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

Berezinskii-Kosterlitz-Thouless Transition and the Haldane Conjecture: Highlights of the Physics Nobel Prize 2016