Hidden $XY$ structure of the bond-charge Hubbard model

The repulsive one-dimensional Hubbard model with bond-charge interaction in the superconducting regime is mapped onto the spin-1/2 $XY$ model with transverse field, after assuming short-ranged antiferromagnetic correlations between electrons. We calculate density correlations and phase boundaries, realizing an excellent agreement with numerical results. The critical line for the superconducting transition is shown to coincide with the analytical factorization line identifying the commensurate-incommensurate transition in the $XY$ model.

Figure 1

(Color online) Comparison between the phase diagram of the HBC chain, calculated numerically in Ref. 6 (symbols with dashed lines) and the phase diagram obtained from the mapping onto the $XY$ model in transverse field (continuous lines). The upper curves correspond to the spin gap transition where spin excitations become gapless while the lower curves mark the transition into the ICSS phase where the charge compressibility diverges.

Figure 2

Analysis of the various contributions [see Eq. (7)] to the static structure factor (Fourier transform) of the density correlation function for the HBC model $\left(L=30\right)$ with the parameters reported in the legend. The vertical line corresponds to $\psi /\pi$ (see Table ).

Figure 3

(Color online) Comparison of the connected real-space correlation functions $N_s\left(R\right)$ (at half filling) and $\rho \left(R\right)$ for the HBC and $XY$ model, respectively (see text for definitions based on singly occupied sites operators). The parameters of the two models are related by mapping as $\gamma =\left(1-X\right)/X$ and $h=U/4X$. All the DMRG calculations for the HBC model and the analytical expressions of the curves for the $XY$ model refer to $L=50$. From top to bottom the data have been offset by $+0.15$, $+0.10$, and $+0.05$ for the sake of clarity.