Kinematics of Persistent Currents in Nanotubes and Tori Time-reversal symmetric gauge field and its application

The basic properties of conducting electrons in carbon nanotubes are reviewed from a theoretical perspective , and studies performed on persistent currents in toroidal carbon nanotubes and on the local energy gap in deformed nanotubes are reported on. 1 Introduction The Aharonov-Bohm (AB) effect shows the importance of the electromagnetic gauge field in quantum mechanics and the fact that a single electron has a fundamental unit of magnetic flux Φ 0 = 2π/e (units for which ¯ h = c = 1 are used here). One direct consequence of the AB effect in solid state physics is persistent currents [1, 2, 3]. Persistent currents are driven by an external gauge field produced when a magnetic flux penetrates a hollow area of a system. The external gauge field breaks the time-reversal symmetry and can induce equilibrium persistent currents in the ground state. In a previous study [4], a twist-induced gauge field was proposed that also affects persistent currents but still preserves the time-reversal symmetry. This gauge field is useful in analyzing persistent currents in toroidal carbon nanotubes as well as in other systems [5]. On the other hand, the effect of lattice deformation in carbon nanotubes on conducting electrons was studied, and it was found that electrons near the Fermi level are affected by a gauge field produced by the deformation [7] (called the " deformation-induced gauge field "). Because a deformation depends on position, the deformation-induced gauge field is also position dependent. The purpose of this paper is to show that the twist-induced gauge field corresponds to a constant part of the deformation-induced gauge field, meaning that a local extension of the twist-induced gauge field appears naturally in the low energy dynamics of conducting electrons in a deformed carbon nanotube. The gauge field presented here provides a unified way of examining the effect of local lattice deformation in carbon nanotubes on the local energy gap. At the conclusion of this paper, some problems associated with the above-mentioned gauge field are discussed.