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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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To simplify the problem, we may neglect the potential energy term $V(r)$, as it is simply irrelevant to our derivation. So we write the Hamiltonian as $$H=\frac{1}{2}(-i\partial_x-A)^2.$$ The ground state is given by minimization of the energy. As the Hamiltonian is a square of $(-i\partial_x-A)$, so it is minimized when $(-i\partial_x-A)=0$. Which means on ...

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First, I will set $e=1$ for simplicity. Let $\psi_0$ denote the wave function that satisfies the free Schrodinger equation: $$i \frac{\partial \psi_0}{\partial t} = -\frac{1}{2m}\mathbf{\nabla}^2 \psi_0 + V \psi_0 \tag{1}$$ Furthermore, let $\psi$ be the wave function that obeys the Schrodinger equation for a non-vanishing vector ...

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The topological manifold of the Lorentz group can be continuously embedded in the metric space $\mathbb{R}^{16}$ together with (metric) topology inherited from $\mathbb{R}$ (direct product topology). The subset of Lorentzian boosts in 1 spatial direction can be parametrized by $\beta =v/c$ and is hence homeomorphic as a topological space with the open unit ...

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