Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders

Abstract

We demonstrate the effects of geometric perturbation on the Tomonaga-Luttinger liquid (TLL) states in a long, thin, hollow cylinder whose radius varies periodically. The variation in the surface curvature inherent to the system gives rise to a significant increase in the power-law exponent of the single-particle density of states. The increase in the TLL exponent is caused by a curvature-induced potential that attracts low-energy electrons to region that has large curvature. PACS numbers: 73.21.Hb, 71.10.Pm, 03.65.Ge Studying the quantum mechanics of a particle confined to curved surfaces has been a problem for more than fifty years. The difficulty arises from operator-ordering ambiguities [1], which permit multiple consistent quantizations for a curved system. The conventional method used to resolve the ambiguities is the confining-potential approach [2,3]. In this approach, the motion of a particle on a curved surface (or, more generally, a curved space) is regarded as being confined by a strong force acting normal to the surface. Because of the confinement, quantum excitation energies in the normal direction are raised far beyond those in the tangential direction. Hence, we can safely ignore the particle motion normal to the surface, which leads to an effective Hamiltonian for propagation along the curved surface with no ambiguity.It is well known that the effective Hamiltonian involves an effective scalar potential whose magnitude depends on the local surface curvature [2,3,4,5]. As a result, quantum particles confined to a thin, curved layer behave differently from those on a flat plane, even in the absence of any external field (except for the confining force). Such curvature effects have gained renewed attention in the last decade, mainly because of the technological progress that has enabled the fabrication of low-dimensional nanostructures with complex geometry [6,7,8,9,10,11,12,13,14]. From the theoretical perspective, many intriguing phenomena pertinent to electronic states [15,16,17,18,19,20,21,22], electron diffusion [23], and electron transport [24,25,26,27] have been suggested. In particular, the correlation between surface curvature and spin-orbit interaction [28,29] as well as with the external magnetic field [30,31,32] has been recently considered as a fascinating subject.Most of the previous works focused on noninteracting electron systems, though few have focused on interacting electrons [33] and their collective excitations. However, in a low-dimensional system, Coulombic interactions may drastically change the quantum nature of the system. Particularly noteworthy are one-dimensional systems, where the Fermi-liquid theory breaks down so that the system is in a Tomonaga-Luttinger liquid (TLL) state [34]. In a TLL state, many physical quantities exhibit a power-law dependence stemming from the absence of single-particle excitations near the Fermi energy; this situation naturally raises the question as to how geometric perturbation affects the TLL behaviors of quasi onedimensional curved sys...

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