Symmetry Meets Topology
Electrons possess the quantum mechanical property known as spin, often viewed in simple terms as the electron rotating. Quantum mechanics tells us that the amplitude of electron spin is restricted to integer multiples of ½ units of the Planck constant, which means that the electron wave function changes sign (from positive to negative and vice versa) after a full rotation of 2π radians. This sign change determines the fermionic statistics of electrons, and similarly for all other leptons and quarks, without which no matter would be able to form. A natural question is whether the half spin (or spin-½ in usual notation) of the electron has to be a fundamental property. In other words, if the universe were populated by particles that had only integer spin, which do not change sign upon full rotation, would it be possible to combine them and make spin-½ particles? Interestingly, the answer to this question, and its generalization, is deeply connected to a seemingly unrelated subject in condensed matter physics—the classification of topological states of matter, the topic addressed by Chen et al. ( 1 ) on page 1604 of this issue.