Many of the most interesting problems in theoretical condensed matter physics involve the study of two- and three-dimensional materials, but this is in general very difficult. On the other hand, one-dimensional quantum systems are much better understood, with exact solutions possible using techniques like bosonization, and with extremely efficient numerical approaches based on matrix product states. Thus one promising route to studying higher-dimensional systems is the application of techniques developed for one-dimensional systems. In this dissertation I discuss two research projects in which I use this approach.

In the first study, motivated by intriguing experiments that have shown two-dimensional polymer films to be promising materials for thermoelectric devices, I consider a two-dimensional material consisting of an array of one-dimensional systems. Each is treated as a strongly-interacting Luttinger liquid, and I assume weak (incoherent) coupling between them, an approximation which I refer to as the ``quasi-atomic limit.'' I find integral expressions for the (interchain) transport coefficients, including the electrical and thermal conductivities and the thermopower, and I extract their power law dependencies on temperature. Luttinger liquid physics is manifested in a violation of the Wiedemann-Franz law; the Lorenz number is larger than the Fermi liquid value by a factor between $\gamma^2$ and $\gamma^4$, where $\gamma\geq 1$ is a measure of the electron-electron interaction strength in the system.

In the second project, motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, I study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, I identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.