Disorder effects in fluctuating one-dimensional interacting systems

The zero-temperature localization of interacting electrons coupled to a two-dimensional quenched random potential, and constrained to move on a fluctuating one-dimensional string embedded in the disordered plane, is studied using a perturbative renormalization group approach. In the reference frame of the electrons the impurities are dynamical and their localizing effect is expected to decrease. We consider several models for the string dynamics and find that while the extent of the delocalized regime indeed grows with the degree of string fluctuations, the critical interaction strength, which determines the localization-delocalization transition for infinitesimal disorder, does not change unless the fluctuations are softer than those of a simple elastic string.

Figure 1

The function $W(x,0)$ for the different models: a rigid string in a parabolic well (solid line), an elastic string (long-dashed line), and a floppy string (dashed line). Both axes are scaled by $l_0=v_c∕{\omega }_0$, $(v_c∕v_u)a$, and $4\pi (v_c∕\gamma )\lambda$, in the three cases, respectively.

Figure 2

The separatrix between the localized and delocalized phases in the $K_c\text{−}{\mathcal{D}}_b$ plane for different values of the string fluctuation frequency: $\varpi =1$ (solid line), $\varpi =0.1$ (long-dashed line), and $\varpi =0.01$ (dashed line). Here $v_c=v_F=1$, $y=0.1$, $K_s=\sqrt{(1+y∕2)∕(1-y∕2)}$, $v_s=\sqrt{1-(y∕2{)}^2}$, and $\mathcal{L}=10$. For infinitesimal disorder the critical point separating the localized and delocalized regimes is identical to the static one. However, for finite disorder the size of the delocalized region is increased by the string fluctuations.