# What happens to topological insulators at finite temperature?

There is a similar question here, but I had a few things I wanted to ask.

So basically pretty much all analysis/ theory of topological insulators is for pure wave-functions and conservative Hamiltonians, ie. systems at the zero temperature limit. People keep saying that we can use topological phases for qubits/ protected electron or photon transport to make fancy new transistors or optical devices, but there isn't much literature on finite-temperature topological insulators which seems like it is needed before anything can be made useful.

When people say robust against disorder, does this include thermal fluctuations? So are topological phases normally robust at room temperature? (My impression is no).

What frameworks are there to study topological insulators at finite-temperatures? So far, I have surmised there is:

• Hill Thermodynamics (Using a framework of thermodynamics where boundary effects are considered, unlike in thermodynamic limit which is where things are of infinite size.) Though I'm confused by their argument (here) that they need to consider boundary effects because there is "a strong dependence of the spectrum on boundary conditions due to bulk boundary correspondence." Maybe I'm making a massive error, but I always interpreted the bulk-boundary correspondence as this magical fact that with a certain bulk invariant you necessarily coincide with edge states. So I interpret this to mean that you don't need to know what's happening at the boundary if you know what's happening in the bulk state (if you trust bulk-boundary correspondence). Therefore, why can't we just use normal thermodynamics on the bulk state? Why do we need to include the boundary states in our thermodynamic description?
• Uhlmann Phase Unlike Berry phase which is for pure states, they use Uhlmann phase which is a generalisation of Berry phase for density matrices
• Numerical Methods (here and here) I think this is when they just numerically bash out the band diagram and see if the gap closes for finite temperature.
• Maybe some conformal field theory thing?

Are there any major limitations to the above methods? There doesn't seem to be a huge amount of literature on the topic.

Might also be dumb but can entanglement entropy be used to find thermodynamic entropy? And can this then be used to calculate properties of the system at a finite temperature? (I'm guessing no or else someone would have done this. What's the main problem with this statement?)

How do experiments account for the fact that the theory is often for zero-temperature limit? Are most experiments on topological insulators mainly done at low-temperatures?