Why topologically non-trivial materials are robust againist any external perturbations or defects? Topologically non-trivial materials are insensitive to perturbations or defects. How can I prove it mathematically?
I thought of making the first-order perturbation term zero.
$$\left< \psi \right|H'\left| \psi \right>=0$$ Where $H'$ is the perturbation applied.
But I am unaware of the starting assumptions or conditions to be applied. Can anyone help me with any hint or answer to prove why topological materials are insensitive to perturbations?
 A: For the sake of this explanation, let's concentrate on systems that have a spectral gap (not the most general scenario but it shall do).
Let $P$ be the Fermi projection of some topological material $H$ such that its Fermi energy is placed inside of a spectral gap of $H$. We have the Riesz formula $$ P = -\frac{1}{2\pi\mathrm{i}}\oint(H-zI)^{-1}\mathrm{d}z $$ where the contour of the integral encloses the spectrum below the gap.
If we perturb $H\mapsto H'$ such that the two Hamiltonians share a common gap, we have the formula (using the same contour) \begin{align} P' &= -\frac{1}{2\pi\mathrm{i}}\oint(H'-zI)^{-1}\mathrm{d}z \\ &=P -\frac{1}{2\pi\mathrm{i}}\oint \left((H'-zI)^{-1}-(H-zI)^{-1}\right) \mathrm{d}z \\
&=P -\frac{1}{2\pi\mathrm{i}}\oint (H'-zI)^{-1}(H-H')(H-zI)^{-1}\mathrm{d}z
\end{align} and so, assuming that $H,H'$ are semibounded (from below) we get the estimate $$ \|P-P'\| \leq C\|H-H'\|\,. $$ where the constant $C$ depends on the size of both gaps and lower bound of the spectrum (of $H$ and of $H'$).
Since we can express the Chern number as a function of $P$ which is moreover continuous w.r.t. this operator norm, this shows the desired stability.
