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<p>Topological materials have been under intense investigation for more than 30 years and have experienced astonishing growth  over the last decade. The two-dimensional electron gas has long served as a model system for the exploration of topological physics, supporting a diverse array of strongly correlated emergent phenomena. Indeed, some of the most stunning topological phases in condensed matter systems are the integer and fractional quantum Hall states forming in two-dimensional electron gases.</p> <p>It was realized early on that electron localization in the bulk has an important role in attaining topological phases, where the sample bulk is well described by randomly localized electrons, known as the Anderson insulator. However, a different type of topological phases forms when charge carriers order in the bulk. Such a charge ordering can only occur in the presence of strong electron-electron interactions and low disorder. Localization of this kind is of a collective nature and differs fundamentally from the single particle physics of the Anderson insulator. The nature of charge ordering, however, is more nuanced than first thought. Indeed, in high Landau levels, Hartree-Fock theories predict the proliferation of numerous exotic bulk insulators, where in the limit of no disorder electrons cluster together and form a hexagonal lattice. Initial observations of these highly correlated insulating phases were limited to low disorder two-dimensional electron gases confined to GaAs/AlGaAs heterostructures. However, recent discoveries of charge ordering in two-dimensional electron gases confined to graphene highlight the universality of this phenomena, irrespective of host material. While progress has been made in understanding the collective insulators residing within integer quantum Hall plateaus, many aspects remain unresolved. In this Dissertation, I discuss the transport properties and energetics of collective insulators forming near integer quantum Hall plateaus in the latest generation of very low disorder two-dimensional electron gases.</p> <p>In chapter  1  I briefly introduce recent developments in our current understanding of the integer quantum Hall effect, where the topological phase is described by both a topological invariant as well as a local order parameter related to the Landau symmetry breaking paradigm. Next, I introduce the basic principles of two-dimensional electron gases confined to semiconductor heterostructures and provide a short summary of recent technological breakthroughs in molecular beam epitaxial growth protocols. The chapter concludes with an introduction to the essential physics of both the integer and the fractional quantum Hall effect.</p> <p>Chapter  2  contains a brief review of the existing literature on the collective insulators forming in sufficiently low disorder two-dimensional electron gases. The primary focus of chapter  2  is on the unique magnetotransport patterns seen at various Landau level filling factors, which support the collective insulator  interpretation. Throughout this chapter I tend to lean on theoretical models that describe these collective phases through the lens of the Hartree-Fock theory; however, it is important to note that both density matrix renormalization group theories and direct diagonalization of small electron systems reach similar conclusions.</p> <p>In chapter  3  I present our data displaying the hallmark transport signatures of a collective insulator residing within the flanks of the nu = 1 integer quantum Hall plateau. Our sample belongs to the latest generation of low disorder 2DEGs confined to GaAs/AlGaAs. I provide a detailed analysis of its development in both temperature and filling factor. The distinct transport signatures we observe strongly overlap in filling factor with prior microwave resonance, surface acoustic wave, compressibility, and tunneling measurements, all of which point to the formation of a collective insulator known as the integer quantum Hall Wigner solid. One puzzling aspect, however, is that while the latter measurements exhibit the integer quantum Hall Wigner solid in older generation samples, transport signatures of this phase appear to be present only in the newest and highest mobility samples. By using distinct features in the magnetoresistance, I propose a stability diagram of the integer quantum Hall Wigner solid in nu −T phase space. Analysis of magnetoresistance profiles at fixed filling factors display sharp peaks within the region of integer quantum Hall Wigner solid phase. It is believed that these sharp peaks are a shared property of collective insulators forming in low disorder two-dimensional electron gases and signal the onset of the electron solid formation. Additional analysis of the magnetoresistance profiles suggests activated transport behavior with a gap energy comparable to that of the plateau center. Lastly, I present large signal measurements of the nu = 1 integer quantum Hall Wigner solid. The data displays strong non-linear behavior in the current-voltage characteristics consistent with the depinning and sliding conduction. However, similar threshold conduction is also seen in the current-voltage characteristics near the center of the integer quantum Hall plateau, where the bulk is an Anderson insulator. Much to our surprise, trends in the threshold current are monotonic in filling factor.</p> <p>In chapter  4  I report on the recent emergence of a newly observed collective insulator residing within the nu = 2 integer quantum Hall plateau and centered at filling factor nu = 1.79. Based on the range of filling factors which stabilizes this collective insulator, we find it distinct from the aforementioned integer quantum Hall Wigner solid. Indeed, the transport behavior is eerily reminiscent to the reentrant insulating phase seen at low filling factors between 1/5 < nu < 2/9. Hence, we term this collective insulator the reentrant integer quantum Hall Wigner solid. Evoking concepts of particle-hole symmetry, we find the reentrant integer quantum Hall Wigner solid to be one member of the larger family of Wigner solids, which is intimately linked through this fundamental symmetry of the system.</p> <p>Lastly in chapter  5 , I explore the energetics of the collective insulators which develop in the N = 2 and N = 3 Landau level, specifically the two- and three-electron bubble phase. We extract the onset temperatures of these exotic bulk insulators from the sharp peaks in the magnetoresistance at fixed filling factor. We compare our measured onset energies with the cohesive energies found from numerical calculations. We find the onset temperatures for the both two- and three-electron bubble phase show an approximately linear trend in filling factor within a single Landau level. In addition, we observe that the three-electron bubble phase has a larger onset temperature than the two-electron bubble phase, a result which is inconsistent with some numerical energy calculations. Thus, our measurements of bubble phase energetics call attention to the importance of the short-range Coulomb interaction in the formation of multi-electron bubble phases and is expected to serve as guide towards the refinement of existing theoretical models.</p>

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