Thermodynamics and Thermometry for Dilute SU(N) Hubbard Models using Automated Algebra

The world around us contains a diverse range of materials whose behavior is determined by the underlying structure at different scales. While the properties of materials we interact with like density, pressure, or temperature are noticeable at a macroscopic scale, the key to understanding where they come from and how they are connected relies on understanding the dynamics of the systems composing them. At the smallest scales, this is a collection of interacting quantum particles. Characterizing these systems is notoriously difficult, especially when the interactions are strong. While a number of methods exist to attack the quantum many-body problem, this dissertation explores application and extensions of methods developed by the Computation Quantum Matter group using automated algebra techniques, a computational hybrid of analytic and numerical approaches. After introducing the main relevant contexts (ultracold atoms and neutron matter) and models (Fermi gasses and the Hubbard model), the bulk of this dissertation details my work on a number of automated algebra projects characterizing quantum matter using the virial expansion, culminating with a virial expansion of the harmonically trapped Hubbard model. My analysis provides a solution to the problem of thermometry in ultracold atoms in optical lattices with arbitrary dimension, N-body interactions, and species related by an SU(N) symmetry.