Statistics-tuned phases of pseudofermions in one dimension

We show that a quadratic system of pseudofermions, with tunable fractionalised statistics, can host a rich phase diagram on a one dimensional chain with nearest and next nearest neighbor hopping. Using a combination of numerical and analytical techniques, we show that that by varying the statistical angle and the ratio of the hopping, the system stabilizes two Tomonaga-Luttinger liquids (TLL) with central charges c = 1 and 2 respectively along with the inversion symmetry broken bond ordered (BO) insulating phase. Interestingly, the two quantum phase transitions in the system - (1) between the two TLLs, and, (2) the c = 1 TLL and BO phase can be engendered by solely tuning the statistics of the pseudofermions. Our analysis shows that both these transition are continuous and novel with the former lacking a local order-parameter based description and the latter of Berezinskii-Kosterlitz-Thouless type. These phases and phase transitions can be of direct experimental relevance in context of recent studies of fermionic cold atoms.