Quantum critical behavior of the one-dimensional ionic Hubbard model

We study the zero-temperature phase diagram of the half-filled one-dimensional ionic Hubbard model. This model is governed by the interplay of the on-site Coulomb repulsion and an alternating one-particle potential. Various many-body energy gaps, the charge-density-wave and bond-order parameters, the electric as well as the bond-order susceptibilities, and the density-density correlation function are calculated using the density-matrix renormalization group method. In order to obtain a comprehensive picture, we investigate systems with open as well as periodic boundary conditions and study the physical properties in different sectors of the phase diagram. A careful finite-size scaling analysis leads to results which give evidence in favor of a scenario with two quantum critical points and an intermediate spontaneously dimerized phase. Our results indicate that the phase transitions are continuous. Using a scaling ansatz we are able to read off critical exponents at the first critical point. In contrast to a bosonization approach, we do not find Ising critical exponents. We show that the low-energy physics of the strong-coupling phase can only partly be understood in terms of the strong-coupling behavior of the ordinary Hubbard model.

Figure 1

The exciton gap ${\Delta }_{\mathrm{E}}$ for finite system sizes $L$ and $\delta =20$. The spin gap ${\Delta }_{\mathrm{S}}$ for $L=300$ is also shown for comparison. (b) The exciton gap, the spin gap ${\Delta }_{\mathrm{S}}$, and the gap to the second excited state ${\Delta }_{\mathrm{SE}}$ for $L=128$.

Figure 2

Finite-size scaling analysis of the minimal value of the exciton gap ${\Delta }_{\mathrm{E}}$. The solid lines are linear fits through the four system sizes shown, $L=256$, 300, 350, 400.

Figure 3

Finite-size scaling analysis of the $U$ value at the minimum of the exciton gap ${\Delta }_{\mathrm{E}}$ for $\delta =20$. The solid line represents a linear least-squares extrapolation of the data yielding $U_{\mathrm{c}1}\approx 21.39$.

Figure 4

The spin gap ${\Delta }_{\mathrm{S}}$ for finite systems $L$ as a function of $U$ and (b) the finite-size scaling analysis for $\delta =20$ for chosen $U$ values. The system sizes $L=64$, 128, 200, 256, 300, 350, 400, 450, and 512 are shown in (b) and are used for a least-squares fit to a third-order polynomial in $1∕L$ (solid lines). The dashed line in (b) shows the value of ${\Delta }_{\mathrm{S}}$ at the largest system size in order to illustrate the nonmonotonic behavior.

Figure 5

The exciton gap ${\Delta }_{\mathrm{E}}$, the spin gap ${\Delta }_{\mathrm{S}}$, and the one-particle gap ${\Delta }_1$ for $\delta =20$ after extrapolating to the thermodynamic limit $L\to \infty$. The inset shows the result for a larger range of $U$.

Figure 6

The one-particle gap ${\Delta }_1$ for $\delta =20$. (a) Results for finite systems with $L=16–512$. (b) The finite-size scaling behavior for $L=64$, 128, 200, 256, 300, 350, 400, 450, 512. The solid lines in (b) show least-squares fits to a third-order polynomial in $1∕L$.

Figure 7

The ionicity $⟨n_A-n_B⟩$ for $\delta =1$, 4, 20. The solid lines indicate analytical results from Eq. (6) and the symbols numerical DMRG results for $L=32$ sites.

Figure 8

The ionicity $⟨n_A-n_B⟩$ for finite systems with $L=16,32,\dots ,512$ for $\delta =20$. The inset shows the $L\to \infty$ extrapolated value.

Figure 9

(a) The bond order parameter $⟨B⟩$ for $\delta =20$ for finite systems as a function of $U$ for various system sizes. (b) The scaling of the data as a function of the inverse system size $1∕L$. The solid lines are least-squares fits to the data as described in the text. The inset shows an expanded view of the scaling for $U$ values near the critical point.

Figure 10

The bond-order parameter ${⟨B⟩}_{\infty }$ in the thermodynamic limit for (a) $\delta =1$, (b) $\delta =4$, and (c) $\delta =20$ plotted as a function of $U$ near the transition points.

Figure 11

The BO parameter $⟨B⟩$ as a function of applied dimerization field $\rho$ for $\delta =20$ and $U=21.42$. The upper inset shows data for $U=19$ and the lower inset data for $U=50$.

Figure 12

The BO susceptibility ${\chi }_{\mathrm{BO}}$ as a function of $U$ for (a) $\delta =1$, (b) $\delta =4$, and (c) $\delta =20$ and different $L$. (d) Finite-size scaling of ${\chi }_{\mathrm{BO}}$ for a $U$ located in the dip region of (a)–(c).

Figure 13

The BO susceptibility ${\chi }_{\mathrm{BO}}$ as a function of $1∕L$ for the ordinary Hubbard model $(\delta =0)$ and $U=10$ and the ionic Hubbard model for $\delta =20$ and $U=50$. DMRG data are indicated by the corresponding symbols and the solid curves represent a least-squares fit to the indicated forms.

Figure 14

A scaling analysis of ${\chi }_{\mathrm{BO}}$ for $\delta =20$.

Figure 15

(a) The electric susceptibility ${\chi }_{\mathrm{el}}$ for $\delta =20$ plotted as a function of $U$. (b) A scaling analysis of the data of (a).

Figure 16

The negative of the density-density correlation function $-[⟨n_i{n}_{i+r}⟩-⟨n_i⟩⟨n_{i+r}⟩]$ for $U>U_{\mathrm{c}2}$. The indicated lines are least-squares fits over a range of $r$ in which the behavior is linear on the log-log scale. The solid line has exponent $-3.27$ and the dashed line exponent $-3.4$.

Figure 17

Crossing points of excited states for PBC’s for $\delta =1$ as a function of inverse system size. The inset shows the extrapolation of ${\Delta }_{\mathrm{S}}\left(U_{\mathrm{xx}}\right)$ as a function of inverse system size.