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The solution of this problem is in principle very easy. Since $$z^{4} + 4\left(\mu + 1\right)z^{2} + 8\lambda\Delta z + 4\left(\mu^{2} + \Delta^{2} - V^{2}\right) = 0\text{,}$$ is a fourth order equation I can solved this equation by the hand or with Mathematica/Maple and get 4 solutions. The interesting case is that with two purely real and two complex ...

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You misunderstood the classification I believe. Let's take an example. In class D and 1D, the classification tells you there are two possible vacua (you understood this apparently). This is the famous $\mathbb{Z}_{2}$ ensemble in the classification. Next the classification tells you also that: at the boundary between the two gapped vacua, a Majorana mode ...

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As mentioned in the comments, $C$ and $M$ are matrices in different spaces. Explicitly showing the flavor space matrices, in the lepton sector perhaps, we have, \begin{align} \overline{\psi} _L M ^\dagger \left( \psi _L \right) ^c & = \overline{\psi} _L M ^\dagger C\bar{ \psi _L} ^T \\ & = \left( \begin{array}{ccc}\overline{ \psi _{e, L} } ...

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Non-conservation of charge in Majorana terms The Dirac mass term is $m\bar\psi \psi$ where one field-factor $\bar\psi$ is complex conjugated (aside from other transpositions included in the Dirac conjugation) and the other is not. So one may assign a fermion number $1$ to $\psi$ which means that $\bar\psi$ automatically carries $-1$ and in the product, the ...

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