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1

"Trivial" is used because the equation reduces to something a bit simpler. Take the equation $E_{\pm}=\pm \sqrt{\left(\frac{p^2}{2m}-\mu\right)^2+|\Delta|^2p^2}$ Setting $\Delta$ to $0$ gets rid of the last term, and making $\mu<0$ means that the first term will reduce to $$E_1=\pm \left(\frac{p^2}{2m}-\mu\right)$$ which is pretty simple. That's what ...

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Concepts of Berry curvature and Chern number are key to make a difference between what is topological and what it is not. Let's consider an hamiltonian $\mathcal{H}$ which is a function of N time-dependant parameters $(\Gamma_1(t)\,...\,\Gamma_j(t)...\,\Gamma_N(t))\equiv\mathbf{\Gamma}(t)$. The evolution of such system is given by the Schrödinger equation : ...

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This issue is a well known problem in high energy physics which is called " Neutrino Billiards". You can find a full description about it in: Ref: Berry, M.V. and R.J. Mondragon, Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1987. 412(1842): p. ...

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