Susceptibility at the Superfluid-Insulator Transition for One-Dimensional Disordered Bosons

A pair of recent Monte Carlo studies have reported evidence for and against a crossover from weak to strong-disorder criticality in the one-dimensional dirty boson problem. The Monte Carlo analyses rely on measurement of two observables: the effective Luttinger parameter K_{eff} and the superfluid susceptibility chi. The former quantity was previously calculated analytically, using the strong-disorder renormalization group (SDRG), by Altman, Kafri, Polkovnikov, and Refael. Here, we use an extension of the SDRG framework to find a non-universal anomalous dimension eta_{sd} characterizing the divergence of the susceptibility with system size: chi ~ L^(2-eta_{sd}). We show that eta_{sd} obeys the hyperscaling relation eta_{sd} = 1/2K_{eff}. We also identify an important obstacle to measuring this exponent on finite-size systems and comment on the implications for numerics and experiments.