Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

One-parameter functionals of the Rényi [Formula: see text] and Tsallis [Formula: see text] types are calculated both in the position (subscript [Formula: see text]) and momentum ([Formula: see text]) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field [Formula: see text] and the Aharonov–Bohm (AB) flux [Formula: see text] and whose potential profile is modeled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. Position (momentum) Rényi entropy depends on the field B as a negative (positive) logarithm of [Formula: see text] , where [Formula: see text] determines the quadratic steepness of the confining potential and [Formula: see text] is a cyclotron frequency. This makes the sum [Formula: see text] a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and satisfies the corresponding uncertainty relation. In the limit [Formula: see text] , both entropies in either space tend to their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient [Formula: see text] 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 Rényi and Tsallis uncertainty relations turn into the identity at [Formula: see text] , which is not necessarily the lowest-energy level. At any coefficient [Formula: see text] , the dependence of the position of the Rényi entropy on the AB flux mimics the energy variation with [Formula: see text] , which, 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 as well.