Topological numbers of quantum superpositions of topologically non-trivial bands

In this Article we address the definition and values of topological numbers of the manifolds of wavefunctions - bands obtained by quantum superposition of the wavefunctions that belong to topologically distinct manifolds. The problem, although simple in essence, can be formulated as a paradox: it may seem that quantum superposition implies non-integer topological numbers. We show that the results are different for superpositions that are created dynamically and for those obtained by stationary mixing of the manifolds. For dynamical superpositions we have found that the observable related to the topological number is non-integer indeed. For static superpositions the resulting bands bear integer topological numbers. We illustrate how the number quantization is restored upon avoided crossing of topologically distinct subbands. The crossings lead to the exchange of topological numbers between the bands. We consider complicated phase diagrams arising from this and show the absence of triple critical points and abundance of quadruple critical points that are rare in thermodynamic phase diagrams. We illustrate these features considering the bilayer Haldane model.