Magnetic orders, stripes, and superconductivity - recent progress in computational studies of the Hubbard model

The Hubbard model is a fundamental model in quantum many-body physics. Since the discovery of high-temperature superconductivity, it has been a focal point of research, in condensed matter and more recently also in the field of ultracold atoms. The physics of this model is often the outcome of a delicate balancing act between multiple competing or co-existing tendencies, which makes it highly challenging to determine or understand. Robust predictions are difficult to obtain. Recently, significant progress has been made via advances in computational methods, and the combined use of complementary methods through collaborative efforts. The results have reached a new level of accuracy and rigor, which challenges a number of long-held notions about methods such as quantum Monte Carlo, and the reach of "classical" computation for simulating quantum systems. I will share some of the lessons learned and insights gained, and what these computations have revealed about the properties of the Hubbard model. In particular, we find [1] superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity is weaker and is accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexists with partially-filled stripe order, which persists into the overdoped region with weaker hole-density modulation.