Spontaneous Peierls dimerization and emergent bond order in one-dimensional dipolar gases

We investigate the effect of dipolar interactions in one-dimensional systems in connection with the possibility of observing exotic many-body effects with trapped atomic and molecular dipolar gases. By combining analytical and numerical methods, we show how the competition between short- and long-range interactions gives rise to frustrating effects which lead to the stabilization of spontaneously dimerized phases characterized by a bond ordering. This genuine quantum order is sharply distinguished from Mott and spin-density-wave phases, and can be unambiguously probed by measuring nonlocal order parameters via in situ imaging techniques.

Figure 1

Numerical phase diagram for dipolar mixtures described by the Hubbard Hamiltonian in Eq. (2). In between a CDW and a SDW phase, a region with dominant bond order develops close to the line $U\simeq V\frac{3\zeta \left(3\right)}{2}$; the CDW-BOW and BOW-SDW transitions are of the Gaussian and Berezinskij-Kosterlitz-Thouless type, respectively. Errors in determining phase transition points are smaller than symbol sizes. The arrows in the cartoons denote the alignment of the dipoles, while green (light gray) and blue (dark gray) points represent the internal spin.

Figure 2

(a) Luttinger parameter $K_c$, for $U=4t$ as a function of $V/t$ for different system sizes $L$. (b) BOW order parameter $\langle B\rangle$ for various $V/t$; in the box is its value in the TDL. (c) Density gap ${\Delta }_c$ for various values of $V/t$; in the box its values in the TDL. Straight lines are guides for the eye, whereas dashed lines fit the numerical data using the appropriate scaling laws.

Figure 3

The difference $\Delta S\left(L\right)=S_L(L/2)-S_{L/2}(L/4)$ for several system sizes for $U=4t$ varying $V/t$. The inset shows the Luttinger parameter $K_c$ in the region $2.35\le V/t\le 2.4$. Straight lines are guides for the eye.

Figure 4

(a) Luttinger parameter $K_s$, for $U=4t$ as a function of $V/t$ for different system sizes $L$. (b) BOW order parameter $\langle B\rangle$ for various $V/t$; in the box is its value in the TDL. (c) Spin gap ${\Delta }_s$ for various values of $V/t$; in the box its values in the TDL; the inset shows the region around the origin. Straight lines are guides for the eye, whereas dashed lines fit the numerical data using the appropriate scaling laws.

Figure 5

(a) Luttinger parameter $K_c$ for $U=1.5t$ as a function of $V/t$: The maximum location indicates the Gaussian transition point, $V_c/t=0.88\pm 0.04$. (b) Luttinger parameter $K_s$ for $U=1.5t$ as a function of $V/t$ indicating a BKT transition (see text). (c) Bond-order parameter finite-size scaling at $U=1.5t$ and different values of the dipolar interaction strength.

Figure 6

Density and spin ($\times 25$) gaps in the thermodynamic limit for various $V/t$.