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So: I assume you want to diagonalize this problem by rewriting the Hamiltonian as $H=\sum E_id_i^\dagger d_i$, where $d_i$ are quasiparticle operators which obey the Fermionic commutation relations. If we only had $c^\dagger c$ terms, we would be able to write H as $$H=H_{ij}c_i^\dagger c_j$$ We could then prove that if $\{c_i\}$ obey the Fermion ...

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First you need to bring it into the following form: $H=\Psi^\dagger h \Psi$ Here $\Psi$ is a big column vector: $\Psi=(\dots, c_{m,n}, \dots, c_{m,n}^\dagger, \dots)^T$ Basically, the first half of $\Psi$ are all annihilation operators, and the second half are all creation ones. If the number of sites is $N$, the size of $\Psi$ is $2N$. So $h$ is a ...

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Superfluid helium finds an application as a coolant in superconducting systems (http://link.springer.com/chapter/10.1007%2F3-540-45542-6_4#page-1 )

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I have no time to check for exactly what you are referring to, but did you try to ask Google Scholar ? -> https://scholar.google.fr/scholar?q=interplay+between+superconductivity+magnetic+layers+thin+film+heterostructures&btnG=&hl=en&as_sdt=0%2C5 It gives general reviews already. Mainly in the bigger field of interplay between superconductivity ...

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You take the problem upside-down. Once you got a Hamiltonian, if it has a particle-hole symmetry $P$ which by definition verifies $\left\{ H,P\right\} =0$ with an anti-unitary operation $P$, you can construct it explicitly, and then you know how it applies on the operator basis and so on. For instance, a so-called s-wave superconductor can be described by ...

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