Dinâmica de Férmions em superfícies de revolução sob torção

Abstract

Twisted surfaces of revolution constitute fundamental theoretical models for the study of nanos tructures, heterostructures, and curved quantum devices. In this work, we investigate the quantum behavior of fermions confined to surfaces of revolution subjected to torsion. The problem is addressed in the continuum limit from two distinct perspectives: via the da Costa formalism (Schrödinger equation in curved space), focusing on cylindrical geometries; and in the relativistic regime, via the Dirac equation on curved surfaces, applicable to materials such as graphene. Initially, we calculate the strain tensor to obtain the induced metric and subsequently derive the geometric quantum potential through the da Costa formalism. We demonstrate that, for both linear and non-linear torsions, the geometry induces a geometric phase in the wavefunction, while the da Costa potential remains invariant. Consequently, the system admits bound states if the cylinder is finite, whose energy spectrum is independent of the torsion parameter. The analysis extends to scattering problems in infinite cylinders, revealing that the transmission probability is insensitive to torsion, yet strongly modulated by the particle’s angular momentum and the cylinder radius. In the relativistic approach, we develop the Dirac equation for the twisted cylinder and for axially symmetric variable-radius objects, obtaining a general equation for the dynamics of the pseudospinor components in (2+1)D. For the constant-radius cylinder case, we confirm that the torsion effect manifests again through a geometric phase. This phase also appears in surfaces with variable radius R(z); however, in these cases, the influence of torsion may manifest in other ways depending on the specific geometry. These results suggest relevant implications for the engineering of quantum devices based on materials with controlled curvature and torsion.