= Bogoliubov--de Gennes Hamiltonian
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{title2=BdG Hamiltonian}
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A <Bogoliubov--de Gennes Hamiltonian> is the <matrix> representation of a <quadratic fermion Hamiltonian> in a <Nambu spinor> combining <annihilation operators> and <creation operators>. It treats ordinary one-particle motion and fermionic pairing in one <eigenvalue problem>.
For a column $c$ of $N$ <fermionic annihilation operators>, let $h=h^\dagger$ and $\Delta=-\Delta^T$. With $\Psi=(c,c^\dagger{}^T)^T$, define
$$
\mathcal H_{\rm BdG}=
\begin{pmatrix}
h&\Delta\\
\Delta^\dagger&-h^T
\end{pmatrix}.
$$
The <canonical anticommutation relations> give the identity
$$
H=\frac12\Psi^\dagger\mathcal H_{\rm BdG}\Psi
+E_0+\frac12\operatorname{Tr}h
=c^\dagger hc+\frac12\left(c^\dagger\Delta c^\dagger{}^T+c^T\Delta^\dagger c\right)+E_0.
$$
The factor $1/2$ compensates for the full particle-hole doubling; the <trace> constant follows when the hole block is <normal-ordered>.
A two-by-two opposite-<spin> pairing block
$$
\begin{pmatrix}\xi&\Delta\\\Delta^*&-\xi\end{pmatrix}
$$
has <eigenvalues> $E_\pm=\pm\sqrt{\xi^2+|\Delta|^2}$, as follows from its <characteristic polynomial>. Pairing opens a <spectral gap> at $\xi=0$. Negative and positive partners in a fully doubled representation are related by <Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian>; they are not independent species of physical <quasiparticles>. A reduced block and a full doubled basis therefore require consistent counting conventions.
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