Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-André model

As opposed to random disorder, which localizes single-particle wave-functions\nin 1D at arbitrarily small disorder strengths, there is a\nlocalization-delocalization transition for quasi-periodic disorder in the 1D\nAubry-Andr\\'e model at a finite disorder strength. On the single-particle\nlevel, many properties of the ground-state critical behavior have been revealed\nby applying a real-space renormalization-group scheme; the critical properties\nare determined solely by the continued fraction expansion of the incommensurate\nfrequency of the disorder. Here, we investigate the many-particle\nlocalization-delocalization transition in the Aubry-Andr\\'e model with and\nwithout interactions. In contrast to the single-particle case, we find that the\ncritical exponents depend on a Diophantine equation relating the incommensurate\nfrequency of the disorder and the filling fraction which generalizes the\ndependence, in the single-particle spectrum, on the continued fraction\nexpansion of the incommensurate frequency. This equation can be viewed as a\ngeneralization of the resonance condition in the commensurate case. When\ninteractions are included, numerical evidence suggests that interactions may be\nirrelevant at at least some of these critical points, meaning that the critical\nexponent relations obtained from the Diophantine equation may actually survive\nin the interacting case.\n