A local criterion of topological phase transitions

A local criterion of topological phase transitions is established based on the Morse theory: a topological phase transition occurs when the count of Morse critical points of the order function changes. The locations in space where this change occurs are referred to as spatial critical points of the topological phase transition. In cases of continuous topological phase transitions, these spatial critical points are identified through the emergence of degenerate Morse critical points, where local maxima and minima of the order function split or merge. This resembles the formation and annihilation of a particle-antiparticle pair. The wide-ranging applicability of this criterion is demonstrated through three case studies that explore topological phase transitions in both configuration space and reciprocal space. Every topological phase transition is linked to a localized physical process that cannot be comprehended solely by studying changes in a global quantity, such as a topological invariant.