Valid Bogoliubov transformation?
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In condensed matter physics, one often encounter a Hamiltonian of the form
H= ∑k ( a
†
k
a−k )( Ak Bk Bk Ak )( ak a
†
−k
),
where ak is a bosonic operator. A Bogoliubov transformation
( ak a
†
−k
)=( coshθk −sinhθk −sinhθk coshθk )( γk γ
†
−k
),
with
tanh2θk=
Bk
Ak
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?
H= ∑k ( a
†
k
a−k )( Ak Bk Bk Ak )( ak a
†
−k
),
where ak is a bosonic operator. A Bogoliubov transformation
( ak a
†
−k
)=( coshθk −sinhθk −sinhθk coshθk )( γk γ
†
−k
),
with
tanh2θk=
Bk
Ak
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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