Automated algebra methods for finite-temperature quantum gasses in arbitrary dimension with the Quantum Thermodynamics Computational Engine (QTCE)

Recent progress in experimental ultracold atomic gasses provides a wide array of physical systems to test calculations from the theory side. Traditional non-perturbative methods like Monte Carlo are limited by the fermionic sign problem and the computational cost of varying lattice sizes and dimensions. The Quantum Thermodynamics Computational Engine (QTCE) provides a semi-analytic non-perturbative avenue for calculating non-interacting expectation values using a quantum cumulant expansion. In the QTCE, both fugacity and dimension enter as arbitrary numerical parameters, enabling study of dimensional crossover and computationally cheap calculations at varying temperature or chemical potential. Focusing on a non-relativistic, spin-½ Fermi system with a contact interaction, we discuss the technical aspects of the QTCE. In addition to reducing the naive high-cost calculation through traditional optimization techniques, we discuss diagram generation and graph theory methods for simplification, as well as semi-analytic integration and extrapolation, renormalization, and resummation.