Dipolar bosons on an optical lattice ring

We consider an ultrasmall system of polarized bosons on an optical lattice with a ring topology, interacting via long-range dipole-dipole interactions. Dipoles polarized perpendicular to the plane of the ring reveal sharp transitions between different density-wave phases. As the strength of the dipolar interactions is varied, the behavior of the transitions is first-order-like. For dipoles polarized in the plane of the ring, the transitions between possible phases show pronounced sensitivity to the lattice depth. The abundance of possible configurations may be useful for quantum-information applications.

Figure 1

The superfluid (SF) fraction $f_s$ (solid black lines) and the condensate fraction $f_c$ (dashed red lines) both represented on the left vertical axis as a function of $V/U$ for $U=1,4,7,10$ [(a)–(d), respectively]. Right axis scale corresponds to the thin dot-dashed green lines that represent the variance of the occupation on different sites. For small $V/U$, we observe an increase of the superfluid (and condensate) fractions with $V$ both in the SF phase [for (a) $U=1$ as well as (d)] where the increase of $V$ causes a transition from a Mott insulator (MI) to a SF phase. Around $V/U=0.5$, a transition to a density-wave state $\left[2,0\right]$ occurs, more suddenly for larger $U$, as manifested by the rapid drops of $f_s$ and $f_c$. For larger $V/U$, sharp transitions between different density waves occur as manifested in the $f_c$ as well as in the variance of occupation $\Delta$.

Figure 2

(a) For a ring-shaped lattice, the arrow indicates the polarization of the dipoles. Due to symmetry, sites 2, 3, 6, and 7 are equivalent (type I) as are sites 1, 4, 5, and 8 (type II). (b) The results for a shallow lattice $U/J=1$. The solid black line shows a SF fraction with a drop around $V/U=0.5$. This drop is accompanied by a similarly rapid increase of the number variance ${\Delta }_I$ (dash-dotted blue line, right axis scale) as well as changes of the condensate fraction $f_c$ (thin dotted red line) and the second eigenvalue of OBDM $g_c$ (thin dashed red line). On the other hand, the mean occupation $n_I$ of type-I sites (dashed blue line, right axis scale) increases slowly and smoothly over the whole studied range of $V/U$. For a discussion, see the text.

Figure 3

Intermediate $U/J$ behavior. (a) SF fraction for $U/J=4$ (dash-dotted black line), $U/J=6$ (dashed red line), and $U/J=8$ (solid green line). (b) The mean occupation (solid line) and the variance of type-I states (dashed line) for $U/J=6$. (c) The $V/U$ dependence (again for $U/J=6$) of the condensate fraction $f_c$, i.e., the largest eigenvalue of the OBDM (solid line) and the second largest eigenvalue, denoted by $g_c$ (dashed line). The sudden drop of the SF fraction occurring at $V/U=0.4$ in (a) is accompanied by a sharp increase of the occupation variance in type-I sites ${\Delta }_I$ [dashed red line in (b)]. The occupation $n_I$ (black line) shows there is a small kink too. At the same $V/U=0.4$, the sharp drop in $f_b$ is observed while $g_b$ shows an increase.

Figure 4

Occupation of type-I sites $n_I$ for $U/J=10$ (solid black line) and $U/J=20$ (thin solid red line). The corresponding variance ${\Delta }_I$ is plotted for $U/J=10$ (dashed black line) and $U/J=20$ (thin dashed red line). For a discussion, see the text.