Quantum Diffusion on Molecular Tubes: Universal Scaling of the 1D to 2D Transition

The transport properties of disordered systems are known to depend critically on dimensionality. We study the diffusion coefficient of a quantum particle confined to a lattice on the surface of a tube, where it scales between the 1D and 2D limits. It is found that the scaling relation is universal and independent of the temperature, disorder, and noise parameters, and the essential order parameter is the ratio between the localization length in 2D and the circumference of the tube. Phenomenological and quantitative expressions for transport properties as functions of disorder and noise are obtained and applied to real systems: In the natural chlorosomes found in light-harvesting bacteria the exciton transfer dynamics is predicted to be in the 2D limit, whereas a family of synthetic molecular aggregates is found to be in the homogeneous limit and is independent of dimensionality.

Figure 1

(a) IPR dependence on the disorder strength in 1D and 2D. The parameters fitted in Eqs. (7) and (8) (solid lines) are $a_1=6.2$ and $(a_2,b_2)=(67,6.7)$. The numerical data are shown in symbols. We use 4900 sites for 1D system (black circles), and 2D systems with $70\times 70$ (blue circles) and $90\times 90$ (blue asterisks) square lattices. (b) Comparison between the results of Eq. (3) and those of Eq. (9). The lower (black) circles and solid line refer to 1D systems and the upper (blue) circles and line represent 2D systems. In both cases, we set $\sigma /J=1$. (c) Radius dependence of $D$ with $\sigma /J=1$ and $\mathrm{\Gamma }/J=10^{-4}$. The solid line is the fitting according to Eq. (10), with the corresponding fitted parameters $R_c$ and $D^{2\mathrm{D}}$ indicated.

Figure 2

Relative diffusion coefficient $\tilde{D}=[D(R)-D^{1\mathrm{D}}]/(D^{2\mathrm{D}}-D^{1\mathrm{D}})$ as a function of rescaled radius $\tilde{R}=(R-1)/R_c$. The solid line is the fitting function $S(x)$ and the dashed line indicates $\tilde{R}=1$. Inset (left) shows data from Eq. (3) before rescaling: From top ($\sigma /J=2$, blue) to bottom ($\sigma /J=5$, red) with 0.5 increment and interpolating color gradient. Inset (right) shows data from the quantum bath calculations with varying temperatures: From top ($T/J=7$, red) to bottom ($T/J=0.7$, blue) with 0.7 decrement.

Figure 3

Schematic illustration of the origin of universal radius scaling of transport rate in tubes.