Figure 1

Top panel: The average density, $\rho =N/L$, as a function of the chemical potential, $\mu$. The vertical blue dashed line shows the Maxwell construction between ${\rho }_1$ and ${\rho }_2$. The brown dot-dashed lines show different phase boundaries at ${\rho }_3$ and ${\rho }_4$ as discussed in the text. The inset shows a snapshot of the density profile at half filling, $\rho ={\rho }_1=1$, with the blue (red) squares indicating $\langle n_i^a\rangle =1$ ($\langle n_i^b\rangle =1$). Bottom panel: The correlated winding (blue circles) and the staggered structure factor (green squares) as a function of $\rho$. The dot-dashed lines show different phase boundaries. All data are for $L=10$, $\beta =80$, $t_a=0.08$, $t_b=1.00$, and $U^{ab}=6$. Error bars are smaller than symbol sizes.Reuse & Permissions

Figure 2

Snapshot of the average local densities for $L=20$ and $\rho =1.44$. Top panel: Open boundary conditions. For $a$ particles (left panel), sites close to the boundary (red) have $\langle n_i^a\rangle \sim 0$, while the occupation of the central (blue) region is $\langle n_i^a\rangle \sim 1$. For $b$ particles (right panel) the density close to the boundary (blue region) is $\langle n_i^b\rangle \sim 1$, and at the center (green region), $\langle n_i^b\rangle \sim 0.60$. Middle panel: The same quantities shown in the top panel but with periodic boundary conditions. Bottom panel: At the left the homogeneous density distribution of both $a$ and $b$ particles for $\rho =1.72$. At the right, a sketch of the density profile for a simulation with periodic boundary conditions.Reuse & Permissions

Figure 3

Top panel: Average density, $\rho$, versus chemical potential, $\mu$, for different temperatures. Bottom panel: The correlated winding as a function of $\rho$ for different temperatures. All data are for system size $L=10$. Error bars are smaller than symbol sizes.Reuse & Permissions

Figure 4

Bottom panel: The temperature, $T$, versus doping, $\delta$, phase diagram with equal population of each species. At half filling, the system is antiferromagnetic (AF) below $T_c^{AF}\sim 0.116$ and normal liquid (NL) at larger temperatures. By increasing the doping a discontinuous transition from AF to superfluid (SF) phase occurs, and an AF/SF phase-separated region develops between ${\delta }_1=0$ and ${\delta }_2=0.06$. Between ${\delta }_3=0.13$ and ${\delta }_4=0.28$ phase separation (PS) occurs with the heavy species becoming Mott while the light one displays regions with either Mott or superfluid behaviors. The inset shows entropy as a function of temperature for three dopings. Top left panel: Scaling behavior of the static staggered structure factor for the continuous transition from AF to NL at half filling, ${\delta }_1=0$ (corresponds to the filled blue diamond in the bottom panel). The inset shows the scaling near the critical temperature with Ising-like critical exponents. Top right panel: Correlated winding as a function of temperature for different system sizes at $\delta =0.07$ (filled red diamond). The inset shows the finite-size scaling to find the SF critical temperature in the thermodynamic limit for the continuous transition at $\delta =0.07$. The data points are based on simulation results, and the lines are guides to the eye.Reuse & Permissions