Graphene, a crystalline atomic monolayer of carbon atoms, is a model system as the regularity and cleanliness of its lattice enables precise descriptions of its properties. Theoretical models predict that the behavior of electrons in a heterostructure consisting of two stacked graphene monolayers with a slight rotational misalignment in their lattices is dictated by interactions and topology. We investigate a twisted bilayer sample with both electrical resistivity measurements and nanoSQUID on tip microscopy. The latter, a novel magnetic imaging method developed as part of this dissertation, is uniquely well matched to studying the dilute magnetic signals expected in twisted graphene heterostructures.

We observe the emergence of a quantized anomalous Hall effect in twisted bilayer graphene aligned to hexagonal boron nitride with Hall resistance is quantized to within 0.1\% of the von Klitzing constant h/e^2 at zero magnetic field. In contrast to magnetically doped (Bi,Sb)_2Te_3 quantum anomalous Hall variants, intrinsic strong correlations polarize the electrons into a single valley resolved miniband with Chern number C=1 arising from inversion symmetry breaking and the formation of a moir e: the system does not host band inversion or spin orbit coupling. The measured transport energy gap $\Delta/k_B\approx 27$ K, the largest observed to date, is almost four times the Curie temperature for magnetic ordering $T_C\approx 7.5$ K. We find that electrical currents as small as 1 nA can be used to controllably switch the magnetic order between states of opposite polarization, forming an electrically rewritable magnetic memory. Magnetic imaging reveals a magnetization primarily orbital in nature dominated by chiral edge state contributions from the topological gap of the quantum anomalous hall phase. Mapping the spatial evolution of field-driven magnetic reversal, we find a series of reproducible micron scale domains, pinned to structural disorder, whose boundaries host chiral edge states.

When interactions between electrons dominate over their kinetic energies, this can lead to emergent collective states with new quantum degrees of freedom described in terms of quasiparticles. These collective states can fall into two distinct categories: either the interactions lead to a breaking of an underlying symmetry, or electrons can form topologically ordered states characterized by a degenerate ground state in the zero temperature limit. The condition for strong interactions is met in two-dimensional electron systems (2DES) immersed in high magnetic fields, where electron kinetic energies quench into a ladder of dispersionless Landau levels broadened by disorder. Landau levels are an example of a topological band, characterized by a Chern number C ∈ Z. Correlated electron states then appear at partial filling of a Landau level — these are called fractional quantum Hall states.

In this thesis, I present a series of magneto-capacitance measurements of the thermo- dynamic density of states of low-disorder dual-gated graphene/boron nitride heterostructures – two-dimensional materials that can be isolated and later reassembled into stacks of designer properties. In such devices, we observe the elusive even-denominator fractional quantum Hall states at half filling of mono- and bilayer graphene. For monolayer graphene, we propose a scenario where the observed states are multicomponent states that incorporate correlations between electrons on different carbon sublattices. In the bilayer graphene case, the observed even-denominator states are single-component and

potentially host non-Abelian excitations, i.e. quasiparticles with fractional statistics, that are neither fermions or bosons.

Moreover, if we introduce a twist between the graphene and boron nitride layers, a

periodic moire potential will appear. As a result of the interplay between a magnetic field and the moire potential, new Hofstadter bands with Chern numbers C ≠ 1 can arise, in which we observe fractional Chern insulators — a generalization of fractional quantum Hall states to bands with a higher Chern index.

Free-space time domain THz spectroscopy accesses the electrodynamic responses of quantum materials at frequencies ideally matched to the energy scales of interacting condensed matter systems. THz spectroscopy, however, is challenging when samples are physically smaller than the diffraction limit of ~0.5 mm, as is typical, for example, in van der Waals materials and heterostructures. We examine foundational electrodynamics and spectroscopic principles relevant at THz frequencies. We present THz engineering challenges and overcome them by designing an on-chip, time-domain THz spectrometer with an interchangeable sample architecture and a bandwidth of 750 GHz. We use this spectrometer and extract the optical conductivity of a 7.5-um wide NbN film across the superconducting transition. We then extensively benchmark the spectrometer's operation in the context of THz engineering principles, and conclude by looking forward to the measurements this technology enables, including superconductivity, magnetism, and charge order in sub-diffraction materials and van der Waals heterostructures.

A successful program for improving the quality of graphene based 2DEGs has been the steady removal of charged impurities near the graphene surface. This was first accomplished by using hBN as a substrate and encapsulant, and then further improved by using atomically uniform graphite as a gate electrode instead of amorphous evaporated metal, resulting in the first all `van der Waals' heterostructure. These improvements lead to the robust observation of even-denominator fractional quantum Hall states in bilayer graphene, as well as many other interesting correlated ground states at zero magnetic field such as superconductivity and ferromagnetism in both twisted and un-twisted graphene multi-layers. Moreover, as many of these states occur at an electron density near charge neutrality, they may be accessed within a single device simply by field-effect gating. However, engineering more complex mesoscopic devices which are designed to interface these exotic phases of electronic matter require fabricating spatially varying potential profiles on the order of the correlation lengths in these systems, typically around 100nm. Additionally, these manufactured potentials need to be energetically uniform, i.e., they must not introduce uncontrolled disorder potentials on the order of the energy gaps of the systems of interest, again typically around 1meV, on the requisite length scales. Here we report on a novel fabrication technique, AFM based local anodic oxidation gate lithography, to pre-pattern graphite gates with critical feature sizes smaller than 100nm which does not introduce any unwanted contaminants or damage to the underlying van der Waals heterostructure. We then use this technique to fabricate two types of mesoscopic devices operated in the fractional quantum Hall regime, a quantum point contact and an edge-state Fabry-Pérot interferometer. We show that our quantum point contact nearly perfectly mimics a simple model which characterizes electron tunneling between nu = 1/3 and nu = 1 edge modes at any tunneling strength. This indicates that these junctions are nearly disorder-free validating our novel fabrication technique. Moreover, we use our nearly defect free junctions to create an edge-state Fabry-Pérot interferometer operating in nu = 1/3. Here we observe sharp phase jumps in the interference pattern consistent with the expected braiding phase of anyons in the nu = 1/3 state. These results taken together show that our method for fabricating mesoscopic devices within the graphene based van der Waals heterostructure platform, developed over the last four years, makes these devices comparable to the most modern III-V semiconductor quantum wells, which took over 30 years of refinement.

In the diverse catalog of `quantum materials' one of the most beguiling parameters is the spin-orbit coupling. This dissertation is concerned with how spin-orbit coupling effects exotic physics in correlated transition metal oxides. The materials described herein are all assembled from corner-sharing octahedra (Ir,Ru)O$_6$, and are hole-doped from the $d^5$ configuration, yet they contain a diversity of electronic and magnetic phases. A combination of resonant X-ray scattering and neutron scattering were employed to try to unravel the nature of the order and fluctuations in these materials.

First we examine the unusual Lifshitz transition in metallic Sr$_2$Ir$_x$Ru$_{1-x}$O$_4$, where we unveil a transition from itinerant to local moment behavior that might speak to quantum criticality. Second we explore the closely related compound Sr$_3$(Ir$_{1-x}$Ru$_x$)$_2$O$_7$, where the antiferromagnetism persists to remarkably high Ru content. Third is Sr$_3$Ir$_2$O$_7$F$_2$, which is like a fully hole-doped and distorted analog to Sr$_3$Ir$_2$O$_7$; there the spin-orbit excitons are thoroughly compared to theory. Last is (Ca$_x$Nd$_{1-x}$)$_2$Ir$_2$O$_7$, a tunable magnetic semimetal with potential for interesting topological states.