Extended Bose-Hubbard model with dipolar and contact interactions

We study the phase diagram of the one-dimensional boson gas trapped inside an optical lattice with contact and dipolar interaction, taking into account next-nearest terms for both tunneling and interaction. Using the density-matrix renormalization group, we calculate how the locations of phase transitions change with increasing dipolar interaction strength for average density $\rho =1$. Furthermore, we show the emergence of pair-correlated phases for a large dipolar interaction strength and $\rho \ge 2$, including a supersolid phase with an incommensurate density wave ordering manifesting the corresponding spontaneous breaking of the translational symmetry.

Figure 1

(a) Values of $V_x$ and $a_s/a$ necessary to get $U/t=2$ and $V/t=1.5$ for different values of $d$. (b) Values of parameters in Hamiltonian (1) in such a case. The inset shows the values of $V/U$ for dipolar-only and contact-only terms.

Figure 2

The values of the string and DW order parameters, critical exponent $K$, and energy gap $\mathrm{\Delta }E$ for $V/U=3/4$, $d=0.02$. The positions of black dashed vertical lines correspond to the critical values of $t/U$ for DW-HI and HI-SF transitions ($t_c^{\mathrm{DW}-\mathrm{HI}}/U\approx 0.175$ and $t_c^{\mathrm{HI}-\mathrm{SF}}\approx 0.82$). The inset shows a logarithmic plot of ${\mathcal{O}}_{\mathrm{string}}$ and $\mathrm{\Delta }E$ near the HI-SF transition.

Figure 3

Critical values of $U/t$ for DW-HI and HI-SF transitions, $V/U=3/4$ (black solid lines), and the same for a model with $V_{\mathrm{nnn}}$, $t_{\mathrm{nnn}}$, and $T$ set to zero (red dashed lines).

Figure 4

The values of the order parameters (6), (7), and (8) for $U/t=3$, $d=0.09$. The positions of the black dashed vertical lines correspond to the critical values of $V/t$ for DW-HI and HI-MI transitions ($V/t_c^{\mathrm{HI}-\mathrm{MI}}\approx 1.94$ and $V/t_c^{\mathrm{DW}-\mathrm{HI}}\approx 2.74$).

Figure 5

The energy gap for different system sizes and $U/t=3$, $d=0.09$. The inset shows an extrapolation for $L\to \infty$, $b\approx 0.56151$.

Figure 6

Critical values of $V/t$ for DW-HI and HI-SF transitions, $U/t=3$ (black solid lines), and the same for a model with $V_{\mathrm{nnn}}$, $t_{\mathrm{nnn}}$, and $T$ set to zero (red dashed lines).

Figure 7

The phases for the system for $d=0.1$ at a fixed ratio $V/U=0.75$. Black lines showing the boundaries of DW phases are the values of ${\mu }_{+}$ and ${\mu }_{-}$ obtained from OBC DMRG ($L=200$). The black squares come from sine-square deformation (SSD) DMRG (see Appendix pp3 for details) for $L=100$ and show the transition points between PSS and PSF (where ${\mathcal{O}}_{\mathrm{DW}}$ vanishes). Blue error bars mark the boundaries of the IPSS phase (and also SSD DMRG, $L=100$). The value of pair-tunneling correlations $C_p$ (10) is plotted as a color map with the scale shown on the right.

Figure 8

OBC DMRG results of $\langle b_i^{†}{b}_{i+r}\rangle$ correlations in the middle of an $L=200$ lattice at (a) $\rho =1.25$, $t/U=0.59$ (SS phase) and (b) $\rho =2.25$, $t/U=0.37$ (PSS phase). Log-log plots of the same correlations are shown in the insets.

Figure 9

Correlations and structure factor values obtained with SSD DMRG for the system in the IPSS phase ($L=100$, $t/U=0.48$, and $\mu =3.7$). (a) Density correlations, (b) pair correlations, (c) structure factor (11), and (d) creation-annihilation correlations. For (a), (b), and (d), black points mark the numerical results, with red lines showing the fits of the functions in Eqs. (13) to (15). The value of the appropriate wave number $q_{\alpha }$ obtained from the fits [or from the position of the $S\left(q\right)$ peak in (c)] is written above each plot.

Figure 10

The results of SSD DMRG for the IPSS phase. (a) The relation between $q_{\langle nn\rangle }$ and $q_{\langle b^{†}b\rangle }$. The linear fit (red) is $q_{\langle b^{†}b\rangle }=0.9991\left(6\right)\pi -0.4984\left(7\right)q_{\langle nn\rangle }$. (b) The relation between $q_{\langle nn\rangle }$ and ${\rho }_{\mathrm{bulk}}$ shown for different values of $t/U$.

Figure 11

Position of the peak in $S\left(q\right)$ (11) computed using $m$ middle sites (black points). The red solid line shows a fit of the form $C+A{m}^{-B}cos(Km+\varphi )$. Here $K\approx 0.342\pi$, which is roughly half of $q_{\langle nn\rangle }=C\approx 0.686\pi$ (shown as a green dashed line). At most 2/3 of all lattice sites have been considered. Data are calculated for $\mu /U=5.2$, $t/U=0.6$, $L=100$, and $N_{\mathrm{cut}}=12$. The damped oscillation amplitude $S\left(q\right)$ position is approximately an order of magnitude smaller than the FWHM of the $S\left(q\right)$ function, which for maximal $m$ is $\approx 0.03\pi$.

Figure 12

(a) Position of the $S\left(q\right)$ peak $q_{\langle nn\rangle }$ (see Fig. 11) calculated for different values of maximum particles per site cutoff $N_{\mathrm{cut}}$ for $L=100$. The red solid line shows a power-law decay to a constant value $q_{e\langle nn\rangle }$ (reached for a finite $N_{\mathrm{cut}}$, here approximately 11.24; shown as a blue dashed line). (b) Values of $q_{e\langle nn\rangle }$ calculated for different system sizes $L$. Data for (a) and (b) were calculated for $\mu /U=5.2$ and $t/U=0.6$. The position of the $S\left(q\right)$ peak for $N_{\mathrm{cut}}\ge 10S\left(q\right)$ and for $L\ge 40$ changes by at least one order of magnitude less than the corresponding FWHM.