Symmetric space description of carbon nanotubes

Abstract

Using an innovative technique arising from the theory of symmetric spaces, we obtain anapproximate analytic solution of the Dorokhov–Mello–Pereyra–Kumar (DMPK) equationin the insulating regime of a metallic carbon nanotube with symplectic symmetry and anodd number of conducting channels. This symmetry class is characterized by the presenceof a perfectly conducting channel in the limit of infinite length of the nanotube. Thederivation of the DMPK equation for this system has recently been performedby Takane, who also obtained the average conductance both analytically andnumerically. Using the Jacobian corresponding to the transformation to radialcoordinates and the parametrization of the transfer matrix given by Takane, we identifythe ensemble of transfer matrices as the symmetric space of negative curvatureSO*(4m + 2)/[SU(2m + 1) × U(1)] belonging to the DIII-odd Cartan class. We rederive the leading-order correction to theconductance of the perfectly conducting channel and its variance Var(ln δg). Our results are in complete agreement with Takane’s. In addition, our approach based onthe mapping to a symmetric space enables us to obtain new universal quantities: auniversal group theoretical expression for the ratio , and as a by-product a novel expression for the localization length for the most general case of a symmetricspace with BCm root system, in which all three types of roots are present.