$A$, an atom of type $B$, or a vacancy, with an obvious condition that the corresponding occupation numbers fulfill $c^A_i+c^B_i+c^V_i = 1$. In accord with the stoichiometry of the $A_3B$ alloys there are exactly three times as many $A$ atoms present as $B$ atoms. The number of vacancies is chosen small enough so that they do not affect static properties of the system.

In our simplified model [@paper1] only nearest neighbor atom–atom interactions are taken into account. The corresponding Hamiltonian then reads:

$$H= \sum_{\langle i,j \rangle}[ V_{AA}c^A_ic^A_j + 
V_{BB}c^B_ic^B_j + V_{AB}(c^A_ic^B_j + c^B_ic^A_j)] ,$$

where the sum is restricted to nearest–neighbor pairs. We are interested here in the transition to the ordered $L1_2$–structure in which minority atoms ($B$) predominantly occupy one of the simple–cubic sublattices of the original fcc–lattice. It is then natural to assume $V_{BB}>0$, $V_{AA}<0$ and $V_{AB}<0$. The ground state of the $L1_2$ phase is fourfold degenerate and corresponds to all $B$–atoms segregated to exactly one of the simple cubic sublattices. For a stoichiometric $A_3B$-alloy the transition occurs at a temperature $T_0 \simeq 1.83|J|/k_B$, with $J=-\frac{1}{4}(V_{AA} + V_{BB} - 2V_{AB})$ as discussed in Ref.  (cf. also Refs. ). The ordered phase is characterized by one conserved, scalar order 