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Abstract
The often disparate fields of quantum mechanics and statistical mechanics meet at the intersection of quantum phase transitions. These transitions, just like those in classical mechanics, can be found through measures of correlation in the system. These correlation functions are the workhorses of statistical mechanics, and provide a simple means of understanding phase transitions (among many other phenomena). In electronic structure theory, on the other hand, the electron density or the many-body wave function is traditionally the fundamental quantity. These measures have their benefits. The density can be scaled agnostically to the system complexity and can, given the universal functional, produce the energy. While the wave function contains all of the information about the system at the cost of combinatorial scaling. A third possible means of describing quantum systems comes in the form of reduced density matrices (RDMs). RDMs are the 2nd-quantized form of correlation functions in quantum systems, and the 2-electron RDM (2-RDM) in particular contains all the necessary correlation information to exactly reproduce the energy of an electronic system. In this text, we will explore the benefits of looking at quantum systems through the lens of the 2-RDM. The second through fifth chapters will discuss how the 2-RDM allows for a simple visualization of quantum and topological phase transitions, while also revealing the presence of strongly correlated phenomenon such as electron-electron and electron-hole condensation. The final two chapters will then discuss how the 2-RDM and methods based around it, provide an efficient and exact means of studying mixed fermion-boson systems on quantum devices.