Spin-orbit coupling (SOC) is a relativistic interaction between the electronic spin and orbital degrees of freedom. The SOC can give rise to a variety of interesting phenomena, most notably at low temperatures in compounds composed of large-Z (atomic number) elements, since the SOC scales as Z^4. In this dissertation, I examine the low-temperature expression of SOC in the transport and magnetic properties of a representative selection of materials, including the heavy fermion system URu2Si2, narrow band gap semiconductor FeSb2, 5d transition metal oxide BaIrO3, linear band CoSb3 and RhSb3 skutterudites, as well as a new class of rare earth materials on a novel kagome lattice with non-collinear Ising axes, called the “tripod” kagome lattice. These compounds all feature unusual many-body properties that are either directly or indirectly linked to the large SOC present in each. In URu2Si2, for example, the large SOC is foundational to the hidden order (HO) phase that arises at T_HO = 17.5 K, where a remarkable magnetic signature is seen not in the linear magnetic susceptibility, χ_1, but in the leading nonlinear term χ_3. Exotic magnetic ground states are also seen in the newly synthesized rare earth tripod kagome systems. The SOC is also an important component of the inverted band structure in Dirac materials, and thus plays a role in the band formation of CoSb3 and RhSb3, which have both been predicted to be near a topological transition. Finally, large SOC may also be related to interesting carrier dynamics in FeSb2 and BaIrO3: FeSb2 displays a unique proportionality between the thermopower, S(T), and Hall mobility, μ_H (T), while BaIrO3 exhibits a non-saturating positive linear magnetoresistance, despite ferromagnetic order, which usually results in a negative saturating magnetoresistance. These examples showcase the importance of SOC across a range of strongly correlated phenomena spanning itinerant to localized electronic degrees of freedom.

This thesis will start by presenting an equivalence between a broad class of interacting disorder-free and disordered non-interacting systems. Such systems include, but are not limited to, nodal semimetals, dilute gases of bosons, trapped-ion systems with long-range interactions, and superconductive films. This equivalence is powerful: on the one hand, it allows one to simulate many-body effects by mapping them to single-body effects in disorder potentials; on the other hand, one can predict new types of phase transitions by mapping existing ones to them.

After establishing the equivalence, I will discuss three examples of such unconventional phase transitions found by duality mapping. For the first example, I will show that the BCS-BEC crossover can be mapped to a disorder-driven transition in nodal-point semimetal using the derived duality. For the second example, I will show that the BCS-superconducting transition can be mapped to a disorder-driven transition in nodal-line semimetals. These two disorder-driven transitions are different from the Anderson metal-insulator transition, which expands the types of disorder-driven transitions in non-interacting systems.

The third example is an unconventional interaction-driven transition found by mapping disorder-driven transitions to them. I will derive a phase transition of a dilute gas of bosons with power-law dispersion at finite temperatures between a phase where the bosons are effectively non-interacting and a phase where the bosons are strongly interacting. I will also discuss an example spin model that exhibits this transition, which is the $d$-dimensional XXZ model with long-range interactions, with the interaction strength decaying as the distance to the power $\delta$. The elementary excitations in this model are magnons with dispersion $k^{\delta-d}$ with attractive interactions. The spin model might be realized, and the phase transition can be detected in trapped-ion experiments.

Finally, in the last part of this thesis, I will discuss the effects of quenched disorder in magnetic materials. I will focus on a special type of quenched disorder --- spin vacancies. Though the spin vacancies are defects achieved by substituting a non-magnetic ion to replace the original magnetic ion, the screening by the surrounding bulk spins can introduce a free-spin degree of freedom, a ``quasispin'', to the vacancies, which leads to a $1/T$ contribution to the magnetic susceptibility. I will derive this quasispin effect for Ising chains with nearest- and next-to-nearest-neighbor interactions. I will also study the effective interactions between the spin vacancies mediated through the bulk spins.

\qquad Topology is a branch of mathematics that describes the connectedness of closed surfaces. In condensed matter, ideas from topology have been used to classify electronic and magnetic states. Here we discuss how topological considerations arise in three different contexts: Bulk transport in a Weyl semimetal, sub-gap excitations in a magnetization plateau, and quantum oscillations in a topological Kondo insulator. In the Weyl semimetal candidate, $Y_2Ir_2O_7$, below $T = 60K$ a bulk transport signature emerges that follows $\rho = \rho_0 T^{-4}$ consistent with the thermally screened charged impurity (TSCI) model for a charge compensated disordered Weyl semimetal. Surprisingly the $T^{-4}$ behavior extends far beyond the Ioffe-Regel criterion $k_fl$ $\mathtt{\sim}$ $O(1)$ similar in spirit to the so-called “bad metals” such as $SrRuO_3$, suggesting $Y_2Ir_2O_7$ is a 'bad' Weyl semimetal. The metamagnet $CeSb$ showcases a manifestation of topological quantization involving only local magnetic moments where sub-gap mini-plateaus that imply broken translational symmetry in a 2D ferromagnetic sublattice emerge within the 1/3rd magnetically ordered plateau state. Additionally, the specific heat increases within the 1/3rd plateau incorporating up to 10\% of the total spin entropy identifying the coexistence of seemingly non-interacting $\Delta S_z = 0$ excitations. We also measured the low temperature high field specific heat of $Al$-flux grown $SmB_6$ to further understand the presence of de Haas van Alphen (dHvA) oscillations and absence of Shubnikov de Haas (SdH) oscillations in this material. We found experimental support for Magnetoquantum Oscillations in the specific heat of heavy excitations, that coexist with light effective mass dHvA oscillations, the amplitude of which grow dramatically below $\mathtt{\sim}$ $T = 0.5 K$. This anomalous growth deviates from Lifshitz-Kosevich behavior as expected from the formation of a magnetoexciton-condensate. Lastly, we developed an apparatus and software for the measurement of non-local resistance signatures as a general probe for the discovery and study of topological transport phenomena.