Solitons | Exactly solvable models

Topological quantum number

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. Recent ideas about the nature of phase transitions indicates that topological quantum numbers, and their associated solutions, can be created or destroyed during a phase transition. (Wikipedia).

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3-sphere | Differential equation | Baryon number | Homotopy group | Wess–Zumino–Witten model | Topology | Isospin | Topological entropy in physics | Soliton | Topological quantum field theory | Thirring model | Quantum topology | Renormalization | Ishimori equation | Phase transition | Quantum invariant | Winding number | Fundamental group | Compactification (mathematics) | Topological defect | Degree of a continuous mapping | Inverse scattering transform | Sine-Gordon equation