulk of the 3D TI whose surface states can be viewed as massless Dirac fermions, I find chiral edge states on the gapped surfaces of the 3D TI, which can be considered as interface states between domains of massive and massless Dirac fermions. Effectively these states are result of splitting of a perfect interface conducting channel. This picture is confirmed by the Landauer-B?ttiker calculations in four-terminal Hall bars. Finally, I propose the concept of topological semi-metals. By calculating the local density of states on the surfaces, I demonstrate that surface states and the gapless Dirac cone already exist in the system although the bulk is not gapped. We show how the uni-axial strain induces an insulating band gap and turn the semi-metal into true TI. We predict existence of quantum spin Hall effect in the thin films made of these materials, which can be significantly enhanced by disorders. / published_or_final_version / Physics / Doctoral / Doctor of Philosophy

Although a number of methods with appropriate approximations, such as mean-field theory, local density approximation, and tight-binding method have been well developed and widely used in solid state physics, they possess strong limitations, and thus physicists never stop trying to find methods that could rigorously solve the models of condensed matter systems. This thesis presents several new exact solutions for electron pairing models with spin-orbit interactions and Zeeman coupling, which have not been studied before. First, a type of electron pairing model with spin-orbit interactions or Zeeman coupling is solved exactly in the framework of the Richardson’s previous work for 2D cases. Based on the exact solutions for the electron pairing model with spin-orbit interactions, it is shown rigorously that the pairing symmetry is of the p+ip wave and the ground state possesses time-reversal symmetry, which are expected by the meanfield theory. And the difference is that such peroration from our framework is valid for any strength of the pairing interactions. Intriguingly, how Majorana fermions can emerge is also elaborated in a ribbon system as well. Condensation energy and critical magnetic field are calculated in two systems with the exact solutions, and compared with the relevant results achieved by the mean-field theory, the differences between our results and the mean-field theory show the significance of the work for exact solutions. Secondly, we generalize our scenario to 3D cases. Several remarks of the 3D case are given following the significant results from the 2D cases. And an unconventional type of Fulde-Ferrel-Larkin-Ovchinnikov ground state is revealed exactly, in which the center-of-mass momentum of the fermion pair is proportional to the Zeeman field. As a by-product, a similar Fulde-Ferrel-Larkin-Ovchinnikov state is also disclosed when the magnetic field is in the same plane of k for 2D case. In addition, applying the transformative Richardson ansatz in bosonic system, we elaborate on the drifting effect of the Zeeman field on the spin-orbit-coupled Bose-Einstein condensed matter as well. Finally, we discuss the application of the exact solutions in quantum entanglement quantification. The entanglement monotone concurrence is calculated with exact solutions for two models. It is found to be a smooth function of pairing interactions, as expected. / published_or_final_version / Physics / Doctoral / Doctor of Philosophy

In recent years, studies of diffusion in random media have been extended to include the effects of media in which the defects fluctuate randomly in time. Typically, the diffusive motion of particles in a static medium persists when the medium is allowed to fluctuate, with the diffusivity (diffusion constant) D depending on the character of the fluctuations. In the present work, we study random walks on lattices in which the bonds connecting vertices open and close randomly in time, and the walker is not allowed to cross a closed bond. Variations of the model studied here have been used to model the diffusion of CO through myoglobin, the transport of ions in polymer solutions, and conduction in hydrogenated amorphous silicon. The major objective in analyzing these systems is to find efficient methods for computing the diffusivity. In this dissertation, we focus mainly on methods of computing the diffusivity in our model. In addition, we study the critical behavior of the model and present a demonstration, valid for a restricted range of model parameters, that the distribution of the displacement converges in time to a Gaussian with width D. To compute the diffusivity, we use a numerical renormalization group (RG) method, power series expansions in model parameters, and Monte Carlo simulations. We choose a model with two parameters characterizing the bond fluctuations--the time scale of fluctuations tau and the mean open-bond density p. We calculate a series expansion of the diffusivity to about 10th order in the parameter nu = exp(.1/τ) on the hypercubic lattice Zᵈ for d = 1, 2, 3, as well as on the Bethe lattice. We compute the same power series expansion to 3rd order in ν for arbitrary d. We compute estimates of the diffusivity on the Bethe lattice using the RG methods and show by comparison to Monte Carlo data that the RG provides excellent quantitative predictions of D when τ is not too large.

<p>We are interested in the transition from an amorphous material to a polycrystalline structure. Previous research done by our group introduced a modified Random Nucleation and Growth model to describe this phenomenon. Besides the nucleation rate, its importance in the grain size distribution (GSD) having long been established, we introduced an effective time dependent growth rate, postulated the analytic form, and ascertained its effect on the GSD to match experimental data. Not only would we like to continue to substantiate this form, we wish also to determine how this rate depends on the fundamental parameters of the model. To achieve this, we have developed a numerical simulation in two dimensions, based on the premises of the model, to simulate crystallization in thin films. Not only does the simulation produce data that matches the analytical results of the nucleation rate and volume fraction available for nucleation, it also fits well with the postulated growth rate. This suggests the validity of the introduced effective growth rate and the functionality of the simulation for studying this process. </p>

The rapid progress of the field of ultracold atoms during the past two decades has set new milestones in our control over matter. By cooling dilute atomic gases and molecules to nano-Kelvin temperatures, novel quantum mechanical states of matter can be realized and studied on a table-top experimental setup while bulk matter can be tailored to faithfully simulate abstract theoretical models. Two of such models which have witnessed significant experimental and theoretical attention are (1) the two-component Fermi gas with resonant $s$-wave interactions, and (2) the single-component Fermi gas with dipole-dipole interactions. This thesis is devoted to studying the non-equilibrium collective dynamics of these systems using the general framework of quantum kinetic theory. / Physics

The fractional quantum Hall effect occurs when an extremely clean 2-dimensional fermion gas is subject to a magnetic field. This simple set of circumstances creates phenomena, such as edge reconstruction and fractional statistics, that remain subjects of experimental study 30 years after the discovery of the fractional quantum Hall effect. This thesis investigates the properties of excitations of the fractional quantum / Physics

Many-body Coulomb interactions make understanding the complete excitation spectra of conjugated polymers a formidable task, and a variety of sophisticated experimental and theoretical techniques have been used in an attempt to elucidate their complicated electronic structure. It is thus crucial to have a intuitive, physical picture in order to interpret the wide range of available experimental data. We present the results of calculations within an exciton basis which allows for a simple pictorial description of all linear and nonlinear excitations in conjugated polymers, and settles a number of longstanding controversies. The exciton basis further allows us to justify the application of single configuration interaction (SCI) techniques to the understanding of nonlinear optical experiments in the low energy region. We show that SCI can give a clear, self-consistent picture of the photoexcitations in poly(para-phenylene vinylene), a conjugated polymer which has attracted much recent interest.

This dissertation describes experimental and theoretical studies of carrier tunneling in semiconductor heterostructures and optical properties of neutron irradiated quantum wells. Unambiguous experimental evidence for the dramatic dependence of hole tunneling rates on in-plane momentum in (Ga,In)As/(Al,In)As asymmetric double quantum wells (ADQWs) is presented. Holes generated near the bandedge tunnel on hundred picosecond time scales, whereas holes excited with large excess energy tunnel on subpicosecond time scales. The mechanism responsible for this increase of three orders of magnitude in the hole tunneling rate is nonresonant delocalization of hole wavefunctions by band mixing in the valence band. The carrier density and temperature dependencies of tunneling dynamics are presented. A simple kinetic model developed for electron LO-phonon assisted tunneling shows good qualitative agreement with experimental data. Exciton tunneling in wide gap, II-VI semiconductors was studied using (Zn,Cd)Se/ZnSe ADQW. The strong Coulomb interaction in II-VI semiconductors makes the tunneling process significantly different from that in III-VI ADQWs. Fast (1 ps) and complete recovery of the narrow well exciton absorption was observed after resonant femtosecond pulse excitation. The observed dynamics contradict the theory of independent electron and hole tunneling. The theory of exciton tunneling was developed. Theoretical analysis shows that tunneling of the exciton as a whole entity with the emission of only one LO-phonon is very slow. Instead, the exciton tunnels via an indirect state in a two-step process whose efficiency is dramatically enhanced by the Coulomb interaction. The optical properties of neutron irradiated GaAs/Ga,Al)As multiple quantum wells are investigated. Sharp room temperature exciton features and a 21 ps carrier lifetime are demonstrated in neutron irradiated multiple quantum wells. Carrier lifetime reduction is consistent with the presence of EL2 defects that are efficiently generated by fast neutrons. The influence of the gamma rays accompanying neutron irradiation is discussed. Neutron irradiation provides a straightforward way to control the carrier lifetime in semiconductor heterostructures with minor deterioration of their excitonic properties.

Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pauli blocking and a simple, phenomenological, uniformly weak two particle interaction potential equivalent to the static screening approximation. The resulting asymytotic closure and quantum kinetic Boltzmann equation are derived in a self consistent manner without resorting to a priori statistical hypotheses or cumulant discard assumptions. We find a new class of solutions to the quantum kinetic equation which are analogous to the Kolmogorov spectra of hydrodynamics and classical weak turbulence. They involve finite fluxes of particles and energy across momentum space and are particularly relevant for describing the behavior of systems containing sources and sinks. We explore these solutions by using differential approximation to collision integral. We make a prima facie case that these finite flux solutions can be important in the context of semiconductor lasers. We show that semiconductor laser output efficiency can be improved by exciting these finite flux solutions. Numerical simulations of the semiconductor Maxwell Bloch equations support the claim.

In this thesis, we have systematically investigated the stability, surface dynamics, electronic transport, and growth of metal nanowires using a semiclassical free energy functional based on the mean-field interacting electron model, which is simple and general enough. In this model, the ionic degrees of freedom of the wire are modeled as an incompressible fluid, and the conducting electrons are treated as a Fermi gas confined within the wire with Dirichlet boundary conditions. In equilibrium, we prove that the electron-electron interaction is a second-order effect to the total grand canonical free energy, while the shell-correction to the noninteracting grand canonical free energy is a first-order effect. To leading order, the electron-electron interactions just renormalize the Weyl parameters, such as the average energy density, surface tension and mean curvature energy, but not the mesoscopic shell effect. This finding for open mesoscopic systems is a generalization of the well-known Strutinsky theorem for finite-Fermion systems. It is for this reason that self-consistent jellium calculations obtain essentially identical equilibrium mesoscopic effects as calculations based on the free-electron model. However, for systems out of equilibrium, the electron-electron interaction plays important roles. First of all, the Strutinsky theorem breaks down in the non-equilibrium case. Secondly, the gauge invariance condition is violated if the electron-electron interaction is not adequately included. We first derive a thermodynamic phase diagram for jellium nanowires, which predicts that cylindrical wires with certain "magic" conductance values are stable with respect to small perturbations up to remarkably high temperatures and high applied voltage. We have shown that Jahn-Teller-distorted wires can be stable. The derived sequence of stable cylindrical and elliptical geometries explains the experimentally observed shell and supershell structures for alkali metals. Highly deformed wires can explain additional conductance peaks in low temperature experiments on alkali metals and in gold. We then study the surface dynamic properties of different phases. Both surface phonons and surface self-diffusion of atoms are included in the linearized surface dynamics. It is found that inertial dynamics (phonons) always dominate the long-wavelength behavior at small time scales, including the critical points. (Abstract shortened by UMI.)