Weak- versus strong-disorder superfluid—Bose glass transition in one dimension

Using large-scale simulations based on matrix product state and quantum Monte Carlo techniques, we study the superfluid to Bose glass transition for one-dimensional attractive hard-core bosons at zero temperature, across the full regime from weak to strong disorder. As a function of interaction and disorder strength, we identify a Berezinskii-Kosterlitz-Thouless critical line with two different regimes. At small attraction where critical disorder is weak compared to the bandwidth, the critical Luttinger parameter $K_c$ takes its universal Giamarchi-Schulz value $K_c=3/2$. Conversely, a nonuniversal $K_c>3/2$ emerges for stronger attraction where weak-link physics is relevant. In this strong-disorder regime, the transition is characterized by self-similar power-law-distributed weak links with a continuously varying characteristic exponent $\alpha$.

Figure 1

Top: phase diagram showing the SF and BG regimes. Red and gray symbols denote the SF-BG critical disorder $h_c$ for various values of the interaction $\mathrm{\Delta }$. Color contours show the value of the power-law coefficient $\alpha$ from MPS simulations for $L=100$, with $-1\le \alpha \le 20$. $\alpha$ decreases for stronger attraction along the critical line, i.e., power-law tails become fatter. Power-law fits fail in the white region below the bandwidth BW (black dashed line). Inset: a random field configuration sketch is compared to BWs within weak (gray) vs strong (green) disorder situations. Bottom: Critical LL parameter $K_c$ (MPS and QMC estimates: green symbols; clean case value $K_0$: dashed blue line) and power-law coefficient $\alpha$ (red symbols) at the SF-BG transition for various values of the interaction $\mathrm{\Delta }$. Red and green lines are guides to the eyes. $\alpha$ diverges at the crossover from the weak-link to GS regime. $L_{\mathrm{MPS}}=300$.

Figure 2

QMC results for the SF density ${\rho }_{\mathrm{SF}}$ (a) left, the compressibility $\kappa$ (a) right, and the LL parameter $K$ [Eq. (5)] (b). The $T=0$ converged data are averaged over $>{10}^3$ independent random configurations. (a) Raw data for the SF density ${\rho }_{\mathrm{SF}}$ (left) and the compressibility $\kappa$ (right) across the SF-BG transition for $\mathrm{\Delta }=-3/4$. ${\rho }_{\mathrm{SF}}$ jumps (black cross) at the transition point $h_c=0.375\left(15\right)$ (gray region) while $\kappa$ remains smooth and featureless. (b) Scaling form of the LL parameter Eq. (6) plotted against disorder strength $h$ for various system sizes $\left(L=32,...,512\right)$ and various interaction strengths $\mathrm{\Delta }$, as indicated on the plot. Crossings (gray area) are obtained using an $O\left(1\right)$ length scale ${\ell }_0$.

Figure 3

Distributions of the correlator $ln\tilde{G}=lnG-\overline{lnG}$ (MPS, lines) and the superfluid stiffness $ln{\tilde{\rho }}_{\mathrm{SF}}=ln{\rho }_{\mathrm{SF}}-\overline{ln{\rho }_{\mathrm{SF}}}$ (QMC, symbols) at the critical disorder $h_c$ for various values of $\mathrm{\Delta }$. For $G$ we show $Q=\overline{P_r}[ln\tilde{G}]$, where error bars are obtained from the dispersion of $P_r$ (see main text). Two regimes can be clearly distinguished. In the weak-disorder regime (right panel) where $h_c<\mathrm{BW}$, the distributions of $ln\tilde{G}$ and $ln{\tilde{\rho }}_{\mathrm{SF}}$ are distinct and self-averaging. Conversely, in the strong-disorder regime $h_c>\mathrm{BW}$, the distributions for both quantities overlap and do not depend on system size, indicating self-similarity. Moreover, they develop an exponential tail at low values Eq. (7) controlled by the weak-link exponent $\alpha$, which decreases as $\mathrm{\Delta }\to -1$ (very wide distributions, left panel) and diverges when approaching the GS regime (narrow distributions, second panel from the right). Note the differing $x$-axis scale on the last panel, indicating much narrower distributions.