Single-site entanglement at the superconductor-insulator transition in the Hirsch model

We investigate the transition to the insulating state in the one-dimensional Hubbard model with bond-charge interaction $x$ (Hirsch model), at half-filling and $T=0$. By means of the density-matrix renormalization group algorithm the charge gap closure is examined by both standard finite-size scaling analysis and looking at singularities in the derivatives of single-site entanglement. The results of the two techniques show that a quantum phase transition takes place at a finite Coulomb interaction $u_{\mathrm{c}}\left(x\right)$ for $x\gtrsim 0.5$. The region $0<u<u_{\mathrm{c}}$ turns out to have a superconducting nature, at least for not too large $x>x_{\mathrm{c}}$.

Figure 1

(Color online) The critical values for the opening of the charge gap in the $u,x$ plane, as obtained by the different methods: exact results (triangles), FSS analysis (dots), and singularities in ${\partial }_{\alpha }\mathcal{S}$, for $\alpha =u,x$ (crosses). M, I, and SC stand for metallic, insulating, and superconducting phases, respectively.

Figure 2

(Color online) Finite-size scaling of the charge gap (3) evaluated at $u=0$ for $x=0.1,0.2,0.3,0.4$, and $L=10–62$ as a function of $1∕L$.

Figure 3

(Color online) Finite-size scaling of the charge gap (3) evaluated at $x=0.7$, $u=2.4,2.45,2.5,2.6$, and $L=10–62$ as a function of $1∕L^2$.

Figure 4

(Color online) ${\partial }_u\mathcal{S}$ as a function of $u$, evaluated at $L=30$ for (left) $x=0.1,$ 0.3, 0.4, 0.5 and (right) $x=0.6,$ 0.7, 0.8, 0.9.

Figure 5

(Color online) Spin gap vs chain length at $x=0.6$ and various $u$ values.

Figure 6

(Color online) Pair correlations at half chain $(\mid i-j\mid =L∕2)$ vs chain length at $x=0.6$ and various $u$ values.