The tight-binding hamiltonian for a two-dimensional vanadium dioxide

[gerryliyana]

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In his book, Quantum Field Theory for the Gifted Amateur, author  Stephen Blundell explains that in the tight-binding model, one considers  a lattice of fixed atoms with electrons moving between them (as shown in Fig. 4.6  above). These electrons can lower their kinetic energies by hopping from  lattice site to lattice site. To deal with the discrete lattice in the  model, we need to work in a basis where the fermion creation operator  $\hat{c}_i^\dagger$ creates a particle at a particular lattice site  labelled by $i$. The kinetic energy saving for a particle to hop between  points $j$ and $i$ is called $t_{ij}$ . Clearly $t_{ij}$ will have some  fundamental dependence on the overlap of atomic contains a sum over all  processes wave functions. The Hamiltonian $H$ in which an electron hops  between sites, and so is a sum over pairs of sites: 
  $\hat{H}_{hopping}=\sum_{ij}(-t_{ij})\hat{c}_i^\dagger\hat{c}_j^\dagger$
  Ok. After reading that book, I am trying to work on the kinetic term  of the Hamiltonian for a $VO_2$ system using tight-binding  approximation, but I'm not quite sure whether my work is right or wrong. 



  Would you all be so kind as to check my work? 


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In my case, $VO_2$ is modelled by taking the (110) plane from its unit  cell as shown in second figure above.  The model unit cell has two basis  oxygen atoms ($O_A$ and $O_B$), vertically located $a \approx 2.87$  angstrom apart, each of which contributing two p orbitals ($p_x$ and  $p_y$) and two vanadium atoms ($V_A$ and $V_B$), vertically located $a  \approx 2.87$ angstrom apart, each of which contributing two d orbitals  ($d_{x^2-y^2}$ and $d_{xy}$). Here, we choose 8 basis orbitals to  construct our hilbert space, which we order as follows: $\left| O_A-p_x  \right\rangle$, $\left| O_A-p_y \right\rangle$, $\left| O_B-p_x  \right\rangle$, $\left| O_B-p_y \right\rangle$, $\left| V_A-d_{x^2-y^2}  \right\rangle$, $\left| V_A-d_{xy} \right\rangle$, $\left|  V_B-d_{x^2-y^2} \right\rangle$, and $\left| V_B-d_{xy} \right\rangle$. 



Using this set of bases and tight-binding approximation, the kinetic part of the hamiltonian can be written in $k$ space as 
  $\hspace{1cm}$
  $H=H_{on-site}+H_{hopping}$
  $\hspace{1cm}$
  where
  $\hspace{1cm}$


$H_{on-site}=\sum_{\bar{k}}\left( \epsilon_{d_{A}}  d_{A_{1\bar{k}}}^{\dagger} d_{A_{1\bar{k}}} + \epsilon_{d_{A}}  d_{A_{2\bar{k}}}^{\dagger} d_{A_{2\bar{k}}} + \epsilon_{d_{B}}  d_{B_{1\bar{k}}}^{\dagger} d_{B_{1\bar{k}}}+\epsilon_{d_{B}}  d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}}  p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}}  p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}}  p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}}  p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}\right)$ $\hspace{1cm}$


and 



$\hspace{1cm}$


$H_{hopping}=-t_{V_A-V_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot a  \hat{y}}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot  (-a  \hat{y})}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\right)-t_{V_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}   +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{V_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}   +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{O_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot  a  \hat{y}}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot  (-a  \hat{y})}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\right)-t_{O_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot  c  \hat{x}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot  (-c  \hat{x})}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\right)-t_{O_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot  c  \hat{x}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot  (-c \hat{x})}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\right)$


$\hspace{1cm}$


  Thus,
  

$\hspace{1cm}$


$H =\sum_{\bar{k}}\left(\epsilon_{d_{A}}d_{A_{1\bar{k}}}^{\dagger}  d_{A_{1\bar{k}}}+\epsilon_{d_{A}}d_{A_{2\bar{k}}}^{\dagger}d_{A_{2\bar{k}}}+\epsilon_{d_{B}}d_{B_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\epsilon_{d_{B}}  d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}}  p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}}  p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}}  p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}}  p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}-2t_{V_{A}V_{B}}  d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\cos{\left( k_{y}a  \right)}-2t_{V_AO_B}\left(d_{A_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_AO_A}\left(d_{A_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_A}\left(d_{B_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_B}\left(d_{B_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{O_{A}O_{B}}  p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\cos{\left( k_{y}a  \right)}-2t_{O_{A}O_{A}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\cos{\left(  k_{x}c  \right)}-2t_{O_{B}O_{B}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\cos{\left(  k_{x}c \right)}\right)$


$\hspace{1cm}$


  Note: $h.c$ = hermitian conjugate
  

$\hspace{1cm}$


In equations above, $d_{A_{1\bar{k}}}^{\dagger}  (d_{A_{1\bar{k}}})$,$d_{A_{2\bar{k}}}^{\dagger} (d_{A_{2\bar{k}}})$,  $d_{B_{1\bar{k}}}^{\dagger} (d_{B_{1\bar{k}}})$,  $d_{B_{2\bar{k}}}^{\dagger} (d_{B_{2\bar{k}}})$,  $p_{A_{x\bar{k}}}^{\dagger} (p_{A_{x\bar{k}}})$,  $p_{A_{y\bar{k}}}^{\dagger} (p_{A_{y\bar{k}}})$,  $p_{B_{x\bar{k}}}^{\dagger} (p_{B_{x\bar{k}}})$, and  $p_{B_{y\bar{k}}}^{\dagger} (p_{B_{y\bar{k}}})$ create (annihilate) an  electron at $V_A$($d_{x^2-y^2}$), $V_A$($d_{xy}$), $V_B$($d_{x^2-y^2}$),  $V_B$($d_{xy}$), $O_A$($p_x$), $O_A$($p_y$), $O_B$($p_x$), and  $O_B$($p_y$) orbitals, respectively, with momentum $\bar{k}$.  $\epsilon_{d_A}$, $\epsilon_{d_B}$, $\epsilon_{p_A}$, and  $\epsilon_{p_B}$  indicate on-site energy of $V_A$, $V_B$, $O_A$, and  $O_B$, respectively. Furthermore, $t_{V_AV_B}$, $t_{V_AO_B}$,  $t_{V_AO_A}$, $t_{V_BO_A}$, $t_{V_BO_B}$, $t_{O_AO_B}$, $t_{O_AO_A}$,  and $t_{O_BO_B}$ being hopping parameter of $V_A$-$V_B$, $V_A$-$O_B$,  $V_A$-$O_A$, $V_B$-$O_A$, $V_B$-$O_B$, $O_A$-$O_B$, $O_A$-$O_A$, and  $O_B$-$O_B$ respectively. Then, a and c being the lattice constant.