Double Mean Field
H0=−∑qϵBz[2ReSq(f+qfq−12)+S+qfqf−q+Sqf+−qf+q]
=−∑qϵBz[2ReSq(f+qfq−12)+i2ImSqf+−qf+q−i2ImSqfqf−q]
Hh=2h(f+0χ0+χ+0f0)
Hv=2Jzχ+0χ0(2f+0f0−1)
HMFv=HHatreev+HFockv
HHatreev=4Jz<χ+0χ0>f0f0+4Jz(<f+0f0>−12)χ+0χ0
=4Jznχf0f0+4Jz(nf−12)χ+0χ0
HFockv=−4Jz<χ+0f0>f+0χ−4Jz<f+0χ0>χ+0f0
=−Δf+0χ−Δ∗χ+0f0
HMF=H0+Hh+HMFv
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=H0+4Jznχf+0f0+4Jz(nf−12)χ+0χ0+hefff+0χ+h∗effχ+0f0
where :
heff=2h−Δ
for
H0
we have
H0=−∑qϵ12BzHq
Hq=⎡⎣⎢⎢⎢⎢2ReSq+2Jz∗(2nχ−1)2iImSq−4Jz∗hop−2∗h4Jz∗pair−2a2iImSq−2ReSq+2Jz∗(2nχ−1)−4Jz∗pair+2a4Jz∗hop+2∗h−4Jz∗hop−2∗h−4Jz∗pair+2a2(2nf−1)∗Jz04Jz∗pair−2a4Jz∗hop+2∗h0−2(2nf−1)∗Jz⎤⎦⎥⎥⎥⎥
left vector:
|ψ>=[fq,f+−q,χq,χ+−q]