Dipolar Bose-Hubbard model in finite-size real-space cylindrical lattices

Recent experimental progress in magnetic atoms and polar molecules has created the prospect of simulating dipolar Hubbard models with off-site interactions. When applied to real-space cylindrical optical lattices, these anisotropic dipole-dipole interactions acquire a tunable spatially dependent component while they remain translationally invariant in the axial direction, creating a sublattice structure in the azimuthal direction. We numerically study how the coexistence of these classes of interactions affects the ground state of hard-core dipolar bosons at half filling in a finite-size cylindrical optical lattice with octagonal rings. When these two interaction classes cooperate, we find a solid state where the density order is determined by the azimuthal sublattice structure and builds smoothly as the interaction strength increases. For dipole polarizations where the axial interactions are sufficiently repulsive, the repulsion competes with the sublattice structure, significantly increasing entanglement and creating two distinct ordered density patterns. The spatially varying interactions cause the emergence of these ordered states in small lattices as a function of interaction strength to be staggered according to the azimuthal sublattices.

Figure 1

(a) Cylindrical lattice formed by 13 octagonal rings giving the parameters $L_z=13$ and $L_c=8$. The separations between adjacent sites along the axis and around the ring are denoted by $r_z$ and $r_c$, respectively. A single boson, represented by the larger sphere, occupies one of the lattice sites. Its dipole moment $\widehat{\mathit{d}}$, shown by the single-headed arrow, is polarized in the $x-z$ plane at an angle $\theta$ to the $z$ axis. (b) A single ring of the lattice, showing azimuthal angle $\varphi$ and $L_c=8$ sites labeled by their $c$ coordinate. The two sublattices (named polar and equatorial) are denoted by white and black spheres, respectively. The different nearest-neighbor in-plane separation vectors are labeled. (c) Relative dipole-dipole interaction strengths between nearest-neighbor (solid lines) and next-nearest-neighbor (dashed lines) sites as a function of polarization angle $\theta$. Each interaction has two markers as a visual aid. Domain is split by black vertical lines into three regions (I, II, III) according to the qualitative ordering mechanism and ground state at strong interaction. Zero interaction strength is shown by the black dotted line.

Figure 2

(a) Finite-size phase diagram of model. Unfilled markers show calculated finite-size transition points, while black lines join neighboring markers as a guide to the eye (the boundary between the PS and TU regions is marked in a dash-dotted line to indicate that the transition is not sharp). Filled triangle shows the value of $\theta$ for transition from SS to PS as $V\to \infty$. Quantitative order parameters are plotted for the dashed lines at $\theta =\{0,\frac{\pi }{4},\frac{\pi }{2}\}$ in Fig. 3 and for the dotted lines at $V=\{2,4\}$ in Fig. 4. Black square markers on the $\theta$ axis mark the boundaries between regions I, II, and III. (b)–(e) Expectation value of on-site density for solid ordered states using the common color scale shown next to (e). Arrows point towards the corresponding parameters in (a) marked by asterisks.

Figure 3

Physical parameters for $\theta =0, \frac{\pi }{4}$, and $\frac{\pi }{2}$ as a function of $V$ corresponding to the black dashed lines in Fig. 2. (a) $S_{\text{ent}}$ on the left-hand $y$ axis and ${\delta }_e$ on the right-hand $y$ axis. (b) Solid order parameters $M_{{\mathrm{\Delta }}_z,{\mathrm{\Delta }}_c}$. Any transition lines from Fig. 2 which are intersected by these graphs are denoted with the corresponding marker.

Figure 4

Physical parameters for $V=2$ and $V=4$ as a function of $\theta$ corresponding to the dotted lines in Fig. 2. (a) $S_{\text{ent}}$ on the left-hand $y$ axis and ${\delta }_e$ on the right-hand $y$ axis. (b) Solid order parameters $M_{{\mathrm{\Delta }}_z,{\mathrm{\Delta }}_c}$. A dotted black line at $M_{{\mathrm{\Delta }}_z,{\mathrm{\Delta }}_c}=0$ is added as a guide to the eye. Any transition lines from Fig. 2 which are intersected by these graphs are denoted with the corresponding marker. The boundaries between regions I, II, and III are denoted by black square markers on the $\theta$ axis.

Figure 5

Further detail for $\theta =\frac{\pi }{2}$. (a) Sublattice-resolved orbital occupation and solid order parameters as a function of $V$. Vertical black dotted lines denote $V=3.4$ and $V=3.6$, for which the on-site density is plotted in (b) and (c), respectively. Plots (b) and (c) use the color scale shown next to (c).

Figure 6

Finite-size scaling data for TU-SS transition at $\theta =0$. (a) Binder cumulant $U_{\text{SS}}$ against rescaled interaction strength for $\nu =0.76$. Inset shows $U_{\text{SS}}$ against $V$. (b) Rescaled square magnetization $L_z^{\left(\frac{2\beta }{\nu }\right)}\langle m_{\text{SS}}^2\rangle$ against rescaled interaction strength for $\beta =0.08$. Inset shows original square magnetization $\langle m_{\text{SS}}^2\rangle$ against interaction strength. Markers for different $L_z$ are plotted according to the legend in (a).

Figure 7

TU-PS transition at $\theta =\frac{\pi }{4}$ for different $L_z$. (a) ${\chi }_{\text{F}}$ as a function of $V$. (b) $\frac{\partial S_{\text{ent}}}{\partial V}$ for the same values of $L_z$ shown in (a). Inset shows $S_{\text{ent}}$ as a function of $V$. (c) $ln\left({\chi }_{\text{F}}^{*}\right)$ against $ln\left(L_z\right)$ for both peaks. The dashed red line is a linear fit for the upper peak data for the seven largest system sizes $L_z=\{40,50,60,70,80,90,100\}$, from which $\nu =0.97$ was extracted. (d) Rescaled fidelity susceptibility $L_z^{(1-\frac{2}{\nu })}{\chi }_{\text{F}}$ against rescaled interaction strength $L_z^{\left(\frac{1}{\nu }\right)}(V-V_{\text{TU-PS}})$ around the upper peak for the seven largest system sizes.

Figure 8

TU-DC transition at $\theta =\frac{\pi }{2}$ for different $L_z$. (a) Binder cumulant $U_{\text{DC}}$ as a function of $V$ for different $L_z$ according to legend in (d). The smallest and largest system sizes ($L_z=12$ and $L_z=20$) are plotted with dashed and solid lines, respectively, between data points for clarity. (b, c) Sublattice-resolved Binder cumulants for the polar and equatorial sublattices, where only the smallest and largest cylinder lengths are shown for clarity. (d) $S_{\text{ent}}$ as a function of $V$.

Figure 9

$\langle {\widehat{n}}_{z,c}\rangle$ for three periods of (a) ${\mathrm{DS}}_3$ and (b) ${\mathrm{DS}}_4$ states.

Figure 10

TU-PS transition at $\theta =\frac{\pi }{4}$ for $L_z=100$ plotted with $N=N_H$ in solid lines, $N=N_H-1$ in dashed lines, and $N=N_H-2$ in dotted lines. (a) $\langle {\widehat{n}}_{z,c=1}\rangle$ for three different values of $V$. Neighboring densities for the discrete site index $z$ are joined by straight lines to guide the eye, while one marker per line is used for identification. (b) $S_{\text{ent}}$ and ${\delta }_e$ as a function of $V$.