  We study the behavior of the R\'enyi entropies for the toric code subject to
a variety of different perturbations, by means of 2D density matrix
renormalization group and analytical methods. We find that R\'enyi entropies of
different index {\alpha} display derivatives with opposite sign, as opposed to
typical symmetry breaking states, and can be detected on a very small subsystem
regardless of the correlation length. This phenomenon is due to the presence in
the phase of a point with flat entanglement spectrum, zero correlation length,
and area law for the entanglement entropy. We argue that this kind of splitting
is common to all the phases with a certain group theoretic structure, including
quantum double models, cluster states, and other quantum spin liquids. The fact
that the size of the subsystem does not need to scale with the correlation
length makes it possible for this effect to be accessed experimentally.
