Figure 1

Here, we illustrate the effect of having a resonant two-particle state away from the single-particle ground state. The hopping amplitudes are essentially given as the overlap of the wave functions of the neighboring sites. Thus within this “two-level” model we can identify three different hopping processes and amplitudes: (a) from a single-particle state to another single-particle state, (b) from a two-particle state to a single-particle state or the reverse process, and (c) from a two-particle state to another two-particle state. Different shades of gray (blue and red online) indicate fermions with opposite spins.Reuse & Permissions

Figure 2

Superfluid long-range order $\Phi$ (diamonds, continuous line) and antiferromagnetic order parameter $M_{AF}$ (dashed line) at different hole densities and coupling $U$ (plots a–c). These figures correspond to the scenario with phase separation. Here, we assumed the full Hamiltonian of Eq. (4) and the interaction parametrized by $U=U_o/{(1+\delta g/t)}^2$ with $U_o=13.33$ in units of $t$ [Eq. (5)]. In the phase-separated region $0<x<x_c$ (vertical dotted line), the order parameters are assumed to depend linearly on $x$, so that $\Phi \left(x\right)=\Phi \left(x_c\right)x/x_c$ and $M_{\text{AF}}\left(x\right)=M_{\text{AF}}\left(0\right)\left[1-x/x_c\right]$. For comparison, we also show $M_{\text{AF}}\left(x\right)$ for the homogeneous phase scenario (circles) at the interaction given by $\delta g/t=0$ (plot a). There is a distinguishable difference for the optimizing superfluid parameter ${\Delta }_d$ between $\delta g/t=0$ and $\delta g/t=-0.5$ (plot d). However, there is only a small difference for ${\Delta }_d$ between $\delta g/t=0$ and $\delta g/t=0.2$ cases; thus we use the same parametrization for both. The optimum value of the antiferromagnetic parameter ${\Delta }_{af}$ was $\sim$$0.35$ at $x=0$. It has a rather weak dependence on $\delta g/t$ (that is, on $U$). As we remark later (see the caption of Fig. 3), the location of the critical doping $x_c$ for $\delta g/t=-0.5$ (plot c) is more affected by finite size corrections compared to the weaker coupling cases.Reuse & Permissions

Figure 3

Energy density per hole $e_{\mathrm{hole}}\left(x\right)$ as a function of the hole density $x$ is shown for the full Hamiltonian model. The interaction strengths corresponding to $\delta g/t=-0.5$ ($U/t=53.32$), 0 ($U/t=U_o/t=13.33$), and $0.2$ ($U/t=9.26$) cases are considered. The results are for $N_{\mathrm{latt}}=82$ (empty symbols) and 170 (full symbols) lattice sites. We notice that with the increased system size, $e_{\mathrm{hole}}\left(x\right)$ becomes flatter around the $x_c$. However, we observe that $e_{\mathrm{hole}}\left(x\right)$ for $\delta g/t=0$ and $=0.2$ continues to show a distinct minimum at $x_c>0$, signaling phase separation in the region $0<x<x_c$. On the other hand, for $\delta g/t=-0.5$, the mixed phase could be washed out in the thermodynamic limit. Thus the tendency for the system to phase separate is reduced at large values of $U$. The unit of energy is $t$.Reuse & Permissions

Figure 4

The ground-state energies of the Hamiltonian in Eq. (4) as a function of doping $x$ for various interaction strengths parametrized by $\delta g/t$: $\delta g/t=-0.5$ ($U/t=53.32$), 0 ($U/t=U_o/t=13.33$), and $0.2$ ($U/t=9.26$). Within the region defined by $x\in (0,x_c)$, we also show the energies corresponding to the first-order (continuous line) and the second-order (triangle) phase transition scenarios. The energies are shown to be degenerate (within the error bars) for the mixed and the homogeneous cases. SF${}_d$ smoothly transitions to the normal Fermi fluid (NFF) at $x\gtrsim x_c^{\prime }$ ($x_c^{\prime }\approx 0.4$) when ${\Delta }_d\to 0$. These data are for 170 lattice sites. The statistical errors are smaller than the size of the symbols. The inset figure shows the ground-state energy of the homogeneous phase for different system sizes. The unit of energy is $t$.Reuse & Permissions

Figure 5

Condensation energy per site at different values of $\delta g/t$ for the Hamiltonian of Eq. (4). Hole densities $x$ between 0 and $x_c$ are considered. In this case, $x=0$ corresponds to the pure antiferromagnet while $x=x_c$ corresponds to the pure superfluid phase. For $0<x<x_c$, the condensation energy is that of the homogeneous solution. The unit of energy is $t$.Reuse & Permissions

Figure 6

Comparison of the energy density per hole $e_{\mathrm{hole}}\left(x\right)$ for the full and the t-J Hamiltonians at different values of $\delta g/t$: $\delta g/t=-0.5$ ($U/t=53.32$), 0 ($U/t=U_o/t=13.33$), and $0.2$ ($U/t=9.26$). The system is composed of 170 lattice sites. The unit of energy is $t$.Reuse & Permissions

Figure 7

Comparison of the ground-state energies for the full and the t-J Hamiltonians at different values of $\delta g/t$: $\delta g/t=-0.5$ ($U/t=53.32$), 0 ($U/t=U_o/t=13.33$), and $0.2$ ($U/t=9.26$). The full Hamiltonian includes three-site hopping terms [Eq. (4)] that lower the energy relative to the t-J model. At zero hole doping, three-site terms are suppressed and both models converge. The unit of energy is $t$. Here, $N_{\mathrm{latt}}=170$. The statistical errors are smaller than the size of the symbols.Reuse & Permissions

Figure 8

Doping dependence of the total, kinetic, and exchange energies for the t-J model. The statistical errors are smaller than the size of the symbols. The unit of energy is $t$.Reuse & Permissions