Symmetric space description of carbon nanotubes

TL;DR

This paper applies symmetric space theory to derive an analytic solution for conductance in insulating metallic carbon nanotubes with symplectic symmetry, confirming previous results and providing new universal quantities.

Contribution

It introduces a symmetric space approach to analyze conductance in carbon nanotubes, deriving new universal ratios and localization length expressions.

Findings

Agreement with Takane's conductance results

Universal ratio of conductance variance to mean

New expression for localization length in symmetric spaces

Abstract

Using an innovative technique arising from the theory of symmetric spaces, we obtain an approximate analytic solution of the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation in the insulating regime of a metallic carbon nanotube with symplectic symmetry and an odd number of conducting channels. This symmetry class is characterized by the presence of a perfectly conducting channel in the limit of infinite length of the nanotube. The derivation of the DMPK equation for this system has recently been performed by Takane, who also obtained the average conductance both analytically and numerically. Using the Jacobian corresponding to the transformation to radial coordinates and the parameterization of the transfer matrix given by Takane, we identify the ensemble of transfer matrices as the symmetric space of negative curvature SO^*(4m+2)/[SU(2m+1)xU(1)] belonging to the DIII-odd Cartan class. We rederive the leading-order correction to the conductance of the perfectly conducting channel <log(delta g)> and its variance Var(log(delta g)). Our results are in complete agreement with Takane's. In addition, our approach based on the mapping to a symmetric space enables us to obtain new universal quantities: a universal group theoretical expression for the ratio Var(log(delta g)/<log(delta g)> and as a byproduct, a novel expression for the localization length for the most general case of a symmetric space with BC_m root system, in which all three types of roots are present.