In condensed matter physics, one often encounter a Hamiltonian of the form H=∑k(a†ka−k)(AkBkBkAk)(aka†−k),
where ak is a bosonic operator. A Bogoliubov transformation (aka†−k)=(coshθk−sinhθk−sinhθkcoshθk)(γkγ†−k),
with tanh2θk=BkAk
is often used to diagonalized such a Hamiltonian. However, this seems to assume that |Ak|>|Bk|. Is this true? If so, how else can the Hamiltonian be diagonalized?
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