Scale-invariant phase transition of disordered bosons in one dimension

The disorder-induced quantum phase transition between superfluid and nonsuperfluid states of bosonic particles in one dimension is generally expected to be of the Berezinskii-Kosterlitz-Thouless (BKT) type. Here, we show that hard-core lattice bosons with power-law hopping decaying with distance as $1/{r}^{\ensuremath{\alpha}}$ with a finite integral over space---corresponding in spin language to an $XY$ model with power-law couplings---undergo a non-BKT continuous phase transition instead. We use exact quantum Monte Carlo methods to determine the phase diagram for different values of the exponent $\ensuremath{\alpha}$, focusing on the regime $\ensuremath{\alpha}>2$. We find that the scaling of the superfluid stiffness with the system size is scale invariant at the transition point for any $\ensuremath{\alpha}\ensuremath{\le}3$---a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By a scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\ensuremath{\nu}\ensuremath{\ge}2$ and demonstrate a universal behavior of disordered bosons in one dimension. For $\ensuremath{\alpha}>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder.