Anomalous suppression of the Bose glass at commensurate fillings in the disordered Bose-Hubbard model

We study the weakly disordered Bose-Hubbard model on a cubic lattice through a one-loop renormalization-group analysis of the corresponding effective-field theory which is explicitly derived by combining a strong-coupling expansion with a replica average over the disorder. The method is applied not only to generic uncorrelated on-site disorder but also to simultaneous hopping-disorder correlated with the differences of adjacent disorder potentials. Such correlations are inherent in fine-grained optical speckle potentials used as a source of disorder in optical lattice experiments. As a result of strong coupling, the strength of the replica-mixing disorder vertex, responsible for the emergence of a Bose glass, crucially depends on the chemical potential and the Hubbard repulsion and vanishes to leading order in the disorder at commensurate boson fillings. As a consequence, at such fillings a direct transition between the Mott insulator and the superfluid in the presence of disorder cannot be excluded on the basis of a one-loop calculation. At incommensurate fillings, at a certain length scale, the Mott insulator will eventually become unstable toward the formation of a Bose glass. Phase diagrams as a function of the microscopic parameters are presented and the finite-size crossover between the Mott-insulating state and the Bose glass is analyzed.

Figure 1

Mean-field phase diagram of the clean BH model as a function of $\mu /U$ and $t/U$ showing the first three Mott lobes with $m=1,2,3$ bosons per lattice site. The phase boundaries are obtained by the sign change in the mass coefficient $R^{\left(0\right)}\left(\mu ,t,U\right)$. The dashed lines indicate the tip positions of the lobes at which the linear time derivative term in the effective-field theory vanishes, $K_1^{\left(0\right)}=0$.

Figure 2

(a) Diagrammatic representation of the interaction vertex $h$ which is diagonal in the replica index and local in imaginary time and the replica-mixing disorder vertex $g$, which depends on two different imaginary times. (b) Diagrams contributing to the renormalization of the mass coefficient $r$. Note that the contraction of the disorder vertex also gives rise to a renormalization of the temporal gradient terms ${\gamma }_1$ and ${\gamma }_2$ as described in the text. In (c) and (d) the diagrams leading to a renormalization of $h$ and $g$, respectively, are shown. We restrict ourselves to one-loop order. Diagrams vanishing in the replica limit $n\to 0$ are not shown.

Figure 3

(Color online) Upper left: phase diagram for the clean system showing the $m=1$, $m=2$ MI, and SF phases. The SF density ${\rho }_s$ is shown as a color gradient and, for comparison, the mean-field phase boundaries are indicated by dashed white lines. Other panels show the increase in the BG regions with the system size $L=a\text{exp}\left(l_{\text{max}}\right)$ for pure on-site disorder with $\Delta /U=0.15$. Along the solid white lines, the effective disorder vanishes, $G=0$. Gray areas are not accessible within the present strong-coupling approach.

Figure 4

(Color online) Enhancement and shift of the $m=1$ MI phase due to correlated hopping disorder (right) compared to the pure on-site disordered case (left). The BG region is slightly enhanced due to the disorder in the hopping. Solid white lines indicate the vanishing of the effective disorder, dashed lines the corresponding mean-field (MF) phase boundaries determined by $R^{\left(0\right)}\left(\mu ,t,U\right)=0$. For the BG phase boundary, no MF prediction is available.

Figure 5

(Color online) Phase diagrams as a function of speckle intensity $\Delta /U$ and relative hopping strength $z\overline{t}/U$ for values $\overline{\mu }/U=0.3$ (left panel) and $\overline{\mu }/U=0.45$ (right panel) of the chemical potential, corresponding to fillings below and slightly above the commensurate value ${\left(\overline{\mu }/U\right)}_1$, respectively. In the upper row, the phase diagrams for pure on-site disorder are shown and contrasted with the ones in the presence of simultaneous correlated hopping disorder with coupling $4z\gamma U=1$ (lower row). The crossover between the MI and the BG is shown for a system size corresponding to $l_{\text{max}}=8$. Note that a reliable determination of the BG/SF transition is not possible within our approach. Alternative phase boundaries consistent with recent optical lattice experiments (Ref. 31) are indicated as dashed lines.

Figure 6

(Color online) Evolution of the SF density as a function of $z\overline{t}/U$ (left) and $\Delta /U$ (right) for pure on-site disorder and with correlated hopping disorder.