Spectra of Lorentz-violating Dirac bound states in a cylindrical well

Abstract

In the presence of the Lorentz-violating ${b}^{\ensuremath{\mu}}$ coefficient, the spectra of bound states for a Dirac particle in a cylindric well are changed. Compared to the Lorentz invariant (LI) spectrum, the Lorentz violation deviation becomes significant when eigenenergy $E$ is sufficiently close to the critical values $\ifmmode\pm\else\textpm\fi{}m$, where $m$ is the particle's mass. The detailed profile of the deviation depends on the observer Lorentz nature of ${b}^{\ensuremath{\mu}}$. We discussed three types of ${b}^{\ensuremath{\mu}}$ configuration. When ${b}^{\ensuremath{\mu}}=(0,0,0,{b}_{Z})$ is parallel to the well axis, the would be degenerate LI spectra split into two subspectra, reminiscent of the Zeeman splitting in the presence of a weak magnetic field. Depending on the relative sign of ${b}_{Z}$ accompanying mass $m$ in the dispersion relation, the spectrum extends or shrinks in the allowed eigenenergy region. When ${b}^{\ensuremath{\mu}}$ is a radial [${b}^{\ensuremath{\mu}}=(0,b\mathrm{cos}\ensuremath{\phi},b\mathrm{sin}\ensuremath{\phi},0)$] or purely timelike vector [${b}^{\ensuremath{\mu}}=({b}^{T},\stackrel{\ensuremath{\rightarrow}}{0})$], the spin-up and down components are coupled together, and there is no splitting. However, the monotonic increasing behavior of well depth ${V}_{0}$ with the decrease of eigenenergy $E$ is slightly changed when $E$ is sufficiently close to $\ensuremath{-}m$.