Thermodynamic anomalies in ultracold quantum gases

Although well-researched as a prototype Hamiltonian for strongly interacting quantum systems, the Bose-Hubbard model has not so far been explored as a fluid system with waterlike anomalies. Water, the substance of life, is known for its myriad of anomalous properties, whose origins are still subject of intense debates. In order to provide a different insight into this problem, we show how its density anomaly can be reproduced using a quantum simulator. In particular, we demonstrate that the Bose-Hubbard model, such paradigm system in quantum mechanics, exhibits an increase in density with temperature at fixed pressure in the regular fluid regime and in the superfluid phase. We propose that the mechanism underlying the anomalies is related to zero point entropies and ground state phase transitions. A connection with the typical experimental scales and setups including confinement effects is also addressed. In this scenario, such finding opens a new pathway for theoretical and experimental studies of waterlike anomalies in the area of ultracold quantum gases. We also discuss in detail the occurrence of anomalous double peaks in their specific heat dependence on temperature. This feature, usually associated with a high geometrical frustration, can also be a consequence of a purely energetic competition. By employing self-energy functional calculations combined to finite-temperature perturbation theory, we propose a mechanism based on ground-state degeneracies expressed as residual entropies. A general decomposition of the specific heat in terms of all possible transitions between the system’s eingenvalues provides an insight on the nature of each maximum. Furthermore, we address how the model parameters modify the structure of these peaks based on its spectral properties and atom-atom correlation function. Regarding the theoretical foundations of the methods employed, we address a deep analysis of the Legendre transformation, and how it can be conceived as extremum principle. We discuss the geometrical implications in a general framework, which includes the techniques explored throughout this thesi