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<p>operator format [OPERATOR.KOP]</p>
<pre class="fragment">The tight-binding operator consists in an initial header where one
</pre><p> defines the geometry where the tight-binding lattice lives, followed by the list of hopping parameters among different orbitals and unit cells. The tight-binding operator assumes that the lattice can be constructed by the means of finite units cells. Therefore, the parameters are defined so to exploit this assumption. The following table is an example of the format that a tight-binding operator should have: </p><a class="anchor" id="multi_row"></a>
<table class="doxtable">
<caption>Tight-binding Operator Format Format</caption>
<tr>
<td colspan="3">Number Of Orbitals </td><td colspan="4">Number of Hoppings </td></tr>
<tr>
<td colspan="7">Lattice Constant </td></tr>
<tr>
<td colspan="2"><img class="formulaInl" alt="${\rm lat}_{1,x}$" src="form_2.png"/> </td><td colspan="2"><img class="formulaInl" alt="${\rm lat}_{1,y}$" src="form_3.png"/> </td><td colspan="3"><img class="formulaInl" alt="${\rm lat}_{1,z}$" src="form_4.png"/> </td></tr>
<tr>
<td colspan="2"><img class="formulaInl" alt="${\rm lat}_{2,x}$" src="form_1.png"/> </td><td colspan="2"><img class="formulaInl" alt="${\rm lat}_{2,y}$" src="form_5.png"/> </td><td colspan="3"><img class="formulaInl" alt="${\rm lat}_{2,z}$" src="form_6.png"/> </td></tr>
<tr>
<td colspan="2"><img class="formulaInl" alt="${\rm lat}_{3,x}$" src="form_7.png"/> </td><td colspan="2"><img class="formulaInl" alt="${\rm lat}_{3,y}$" src="form_8.png"/> </td><td colspan="3"><img class="formulaInl" alt="${\rm lat}_{3,z}$" src="form_9.png"/> </td></tr>
<tr>
<td><img class="formulaInl" alt="${\rm orbital}_{i}$" src="form_10.png"/> </td><td><img class="formulaInl" alt="${\rm orbital}_{j}$" src="form_11.png"/> </td><td><img class="formulaInl" alt="${\rm Re}(t_{i,j})$" src="form_12.png"/> </td><td><img class="formulaInl" alt="${\rm Im}(t_{i,j})$" src="form_13.png"/> </td><td><img class="formulaInl" alt="$\delta_{i,j,0}$" src="form_17.png"/> </td><td><img class="formulaInl" alt="$\delta_{i,j,1}$" src="form_18.png"/> </td><td><img class="formulaInl" alt="$\delta_{i,j,2}$" src="form_19.png"/> </td></tr>
</table>
<p>Here, The Number of Orbitals represents the number of orbitals within a unit cell, Number of Hoppings represents all hoppings going in and out of a unit cell. are The Lattice vectors which will be normalized and rescaled with lattice constant. <img class="formulaInl" alt="${\rm orbital}_{i}$" src="form_10.png"/> and <img class="formulaInl" alt="${\rm orbital}_{j}$" src="form_11.png"/> represent the initial and final orbital label. The vectors <img class="formulaInl" alt="$(\delta_{i,j,0},\delta_{i,j,1},\delta_{i,j,2})$" src="form_20.png"/> represent the traslational shift one need to perform in order to connect a orbital from the reference unit cell, to its neighbors.</p>
<p>For example, imagine a linear chain, where we choose the center site as the reference unit cell we then label the left and right site as -1, and , 1 respectively</p>
<table class="doxtable">
<tr>
<td><img class="formulaInl" alt="$\dots$" src="form_21.png"/></td><td><img class="formulaInl" alt="$i-1$" src="form_22.png"/></td><td><img class="formulaInl" alt="$i$" src="form_23.png"/></td><td><img class="formulaInl" alt="$i+1$" src="form_24.png"/></td><td><img class="formulaInl" alt="$\dots$" src="form_21.png"/> </td></tr>
</table>
<p>Given that there is only one orbital in the unit cell, we name this orbital 0. Therefore, a tight-binding operator for the hamiltonian with complex hopping <img class="formulaInl" alt="$t_{i,j}= 1+2i$" src="form_25.png"/>, and onsite energi <img class="formulaInl" alt="$t_{i,j}=3$" src="form_26.png"/> will be:</p>
<div class="fragment"><div class="line">1 3</div><div class="line">1</div><div class="line">1.0 0.0 0.0</div><div class="line">0.0 1.0 0.0</div><div class="line">0.0 0.0 1.0</div><div class="line">0 0 1.0 2.0 1 0 0</div><div class="line">0 0 3.0 0.0 0 0 0</div><div class="line">0 0 1.0 2.0 -1 0 0</div></div><!-- fragment --><div class="fragment"></div><!-- fragment --> </div><!-- contents -->
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