# Topological numbers of quantum superpositions of topologically non-trivial bands

## TL;DR

This paper investigates how topological numbers behave in quantum superpositions of bands with different topologies, revealing differences between dynamic and static superpositions and the effects of band crossings.

## Contribution

It clarifies the topological number behavior in superpositions, distinguishing between dynamic and static cases, and explores the impact of band crossings on topological quantization.

## Key findings

- Dynamic superpositions yield non-integer topological numbers.
- Static superpositions maintain integer topological numbers.
- Band crossings exchange topological numbers and affect phase diagrams.

## Abstract

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.