Question

How can the Quantum Hall Effect consist of Topological Phase Transitions when there is no changing Topological Invariant?

Answer 1

This article about Topology in Physics and had some problems understanding the Topological Phase Transition of the Quantum Hall Effect. The way the Topological Phase Transition, it is a change in the topological invariant. The article mentions the following things:-

1. Every Vortex has a winding number.
2. Winding Numbers are Topological Invariants.
3. The total sum of Winding Numbers in a system can not change Chen-Number.
4. A new Vortex always appears with an Anti-Vortex.

Answer 2

Hello mate here is your answer.

Every Vortex has a winding number.

Winding Numbers are Topological Invariants.

The total sum of Winding Numbers in a system can not change (Chen-Number).

A new Vortex always appears with an Anti-Vortex.

Now the explanation for the Quantum Hall effect shall be, that the underlying shape of the inspected matter was (abstract) like a doughnut. When the magnetic field ramps up, Vortices are appearing. So there is to be a Topological Phase transition every time the electrical resistance jumps (right?). This Topological Phase Transition occurs because of the new Vortices.

Hope it helps you.