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 $K_{\mathrm{eff}}$ 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 nonuniversal anomalous dimension ${\eta }_{\mathrm{sd}}$ characterizing the divergence of the susceptibility with system size, $\chi \sim L^{2-{\eta }_{\mathrm{sd}}}$. We show that ${\eta }_{\mathrm{sd}}$ obeys the hyperscaling relation ${\eta }_{\mathrm{sd}}=1/2K_{\mathrm{eff}}$. We also identify an important obstacle to measuring this exponent on finite-size systems and comment on the implications for numerics and experiments.

Figure 1

In the main figure, the solid line shows the anomalous exponent ${\eta }_{\mathrm{sd}}\left(y_i\right)$, as approximated in the small $y_i$ regime by (3). The dashed reference line shows the anomalous exponent of Giamarchi and Schulz (Ref. 1), and the crossing presumably indicates the crossover from strong-disorder to weak-disorder criticality. The inset shows the system size at which we can expect to cleanly observe the strong-disorder exponent ${\eta }_{\mathrm{sd}}$. The two reference lines show, at increasing values of $L$, the typical system size probed by Monte Carlo (Ref. 8) and the system sizes available to numerical SDRG.

Figure 2

In (a), the site decimation RG step, and in (b), the link decimation RG step.

Figure 3

In the main figure, a check of the result (21) for $\overline{R}(\zeta ,\Gamma )$. Note that $\alpha =[\exp (y_i-\frac{2}{\gamma \left(\Gamma \right)})-1]$. We stop the numerical RG for $y_i=0.1$ when only two sites are remaining and pool ${10}^3$ samples. In the inset, a similar check of the result (22) for $\overline{s}(\zeta ,\Gamma )$. Here, $\lambda =\frac{{\Omega }_i{y}_i^2}{2}$. The reference lines show the analytical predictions, and the inset blocks one outlier of the main figure.