Elastic field causing noncommutativity

TL;DR

This work investigates how a uniform torsion background from a distribution of screw dislocations induces effective spatial noncommutativity and reshapes the spectrum of a free quantum particle. By embedding the torsion in the metric via the geometric theory of defects, the planar dynamics in fixed $p_z$ acquires a magnetic-like coupling with $B_{eff} = \frac{c}{e} p_z \Omega$, yielding Landau-like levels with controllable tilt and shift. The noncommutativity emerges as $([\hat{x},\hat{y}] = i \frac{\hbar}{p_z\Omega})$, linking the noncommutativity scale to the dislocation density and longitudinal momentum, and the energy spectrum reduces to Landau levels perturbed by $1/\Lambda$ and $1/\Lambda^{2}$ terms. The results provide a geometric, covariant route to noncommutative plane physics and clarify how torsion determines the approach to the Landau limit.

Abstract

We study how a uniform torsion background, modeling a continuous density of screw dislocations and induces effective spatial noncommutativity and reshapes the energy spectrum of a free quantum particle. Within the geometric theory of defects, the metric yields a first-order (magnetic-like) coupling in the transverse dynamics, equivalent to an effective magnetic field $B_{eff}$ proportional to $p_z Omega$, where $Omega$ encodes the torsion strength. In the strong-coupling (Landau) regime, the planar coordinates obey [x,y] != 0 and the spectrum organizes into Landau-like levels with a slight electric-field-driven tilt and a uniform shift. Thus, increasing $Omega$ drives the system continuously toward the familiar Landau problem in flat space, with torsion setting the noncommutativity scale and controlling the approach to the Landau limit.