Title: Scaling laws of Rydberg excitons,
Abstract: Rydberg atoms have attracted considerable interest due to their huge
interaction among each other and with external fields. They demonstrate
characteristic scaling laws in dependence on the principal quantum number $n$
for features such as the magnetic field for level crossing. While bearing
striking similarities to Rydberg atoms, fundamentally new insights may be
obtained for Rydberg excitons, as the crystal environment gives easy optical
access to many states within an exciton multiplet. Here we study experimentally
and theoretically the scaling of several characteristic parameters of Rydberg
excitons with $n$. From absorption spectra in magnetic field we find for the
first crossing of levels with adjacent principal quantum numbers a $B_r \propto
n^{-4}$ dependence of the resonance field strength, $B_r$, due to the dominant
paramagnetic term unlike in the atomic case where the diamagnetic contribution
is decisive. By contrast, in electric field we find scaling laws just like for
Rydberg atoms. The resonance electric field strength scales as $E_r \propto
n^{-5}$. We observe anticrossings of the states belonging to multiplets with
different principal quantum numbers. The energy splittings at the avoided
crossings scale as $n^{-4}$ which we relate to the crystal specific deviation
of the exciton Hamiltonian from the hydrogen model. We observe the exciton
polarizability in the electric field to scale as $n^7$. In magnetic field the
crossover field strength from a hydrogen-like exciton to a magnetoexciton
dominated by electron and hole Landau level quantization scales as $n^{-3}$.
The ionization voltages demonstrate a $n^{-4}$ scaling as for atoms. The width
of the absorption lines remains constant before dissociation for high enough
$n$, while for small $n \lesssim 12$ an exponential increase with the field is
found. These results are in excellent agreement with theoretical calculations.