  The 2016 Physics Nobel Prize honors a variety of discoveries related to
topological phases and phase transitions. Here we sketch two exciting facets:
the groundbreaking works by John Kosterlitz and David Thouless on phase
transitions of infinite order, and by Duncan Haldane on the energy gaps in
quantum spin chains. These insights came as surprises in the 1970s and 1980s,
respectively, and they have both initiated new fields of research in
theoretical and experimental physics.
