What does it mean in terms of band structure when we say that any topological invariant of some system is non-zero? For example what does it mean when we say that Chern number=1 in case of IQHE?
Does it mean that there are topological invariant times bands that are crossing the fermi level?


Dynamics of the physical systems are governed by differential equations. As such, usually it means that evolution is a continuous and unique function of initial or boundary parameters, and time. Trajectories do not cross, different solutions do not coincide etc. Topology is a part of mathematics which deals with continuous changes of objects. For example you may have a piece of paper and fold origami. As long as you do not use scissors and don't tear off paper, you are doing continuous transformations. Maybe even better you should think about rubber origami stretched and compressed, but not glued or teared. In such systems, if you start with some peculiar topological configuration, for example a ring ( think: closed, periodic trajectory) it may change very much, becoming even complicated and chaotic, but it is still a ring, folded in very complicated way. In fact one of the simplest topological conserved number is winding number counting how many times ring is winding itself ( think about the system where in the centre of the rings we are talking about is something which cannot be passed, like a point of zero energy in the system which cannot relax its energy to environment for example. Remember: we are talking about phase space, not configuration space! ). In such systems, winding number, let's say equal to 1, means trajectory revolves one time around such centre in phase space. Winding number equal to 2, revolves 2 times etc. It is possible, that different winding numbers means different energy! For example you can think about a steel belt. Winding means it form kind of helix. If winded enough time it may have some energy. Definitely different winding numbers will have different amounts of mechanical energy stored.

But as you agree that you have to wind it particular number of times ( probably because you have to tie its ends in some particular way, preventing unwinding) this system will have discrete energy spectrum! And energy bands separated one from another.

So topological invariants can lead to separation of trajectories in phase space, in such way that some additional order is created. It even is called topological phase transition, because if dynamics is changing, this order may break giving drastic change of system behaviour.

The band structure you are mentioned is one of the signals of such additional order. It may show as energy, momentum, space etc ordering/ bands.

Very similar phenomena can occur in crystal networks, semiconductors, quantum systems, nonlinear dynamics, polymer dynamics, low temperature phases where energy per particle is so small and systems are isolated so it cannot break quantum engagement etc

The most important thing is that topological conserved quantities usually are independent of various physical characteristics, depending usually only on very basic structure of the system: it's topological configuration. And of course, usually this depends on continuous relations defined in the system. If any of the parameters became non continuous for any reasons, topological conservation breaks.

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