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We investigate the phase diagrams of a one-dimensional lattice model offermions and of a spin chain with interactions extending up to next-nearestneighbour range. In particular, we investigate the appearance of regions withdominant pairing physics in the presence of nearest-neighbour andnext-nearest-neighbour interactions. Our analysis is based on analyticalcalculations in the classical limit, bosonization techniques and large-scaledensity-matrix renormalization group numerical simulations. The phase diagram,which is investigated in all relevant filling regimes, displays a remarkablyrich collection of phases, including Luttinger liquids, phase separation,charge-density waves, bond-order phases, and exotic cluster Luttinger liquidswith paired particles. In relation with recent studies, we show severalemergent transition lines with a central charge $c = 3/2$ between theLuttinger-liquid and the cluster Luttinger liquid phases. These results couldbe experimentally investigated using highly-tunable quantum simulators.

Pairing in spin chains and spinless fermions with next-nearest neighbour interactions