  The existence of a self-localization transition in the polaron problem has
been under an active debate ever since Landau suggested it 83 years ago. Here
we reveal the self-localization transition for the rotational analogue of the
polaron -- the angulon quasiparticle. We show that, unlike for the polarons,
self-localization of angulons occurs at finite impurity-bath coupling already
at the mean-field level. The transition is accompanied by the
spherical-symmetry breaking of the angulon ground state and a discontinuity in
the first derivative of the ground-state energy. Moreover, the type of the
symmetry breaking is dictated by the symmetry of the microscopic impurity-bath
interaction, which leads to a number of distinct self-localized states. The
predicted effects can potentially be addressed in experiments on cold molecules
trapped in superfluid helium droplets and ultracold quantum gases, as well as
on electronic excitations in solids and Bose-Einstein condensates.
