Interacting hard-core bosons with anisotropic hopping: Checkerboard supersolid, order by disorder, and first-order phase transitions

Using extensive quantum Monte Carlo simulations, we study a minimum model of interacting hard-core bosons on a square lattice with hoppings of different lengths, featuring nearest-neighbor hopping ($t_1$), anisotropic next-nearest-neighbor hopping ($t_2^{\prime }$), and nearest-neighbor repulsion ($V_1$). The paradigmatic checkerboard supersolid (CSS) phase emerges as $t_2^{\prime }$ turns on, with the solid order being characterized by ordering vector $\mathbf{q}=(\pi ,\pi )$. This serves as a rare example of the CSS phase which is obtained by doping a checkerboard solid and harbors spontaneously broken gauge and translational symmetries. A first-order supersolid-to-superfluid transition is observed. Moreover, we find a solid order-by-thermal disorder behavior together with a superfluid-to-solid transition upon increasing temperature. The underlying picture of the order-by-disorder phenomenon is figured out within the framework of the entropy effect.

Figure 1

Left: An illustration of model (1). The balls denote bosons which can hop between NN sites ($t_1$, red arrow) and between NNN sites along the $\left[11\right]$ direction ($t_2^{\prime }$, purple arrow) of the lattice. The bosons interact via a NN repulsion ($V_1$, green arrow). Right: A sketch of typical particle distributions in the CSS phase, with ball size denoting particle density.

Figure 2

Ground-state phase diagram of the $t_1-t_2^{\prime }-V_1$ model with symmetric hopping $t_1=t_2^{\prime }$ (say $t$) on the square lattice. It contains superfluid (SF), checkerboard solid (CS), and checkerboard supersolid (CSS) phases, as well as a trivial phase where lattice sites are fully occupied. The inset focuses on the CSS region. Dots represent simulation results and lines are eye guides (same for following plots). The complete phase diagram is exactly symmetric with respect to the horizonal axis, such that the part for $\mu <2$ can be straightforwardly obtained.

Figure 3

Finite-temperature phase diagram on the $\mu -T$ plane for $t=0.1$ and on the $t-T$ plane for $\mu =3.75$. The black and blue dots indicate the dissolving of crystalline order, while the red dots mark the vanishing of SF density. The central vertical axis is for $t=0.1$ and $\mu =3.75$, the left and right sides of which are, respectively, for varying $\mu$ and $t$. In addition to the CSS, SF, and CS phases which can be found on the ground-state phase diagram and survive at low temperatures, a normal liquid (NL) phase appears only at finite temperature.

Figure 4

Quantities $\rho$ (a), ${\rho }_s$ (b), and $S(\pi ,\pi )$ (c) vs $\mu$ at $t=0.1$. The two vertical lines mark phase transition points. The insets are zoom-in plots around the estimated CSS-SF transition point, showing additional Monte Carlo data for $L=64$.

Figure 5

Quantities ${\rho }_s$ (a) and $S(\pi ,\pi )$ (b) vs $t$ at $\mu =3.75$. The two vertical lines mark phase transition points. The insets are zoom-in plots for $\rho ,{\rho }_s$, and $S(\pi ,\pi )$ around the estimated CSS-SF transition point, showing additional Monte Carlo data for $L=64$.

Figure 6

Quantities ${\rho }_s$ (a) and $S(\pi ,\pi )$ (b) as a function of $T$ at $t=0.1$ and $\mu =3.75$. The insets show scaling analyses for CSS-CS and CS-NL phase transitions which belong to KT and Ising types, respectively (see text).

Figure 7

Quantities ${\rho }_s$ (a), $S(\pi ,\pi )$ (b), and $\rho$ (c) as a function of $T$ at $t=0.104$ and $\mu =3.75$. The central inset of (b) shows scaling analysis for the Ising-type CS-NL phase transition, while the upper right inset displays a histogram analysis of the structure factor at the estimated SF-CS transition point $T=0.036$, with magenta and cyan dots, respectively, for $L=128$ and 192.