Topological photonics/phononics, inspired by the discovery of topological insulators, is a prosperous field of research, in which remarkable one-way propagation edge states are robust against impurities or defect without backscattering. This dissertation discusses the implementation of multiple topological phases in specific designed photonic and phononic crystals. First, it reports a tunable quantum Hall phase in acoustic ring-waveguide system. A new three-band model focused on the topological transitions at the Γ point is studied, which gives the functionality that nontrivial topology can be tuned by changing the strengths of the couplings and/or the broken time-reversal symmetry. The resulted tunable topological edge states are also numerically verified. Second, based on our previous studied acoustic ring-waveguide system, we introduce anisotropy by tuning the couplings along different directions. We find that the bandgap topology is related to the frequency and directions. We report our proposal on a frequency filter designed from such an anisotropic topological phononic crystal. Third, motivated by the recent progress on quantum spin Hall phases, we propose a design of time-reversal symmetry broken quantum spin Hall insulators in photonics, in which a new quantum anomalous Hall phase emerges. It supports a chiral edge state with certain spin orientations, which is robust against the magnetic impurities. We also report the realization of the quantum anomalous Hall phase in phononics.

Two major problems in viscoelastic turbulence, the effects of polymers on the laminar-turbulent transition dynamics and the origin of the maximum drag reduction asymptote, can be both better understood in the regime near the margin of turbulence. In the first part of this thesis, direct numerical simulation trajectories initiated from the edge state are used to follow its unstable manifold into the turbulent basin. In Newtonian flow, the growth of turbulence starts with the intensification of velocity streaks and a sharp rise in the Reynolds shear stress. It is followed by a quick breakdown into high-intensity small-scale fluctuations before entering the core of turbulence. Adding drag-reducing polymers does not affect the initial growth of turbulence but stabilizes the primary streak-vortex structure, which help the flow circumvent the breakdown stage. Throughout the process, polymers act in reaction to the growing turbulence and do not drive the instability. This part not only reveals the transition dynamics into turbulence but also presents a comprehensive view of the bursting stage observed in the near-wall self-sustaining cycle, which starts as the flow leaves hibernating turbulence and is redirected towards the turbulent basin by the unstable manifold of the edge state. On the other hand, this thesis also discusses the effects of polymer addition on the laminar-turbulent transition in extended domains. Localized turbulent spot can be clearly observed in the large box, and this turbulent region will spread as well as tend to “split” but finally fill up the whole domain before it is separated. Polymers do not affect the flow dynamics until the burst. Similarly, vortex structures rapidly break down into small scales after the first bursting of Reynolds shear stress, but polymer additives depress this process. The thesis offers a clear and comprehensive overview of the transition into turbulence in the presence of drag-reducing polymers. Future work remains in two major directions. The first is to pinpoint the flow states responsible for the quantitative origin of the universal upper limit of drag reduction observed in experiments. The second is to determine the role, if any, of elasticity-driven instabilities in the transition. / Thesis / Master of Applied Science (MASc) / Turbulence exists everywhere and can be observed in most fluid flows occurring in nature. To reduce the energy consumption, frictional resistance in the turbulence must be considered in fluid transportation. It has been known since the 1940s that a small amount of long-chain polymer additives can dramatically reduce such drag. The mechanism of drag reduction has attracted extensive attention. Two problems of particular interest are the upper limit of drag reduction (termed maximum drag reduction) and the polymer effects on the laminar-turbulent transition. In this thesis, full transient trajectories from marginal turbulent states towards sustained turbulence in both Newtonian and polymeric flows are monitored by direct numerical simulations. It is observed that polymer additives do not affect the initial growth of turbulence but prevent flows from breaking into strong but small-scale fluctuations afterwards. In a more extended domain, turbulence starts as localized spots which spread across the channel. Adding polymers changes the dynamics of turbulence propagation as well. In addition to the aforementioned problems, this study also sheds lights on the so-called bursting events intermittent surges in turbulent activities observed in experiments.

In this thesis, the problem of partitioning a network into interconnected sub-networks is addressed. The goal is to achieve a partitioning which satisfies a set of specific engineering constraints, imposed in this case, by the requirements of the decomposed state-estimation (DSE) in electrical power systems. The network-partitioning problem is classified as NP-hard problem. Although many heuristic algorithms have been proposed for its solution, these often lack directness and computational simplicity. In this thesis, three new partitioning techniques are described which (i) satisfy the DSE constraints, and (ii) simplify the NP-hard problem by using the natural graph properties of a network. The first technique is based on partitioning a spanning tree optimally using the natural property of the spanning tree branches. As with existing heuristic techniques, information on the partitioning is obtained only at the end of the partitioning process. The study of the DSE constraints leads to define conditions of an ideal balanced partitioning. This enables data on the balanced partitioning to be obtained, including the numbers of boundary nodes and cut-edges. The second partitioning technique is designed to obtain these data for a given network, by finding the minimum covering set of nodes with maximum nodal degree. Further simplification is then possible if additional graph-theoretical properties are used. A new natural property entitled the 'edge state phenomenon' is defined. The edge state phenomenon may be exploited to generate new network properties. In the third partitioning technique, two of these, the 'network external closed path' and the 'open internal paths', are used to identify the balanced partitioning, and hence to partition the network. Examples of the application of all three methods to network partitioning are provided.

We present studies of edge states induced by the Aharonov-Casher vector potential or Rashba-type spin-orbit interaction using quantum transport in InGaAs/InAlAs herterostructures. The Aharonov-Casher effect is electromagnetically dual to the Aharonov-Bohm effect and is predicted to lead to edge states in a parabolic confinement at two-dimensional sample edges. As a narrow gap material, InGaAs has a low effective mass, high mobility, and strong spin-orbit interaction, which indicate that it can be used as a good material to detect the Aharonov-Casher effect or SOI interaction. Using InGaAs, we measured the magnetoresistance in a quantum antidot in narrow short channels in a tilted magnetic field. The fine structure (mT spacing) observed in the magnetoresistance indicate a probable energy spacing between AC edge states. We also fabricated side-gate channel structures in InGaAs/InAlAs quantum wells and investigated the values of the Rashba spin-orbit coupling constant α using the weak antilocalization analysis as a function of the side-gate voltage. We take the effect of the finite width into account and find the corrected values. With the simulation of electric fields in the wide channel and narrow channel, we found that the electric field components can be changed using side-gate voltages. While our results do not indicate which electric field component is responsible, the data indicate that the deduced spin-orbit strength values in a narrow channel are tunable by the side-gate voltage. / Ph. D.

The focus of this thesis is the numerical study of subcritical transition to turbulence in boundary-layer flows. For the most part, boundary layers with uniform suction are considered. Constant homogeneous suction counteracts the spatial growth of the boundary layer, rendering the flow parallel. This enables research approaches which are not feasible in the context of spatially developing flows. In the first part, the laminar–turbulent separatrix of the asymptotic suction boundary layer (ASBL) is investigated numerically by means of an edge-tracking algorithm. The obtained edge states experience recurrent dynamics, going through calm and bursting phases. The self-sustaining mechanism bears many similarities with the classical regeneration cycle of near-wall turbulence. The recurrent simple structure active during calm phases is compared to the nucleation of turbulence events in bypass transition originating from delocalised initial conditions. The implications on the understanding of the bypass-transition process and the edge state's role are discussed. Based on this understanding, a model is constructed which predicts the position of the nucleation of turbulent spots during free-stream turbulence induced transition in spatially developing boundary-layer flow. This model is used together with a probabilistic cellular automaton (PCA), which captures the spatial spreading of the spots, correctly reproducing the main statistical characteristics of the transition process. The last part of the thesis is concerned with the spatio-temporal aspects of turbulent ASBL in extended numerical domains near the onset of sustained turbulence. The different behaviour observed in ASBL, i.e. absence of sustained laminar–turbulent patterns, which have been reported in other wall-bounded flows, is associated with different character of the large-scale flow. In addition, an accurate quantitative estimate for the lowest Reynolds number with sustained turbulence is obtained / <p>QC 20160429</p>

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.