Broken symmetries and local measurements in two-dimensional materials

In two dimensions, correlated electron systems display a rich variety of phases. In the last few decades, new surface probes such as the scanning tunneling microscope (STM) have played an important role in bringing the study of these systems to prominence. With the increased availability of data that has accompanied this work has come the need for new methods of analysis.; In this dissertation, we propose a number of new theoretical tools for the characterization of two-dimensional systems. This work consists of three primary sections: In the first, we propose new methods of elucidating the underlying "clean" charge order in systems in which a spatial symmetry-breaking phase transition has been rounded by the presence of quenched disorder. We test our methods against simulated data, and then apply them to STM data from the superconductor Bi2Sr2CaCuO8+delta in an attempt to resolve an ongoing debate on the nature of the charge order present in this and related materials. We find that in many cases, the experiments indicate the presence of unidirectional charge order. In other cases, the answer is presently uncertain. In the second section, we work on a complimentary problem: we explore the phase diagram of the quasi-two-dimensional rare earth tri-telluride materials via a Landau-Ginzburg free energy derived from a microscopic model. We account for the charge order seen in experiment and make predictions about the existence of other phases in these materials. In the final section, we develop a novel, highly spatially-localized technique for measuring optimal model parameters for an effective Hamiltonian from STM data. This technique exploits the availability of both positive and negative bias voltage data to form a new set of expressions analogous to the famous f-sum rule. We apply this technique to the same cuprate as in the first section and comment on the resolution of competing interpretations of the spatial inhomogeneities in the superconducting gap near optimal doping. We find that the electro-chemical potential is remarkably uniform in these materials, however the underlying variations in the superconducting gap parameter are larger than are observed in measurements.