Exact diagonalization of a one-dimensional Hubbard model at density $\rho =0.4$: Effects of Coulomb repulsions and distant transfer

An extended Hubbard model that includes not only on-site but also intersite Coulomb repulsion and distant transfer is numerically investigated using the exact Lanczos diagonalization method for finite-size systems up to $L=20$ sites. The aim is to study the charge order and unconditional dimerization of a chain at density $\rho =0.4$. From the analysis of the spin and charge correlation functions, we deduce the formation of a dimer insulating state which is a Wigner lattice-type charge ordered state. The next-nearest-neighbor hopping $t_2$ enhances the intradimer correlations and weakens the interdimer correlations. Implications for the $\mathrm{Cu}{\mathrm{O}}_2$ chains in ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ are discussed.

Figure 1

Schematic representation of a model of interacting antiferromagnetic dimers along the chains of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$, with lattice parameter $c$, suggested by INS (Refs. 14, 15, 16).

Figure 2

Schematic representation of the single-chain EHM.

Figure 3

Charge structure factor $C\left(q\right)$ for $L=20$. (a) and (b) at various $V{h}_t∕t_1$ $(V{h}_r∕t_1)$ values, respectively. The $V{h}_t∕t_1$ and $V{h}_r∕t_1$ insets in (a) and (b), respectively, show the dependence of $C\left(2k_F\right)$ of the system size $L$. (c) The behavior of $C\left(q\right)$ for $r\ne 4$.

Figure 4

$C\left(q\right)$ for $L=20$ and for both interactions: the bare hole-hole repulsion $(V{h}_t∕t_1)$ and e-e repulsion $(V{h}_e∕t_1)$, respectively.

Figure 5

Static charge structure factor $C\left(q\right)$ for $L=20$. (a) $V{h}_t∕t_1=10$ and different NNN hopping $t_2$. The inset of (a) shows the behavior of $C\left(q\right)$ for $V{h}_t∕t_1=40$ at various $t_2$ values. (b) The behavior of $C\left(q\right)$ for $V{h}_t∕t_1=40$ at various $t_2$ values. The inset shows the behavior of $C\left(2k_F\right)$ in more detail. (c) $C\left(q\right)$ for $V{h}_r∕t_1=10$ for different $t_2$. The inset of (c) shows the behavior of $C\left(q\right)$ for $V{h}_r∕t_1=40$ for different $t_2$.

Figure 6

Spin structure factor $S\left(q\right)$ for $L=20$ and for the bare repulsion. (a) Various $V{h}_t∕t_1$ at $t_2∕t_1=0$. [(b), (c), and (d)] Different $t_2$ for $V{h}_t∕t_1=10,V{h}_t∕t_1=40$, and $V{h}_t∕t_1=100$, respectively.

Figure 7

Spin structure factor $S\left(q\right)$ for $L=20$ and for the truncated repulsion for $r=4$. (a) Various $V{h}_r∕t_1$ at $t_2∕t_1=0$. [(b), (c), and (d)] Different $t_2$ for $V{h}_r∕t_1=10,V{h}_r∕t_1=40$, and $V{h}_r∕t_1=100$, respectively.

Figure 8

Spin structure factor $S\left(q\right)$ for the long-range e-e repulsion $V{e}_t∕t_1$ for $L=20$ at $t_2∕t_1=0$. The inset shows in more detail the behavior of $S\left(q\right)$ for $L=20$ at $V{e}_t∕t_1=40$ and $V{e}_t∕t_1=100$, respectively.

Figure 9

Spin-spin correlation function $⟨S_0{S}_l⟩$ for $L=20$ at various $t_2∕t_1$ values. (a) The bare repulsion of magnitude $V{h}_t∕t_1=100$. (b) The truncated repulsion of magnitude $V{h}_r∕t_1=100$.

Figure 10

Spin structure factor $S\left(q\right)$ for $L=20$ and $V{h}_r∕t_1=100$ for $t_2∕t_1=2$. The inset represents $S\left(q\right)$ for $t_2∕t_1=1$. The full line in both plots represents a fit to an isolated dimers made up by two NNN spins.

Figure 11

Static spin structure factor $S\left(q\right)$ for $L=20$ and $V{h}_r∕t_1=40,t_2∕t_1=2$, and various $V_e∕t_1$. The inset shows the behavior of $S\left(q\right)$ for $t_2∕t_1=1$ and various $V_e∕t_1$. Clearly, $V_e$ has no effect on $S\left(q\right)$ for $t_2∕t_1=1$.

Figure 12

Charge gap ${\Delta }_c\left(L\right)$ as defined in Eq. (4). (a) and (b) show the dependence of ${\Delta }_c\left(L\right)$ of the system size $L$ for $t_2∕t_1=1$ and $t_2∕t_1=2$, respectively, for different $V{h}_r∕t_1$ values. [(c) and (d)] ${\Delta }_c\left(L\right)$ for $L=20$ and large $V{h}_r∕t_1$ values: (c) $t_2∕t_1=1$ and (d) $t_2∕t_1=2$. The full and dashed lines represent LF and QF, respectively, to large limit.

Figure 13

Spin gap ${\Delta }_s\left(L\right)$ as defined in Eq. (5). (a) The dependence of ${\Delta }_s\left(L\right)$ of the system size $L$ for $t_2∕t_1=1$ and $t_2∕t_1=2$, respectively, for different $V{h}_r∕t_1$ values. (b) ${\Delta }_s\left(L\right)$ for $L=20$ and large $V{h}_r∕t_1$ values: (c) $t_2∕t_1=1$ and (d) $t_2∕t_1=2$. The full line represents the fit to large limit. The dotted line in the inset is guide to the eye.