Topological quantum number

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In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution, or, more precisely, it is the degree of a continuous mapping.^[1]

The concept of topological quantum numbers being created or destroyed during phase transitions emerged in condensed matter physics in the 1970s.The Kosterlitz-Thouless Transition demonstrated how topological defects, like vortices, could be created and annihilated during phase transitions in two-dimensional systems.^[2] Concurrently, in quantum field theory the 't Hooft-Polyakov monopole model demonstrated how topological structures, such as magnetic monopoles, could appear or disappear depending on the phase of a field, linking phase transitions to shifts in topological quantum numbers.^[3] In the 1980s, Haldane's theoretical model demonstrated that materials can possess topological quantum numbers like the Chern number, which can lead to different phases of matter. This concept was further explored with the development of topological phases, including the fractional quantum Hall effects and topological insulators.^[4][5]