Ground-state phase diagram of the one-dimensional Hubbard model with an alternating chemical potential

We investigate the ground-state phase diagram of the one-dimensional half-filled Hubbard model with an alternating potential—a model for the charge-transfer organic materials and the ferroelectric perovskites. We numerically determine the global phase diagram of this model using the level-crossing and the phenomenological renormalization-group methods based on the exact diagonalization calculations. Our results support the mechanism of the double phase transitions between Mott and band insulators pointed out by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83, 2014 (1999)]: We confirm the existence of the spontaneously dimerized phase as an intermediate state. Further we provide numerical evidence to check the criticalities on the phase boundaries. Especially, we perform the finite-size-scaling analysis of the excitation gap to show the two-dimensional Ising transition in the charge part. On the other hand, we confirm that the dimerized phase survives in the strong-coupling limit, which is one of the resultants of competition between the ionicity and correlation effects.

Figure 1

The $\Delta$ dependence of $x_{\sigma ,i}$ at $u=0.6$ for the 16-site system $[u=U∕(U+4)]$. The spin-gap transition point ${\Delta }_{\sigma }(U,L)$ is estimated from the level crossing between the singlet (circles) and triplet (triangles) spin excitations. The squares plot $x_{\mathrm{av}}=(x_{\sigma ,1}+3x_{\sigma ,2})∕4$, and the inset shows the $L$-dependence of $x_{\mathrm{av}}$ at $\Delta =1.0$, where a least-squares-fitting line to the data of $L=12–16$ is given.

Figure 2

The $L$ and $\Delta$ dependences of the scaled gap $L\Delta E_{\rho ,1}$. From left to right, $u=0.12$, 0.60, and 0.72, respectively. The correspondence between marks and system sizes is given in the figure. Crossing points give the $L$-dependent transition points ${\Delta }_{\rho }(U,L+1)$.

Figure 3

The $\Delta$ dependence of the charge-excitation-gap ratio $R=\Delta E_{\rho ,1}(U,\Delta ,L)∕\Delta E_{\rho ,2}(U,\Delta ,L)$ for $L=10–16$ at $u=0.72$. The arrow shows the transition point ${\Delta }_{\rho }\left(U\right)$. The inset plots the $L$-dependence of $R$ at ${\Delta }_{\rho }\left(U\right)$ with a least-squares-fitting line.

Figure 4

The finite-size-scaling plots of the charge-excitation gap $\Delta E_{\rho ,1}$ for systems of $L=14–18$ at $u=0.72$ and 0.80. We use the 2D-Ising critical exponent $\nu =1$. A dotted line (the slope 1) is given as a guide to the eye.

Figure 5

The ground-state phase diagram of the 1D Hubbard model with the alternating potential. The open circles (squares) with a fitting curve show the spin-gap (2D-Ising) transition line in the spin (charge) part. The stable regions of the MI, SDI, and BI phases are given in the 2D parameter space $(u,\delta )$ [$u=U∕(U+4)$ and $\delta =\Delta ∕(\Delta +2)$]. Insets (a) and (b) show the extrapolations of the $L$-dependent transition points in the spin and the charge parts, respectively. For comparison, we also plot the DMRG calculation results given in Ref. 19 by using the filled squares ($U_{c1}$ in their notation) and the filled circles $\left(U_{c2}\right)$.

Figure 6

The deviations of boundaries from the $2\Delta =U$ line, ${\Delta }_{\nu }(U,L)-U∕2$. We use $u=U∕(U+4)$ as the $x$ axis. The correspondence between marks and system sizes is given in the figure. Marks with solid (dotted) curves exhibit the deviations in the spin (charge) part.

Figure 7

Behavior of the ground-state expectation value of the twist operator $z_{\nu }$ $(\nu =\rho ,\sigma )$ near the $2\Delta =U$ line. The correspondence between marks and system sizes are given in the figure.

Figure 8

Comparisons of the system-size dependences of the transition points obtained by the LC and PRG methods vs by the condition $z_{\nu }=0$. The fitting curves show the extrapolations of data to the thermodynamic limit.