1, 2$ shown in Fig. \[figure1\]. Column height $z_i$ at site [*i*]{} in the [*l*]{}th sublattice is defined as $z_i = h_i a\sqrt{3}/6$ where [*a*]{} is bcc lattice constant, $h_i=3 n_i +l$ and $n_i$ is the number of atoms in [*i*]{}th column. The assumption on constant coverage equal to 1PML means that there is exactly one A atom in each column placed on its top. As we are going to study reconstruction only, not desorption or crystal melting, the typical restriction of SOS models on columns heights will be assumed. The $h_i$ difference between nearest neighbor sites are forced to be $\pm 1, \pm 2$.

Let us define the Hamiltonian as surface formation energy of A material on B (111) surface $$\label{esurf}
 H  =E_{A/B}(N_A, N_B) -E^{bulk}_{A}(N_A)-E^{bulk}_{B}(N_B),$$ where $E_{A/B}(N_A, N_B)$ denotes energy of $N_A$ atoms A on $bcc(111)$ surface consisting of $N_B$ atoms B, $E^{bulk}_{A}(N_A)$, $E^{bulk}_{B}(N_B)$ are energies of A, B atoms in their bulk environments. Now taking into account above assumptions we can express Hamiltonian Eq. (\[esurf\]) in terms of column heights $h_i$ in the following form

$$\label{hsos}
\begin{array}{lll}
H &=& 
  \frac{1}{2}\sum\limits_i 
    \left\{  
      \sum\limits_{j_1} 
         {\left[ 
           J_1 \delta 
             \left( 
                 \left| h_i  - h_{j_1 } \right| - 1 
             \right)
             + K_1 \delta 
             