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The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism, ferromagnetism, superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations. 

The nodal-line semiconductor Mn3Si2Te6 is generating enormous excitment due to the recent discovery of a field-driven insulator-to-metal transition and associated colossal magnetoresistance as well as evidence for a new type of quantum state involving chiral orbital currents. Strikingly, these qualities persist even in the absence of traditional Jahn-Teller distortions and double-exchange mechanisms, raising questions about exactly how and why magnetoresistance occurs along with conjecture as to the likely signatures of loop currents. Here, we measured the infrared response of Mn3Si2Te6 across the magnetic ordering and field-induced insulator-to-metal transitions in order to explore colossal magnetoresistance in the absence of Jahn-Teller and double-exchange interactions. Rather than a traditional metal with screened phonons, the field-driven insulator-to-metal transition leads to a weakly metallic state with localized carriers. Our spectral data are fit by a percolation model, providing evidence for electronic inhomogeneity and phase separation. Modeling also reveals a frequency-dependent threshold field for carriers contributing to colossal magnetoresistance which we discuss in terms of polaron formation, chiral orbital currents, and short-range spin fluctuations. These findings enhance the understanding of insulator-to-metal transitions in new settings and open the door to the design of unconventional colossal magnetoresistant materials.