Anderson localization versus charge-density-wave formation in disordered electron systems

We study the interplay of disorder and interaction effects including bosonic degrees of freedom in the framework of a generic one-dimensional transport model: the Anderson-Edwards model. Using the density-matrix-renormalization group technique, we extract the localization length and the renormalization of the Tomonaga-Luttinger-liquid parameter from the charge-structure factor by a elaborate sample-average finite-size scaling procedure. The properties of the Anderson localized state can be described in terms of scaling relations of the metallic phase without disorder. We analyze how disorder competes with the charge-density-wave correlations triggered by the bosons and give evidence that disorder will destroy the long-range charge-ordered state.

Figure 1

DMRG metal-insulator phase boundary for the 1D half-filled Edwards model without disorder (solid line). CDW order is suppressed if the background fluctuations dominate [${\omega }_0<1$] or if the system's ability for relaxation is high [$\lambda >{\lambda }_c\left({\omega }_0\right)$]. The blue crosses denote the parameter sets considered in this paper for the disordered Edwards model.

Figure 2

Charge-structure factor of the AEM with $\lambda =0.1$ and ${\omega }_0=2$ (left panel) and the A$tV$M with $V/t_f=1.5$ (right panel) sampled over 300 and 500 disorder realizations, respectively. Dashed lines give the finite-size scaling of ${\tilde{C}}_{av}\left(q\right)$ according to (7).

Figure 3

Left panels: Log-log plot of the localization length vs disorder strength for the AEM at (a) fixed $\lambda =0.1$, (c) fixed ${\omega }_0=2.5$, and (e) for the A$tV$M. Dashed lines are fits to Eq. (8) with (a) ${\xi }_0=620$, 98, 50 and $\gamma =1.75$, 1.65, 1.22 for $\lambda =1$, 0.2, 0.1; (c) ${\xi }_0=500$, 65, 40 and $\gamma =1.81$, 1.35, 1.1 for ${\omega }_0=1$, 2, 3; and (e) ${\xi }_0=440$, 350, 230, 190, 150 and $\gamma =1.95$, 1.5, 1.3, 1.1, 0.95 for $V/t_f=0$, 0.5, 1. 1.5, 2 (from top to bottom), respectively. Right panels: Corresponding results for the modified TLL parameter $K_{\rho }^{*}$ in the (b), (d) AEM and (f) A$tV$M.

Figure 4

Left panels: Decay length ${\xi }_0$ as a function of the (effective) Coulomb repulsion $V_{\left(\mathrm{eff}\right)}$ for the A$tV$M and AEM. Right panel: Corresponding $\gamma$ exponent vs $K_{\rho }$, in comparison to the scaling relation (10) (dashed line). For further explanation, see text.

Figure 5

Local densities of fermions $\langle f_i^{†}{f}_i^{}\rangle$ (blue circles) and bosons $\langle b_i^{†}{b}_i^{}\rangle$ (red open squares) in the central part of an open AEM chain without ($\Delta =0$) and with ($\Delta =2$) disorder. Results are given for a single realization $\left\{{\varepsilon }_i\right\}$ (black triangles). Model parameters are $\lambda =0.01$, ${\omega }_0=2.5$, and $L=128$ (OBC).