Lifshitz transitions in multi-band Hubbard models for topological superconductivity in complex quantum matter

How the macroscopic quantum coherence can resist to the decoherence attacks of high temperature is a major challenge for the science of the 21st century. Superstripes 2017 conference held in Ischia on June 2017 has been focused on the new physics of high $T_c$ superconductors made of complex quantum matter. Today the standard model of high $T_c$ superconductivity which grabs the physics of complex quantum matter is the multi-band Hubbard model where the dome of $T_c$ occurs by driving the chemical potential in the proximity of a topological Lifshitz transition. The multi-gap superconductivity in the $T_c$ dome is driven by exchange interaction between a condensate in the BEC-BCS crossover which coexists with second BCS condensates. The proximity to Lifshitz transitions in correlated electronic systems gives the ubiquitous arrested phase separation observed in all high temperature superconductors. Non Euclidean filamentary hyperbolic geometry is needed for the space description of superstripes textures produced by the coexistence of short range CDW puddles, hole poor SDW puddles and self organized dopants rich puddles. A road map to room temperature superconductors in particular organic compounds made of superlattices of quantum wires driven by Fano resonances with one of the condensates in the BEC-BCS crossover has been proposed.