Nanoscale phase separation and superconductivity in the one-dimensional Hirsch model

We investigate numerically at various fillings the ground state of the one-dimensional Hubbard model with correlated hopping $x$ (Hirsch model). It is found that, for a large range of filling values $n$ around half filling, and for repulsive Coulomb interaction $u\le u_c\left(x,n\right)$, phase separation at a nanoscale (NPS phase) between two conducting phases at different densities occurs when $x\gtrsim 2/3$. The NPS phase is accompanied by the opening of a spin gap and the system behaves as a Luther-Emery liquid with dominant superconducting correlations. Close to half filling, an anomalous peak emerges in the charge structure factor related to the density of doubly occupied sites, which determines the size of the droplets in the NPS phase. For $1/2\lesssim x\lesssim 2/3$ a crossover to a homogeneous phase, still superconducting, takes place.

Figure 1

$T=0$ phase diagram of the one-dimensional Hirsch model at half filling (Ref. 12). In each phase the gapped sectors (charge, ${\Delta }_c$ and/or spin ${\Delta }_s$) are reported; SDW stands for spin-density wave, while BOW stands for bond ordered wave.

Figure 2

(Color online) Discrete derivative $\left[E_0\left(n+\delta n\right)-E_0\left(n-\delta n\right)\right]/2\delta n$ of the DMRG ground-state energy $E_0$ at $u=1$: $x$ increases from top to bottom with reference to $n=1$. The continuous lines represent the exact results at $s_x=0$ (Ref. 8). The numerical phase diagram is given in the inset: red diamonds are determined from the end points of the plateaux and black dots refer to the analysis of $K_{\rho }$ specified in the text. In the SC-LEL phase the shaded region is characterized by NPS.

Figure 3

(Color online) Same as Fig. 2 but with fixed $x=0.8$ and varying $u$ (increasing from bottom to top). In the inset the continuous line is the analytical estimate of the phase diagram according to Sec. 2B, while the crosses $\left(“\text{n}+\text{a}”\right)$ mark the transition points obtained by intersecting the analytical curves out of the plateaux with the numerical values of the chemical potential within the plateau. The shaded region is characterized by NPS. The jump at $n=1$ and $u=4$ corresponds to the nonvanishing charge gap of the insulating phase.

Figure 4

(Color online) Charge-charge (black circles) and pair-pair (red squares) correlations between site $L/3$ and site $2L/3$ (open chains) versus filling $n$. Here $x=0.8$ and $u=1$. In the two insets we have reported the finite-size spin gaps computed as ${\Delta }_s=E_0\left(S_z=1\right)-E_0\left(S_z=0\right)$ ($S_z$ being the total $z$ component of the spin) for (a) $n=5/6>n_l$ $\left({\Delta }_s\ne 0\right)$ and (b) $n=9/6\gtrsim n_h$ $\left({\Delta }_s\to 0\right)$.

Figure 5

(Color online) Static charge structure factor at half filling for $x=0.8$ and various $u$. The slope of the dashed line is 1.6. For each curve, dotted lines are drawn in correspondence with $q=2\pi n_d$.

Figure 6

(Color online) Static charge structure factor at $x=0.8$ and $u=1.0$ for several fillings $n\left(L=120\right)$. The continuous lines represent the analytical results obtained within the 2SF model in the low- and high-density regimes at the same $n$ and $t_x\left(=0.6\right)$ values.

Figure 7

(Color online) Static charge structure factor $N\left(q\right)$ at half filling and $u=0.0$ for several $x$ values. Inset: $K_{\rho }$ derived from $N\left(q\right)$ as a function of $x$ for three different values of $u$. The lines are just guides for the eyes.

Figure 8

(Color online) Static charge structure factor at filling $n=5/12$ and $u=1.0$ plotted for several values of $x$. Inset: static charge structure factor at half filling in the insulating regime $\left(u=4.0\right)$ for several values of $x$.

Figure 9

(Color online) $K_{\rho }$ at $x=0.8$ and $u=1.0$ as a function of the filling $n\left(L=120\right)$. The line is just a guide for the eyes and the crosses mark $n_l$ and $n_h$ as determined in Sec. 2.

Figure 10

(Color online) $K_{\rho }$ vs $n$ at $x=0.6$ and $u=1$. Inset: pair-pair correlations between site $L/3$ and site $2L/3$ (open chains) in same case.