hamiltonian-all.tex

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\begin{document}
\begin{eqnarray*}
H & = & -\sum_{<i,j>\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}-J\sum_{i\in D}\vec{S}_{Di}\cdot\vec{S}_{i}+\sum_{\sigma,i\in D}\varepsilon_{\mathrm{d}i\sigma}d_{i\sigma}^{\dagger}d_{i\sigma}+\frac{\omega_{0}}{2}\sum_{\left\langle i,j\right\rangle }\left|\vec{q}_{i}-\vec{q}_{j}\right|^{2}\end{eqnarray*}
where $S_{Di}^{z}=\left(d_{i\uparrow}^{\dagger}d_{i\uparrow}-d_{i\downarrow}^{\dagger}d_{i\downarrow}\right)/2$,
$S_{i}^{+}=c_{i\uparrow}^{\dagger}c_{i\downarrow}$, $S_{Di}^{+}=d_{i\uparrow}^{\dagger}d_{i\downarrow}$,
and \[
t_{ij}=t_{0}-\alpha\left|\vec{q}_{i}-\vec{q}_{j}\right|\]


Then\begin{eqnarray*}
\left\langle H_{U}\right\rangle  & = & U\sum_{i}\left(\left\langle n_{i\uparrow}\right\rangle n_{i\downarrow}+n_{i\uparrow}\left\langle n_{i\downarrow}\right\rangle -\left\langle n_{i\uparrow}\right\rangle \left\langle n_{i\downarrow}\right\rangle \right)\\
\left\langle H_{J}\right\rangle  & = & -J\sum_{i\in D}\left(\left\langle S_{Di}^{z}\right\rangle S_{i}^{z}+S_{Di}^{z}\left\langle S_{i}^{z}\right\rangle -\left\langle S_{Di}^{z}\right\rangle \left\langle S_{i}^{z}\right\rangle \right)-\frac{J}{2}\sum_{i\in D,\sigma}V_{i}\left(c_{i\sigma}^{\dagger}d_{i\sigma}+h.c.\right)-\frac{J}{2}\sum_{i\in D}V_{i}^{2}\end{eqnarray*}
where $S_{i}^{z}=\left(c_{i\uparrow}^{\dagger}c_{i\uparrow}-c_{i\downarrow}^{\dagger}c_{i\downarrow}\right)/2$,
$S_{Di}^{z}=\left(d_{i\uparrow}^{\dagger}d_{i\uparrow}-d_{i\downarrow}^{\dagger}d_{i\downarrow}\right)/2$,
$S_{i}^{+}=c_{i\uparrow}^{\dagger}c_{i\downarrow}$, $S_{Di}^{+}=d_{i\uparrow}^{\dagger}d_{i\downarrow}$
and \[
V_{i}=-\left\langle c_{i\uparrow}^{\dagger}d_{i\uparrow}+d_{i\downarrow}^{\dagger}c_{i\downarrow}\right\rangle =-\left\langle c_{i\downarrow}^{\dagger}d_{i\downarrow}+d_{i\uparrow}^{\dagger}c_{i\uparrow}\right\rangle .\]
 
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