Analysis of the $t_2-V$ model

We study a model (i.e., the $t_2\text{−}V$ model involving next-nearest-neighbor hopping and nearest-neighbor repulsion) in one dimension that generically depicts the dominant transport mechanism in cooperative strong electron-phonon interaction systems. Using analytic and numerical approaches, hard-core bosons are shown to typically undergo a striking discontinuous transition from a superfluid to a supersolid. Topological inequivalence of rings with even and odd number of sites is manifested through observable differences (in structure factor peaks) at the transition. Connections are also identified between the $t_2\text{−}V$ model and other topologically interesting models.

Figure 1

Plots of rescaled structure factor $S^{*}\left(\pi \right)$ and superfluid fraction $n_s$ at various filling factors $f$ obtained using modified Lanczos technique. The calculations were at $f=1/2,1/4$ with system size $N_s=16$ and at $f=1/3$ with $N_s=12$. At a critical repulsion there is a striking discontinuous transition in both $S^{*}\left(\pi \right)$ and $n_s$; while $S^{*}\left(\pi \right)$ jumps from its minimum to maximum, there is a significant drop in $n_s$.

Figure 2

Plot of $V_c\left(\infty \right)$ (critical repulsion for an infinite system) obtained from Green's-function analysis for half-filled $\left(f=1/2\right)$ and two HCB systems $\left(\mathrm{f}\to 0\right)$ and from finite size scaling at various other fillings $f$.

Figure 3

Plots of rescaled structure factor $S^{*}\left(Q\right)\equiv S\left(Q\right)/{\left[S\left(Q\right)\right]}_{\max }$ (with $Q=\pi \pm \frac{\pi }{2N+1})$ and superfluid fraction $n_s$ at various fillings $f$ obtained using modified Lanczos method. At a critical repulsion there is a sharp rise in $S^{*}\left(Q\right)$ with a concomitant significant drop in $n_s$.

Figure 4

Plots of energy $E$ vs repulsion $V$ for (a) the lowest energy states with $\left(m,6-m\right)$ particles in e ring with $N_s=16$ and $N_p=6$ (non-half-filled case); and (b) the lowest energy states with $\left(m,8-m\right)$ particles in e ring with $N_s=16$ and $N_p=8$ (half-filled case).

Figure 5

Plots of energy $E$ vs repulsion $V$ for (a) the lowest energy states with $\left(m,4-m\right)$ particles in e ring with $N_s=12$ and $N_p=4$ (non-half-filled case), and (b) the lowest energy states with $\left(m,6-m\right)$ particles in e ring with $N_s=12$ and $N_p=6$ (half-filled case).

Figure 6

Plots of energy $E$ vs repulsion $V$ for (a) the three lowest energy states in o ring with $N_s=17$ and $N_p=6$; and (b) the five lowest energy states in o ring with $N_s=17$ and $N_p=8$.

Figure 7

Plots of energy $E$ vs repulsion $V$ for (a) the three lowest energy states in o ring with $N_s=13$ and $N_p=4$, and (b) the four lowest energy states in o ring with $N_s=13$ and $N_p=6$.

Figure 8

Plots of energy $E$ vs repulsion $V$ for ground state at various fillings $f=N_p/N_s$.

Figure 9

Plots of the lowest energy $E$ (obtained using modified Lanczos in a system with $N_s=16$ sites) for the extremely anisotropic NNN Heisenberg model at various normalized magnetizations $(N_s-2N_p)/N_s$ corresponding to fillings $f=N_p/N_s$ for the $t_2\text{−}V$ model.