  The challenge of understanding the collective behaviors of social systems can
benefit from methods and concepts from physics [1-6], not because humans are
similar to electrons, but because certain large-scale behaviors can be
understood without an understanding of the small-scale details [7], in much the
same way that sound waves can be understood without an understanding of atoms.
Democratic elections are one such behavior. Over the past few decades,
physicists have explored scaling patterns in voting and the dynamics of
political opinion formation, e.g. [8-13]. Here, we define the concepts of
negative representation, in which a shift in electorate opinions produces a
shift in the election outcome in the opposite direction, and electoral
instability, in which an arbitrarily small change in electorate opinions can
dramatically swing the election outcome, and prove that unstable elections
necessarily contain negatively represented opinions. Furthermore, in the
presence of low voter turnout, increasing polarization of the electorate can
drive elections through a transition from a stable to an unstable regime,
analogous to the phase transition by which some materials become ferromagnetic
below their critical temperatures. Empirical data suggest that United States
presidential elections underwent such a phase transition in the 1970s and have
since become increasingly unstable.
