R\'{e}nyi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields

Abstract

One-parameter functionals of the R\'{e}nyi $R_{\rho,\gamma}(\alpha)$ and Tsallis $T_{\rho,\gamma}(\alpha)$ types are calculated both in the position (subscript $\rho$) and momentum ($\gamma$) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field $\bf B$ and the Aharonov-Bohm (AB) flux $\phi_{AB}$ and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius $r$. Position (momentum) R\'{e}nyi entropy depends on the field $B$ as a negative (positive) logarithm of $\omega_{eff}\equiv\left(\omega_0^2+\omega_c^2/4\right)^{1/2}$, where $\omega_0$ determines the quadratic steepness of the confining potential and $\omega_c$ is a cyclotron frequency. This makes the sum ${R_\rho}_{nm}(\alpha)+{R_\gamma}_{nm}(\frac{\alpha}{2\alpha-1})$ a field-independent quantity that increases with the principal $n$ and azimuthal $m$ quantum numbers and does satisfy corresponding uncertainty relation. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient $\alpha$ where the momentum entropies exist reveals that it depends on the ring geometry, AB intensity and quantum number $m$. It is proved that there is the only orbital for which both R\'{e}nyi and Tsallis uncertainty relations turn into the identity at $\alpha=1/2$ and which is not necessarily the lowest-energy level. At any coefficient $\alpha$, the dependence of the position R\'{e}nyi entropy on the AB flux mimics the energy variation with $\phi_{AB}$ what, under appropriate scaling, can be used for the unique determination of the associated persistent current. Similarities and differences between the two entropies and their uncertainty relations are discussed too.