Invertible and non-invertible alloy ising problems

Physical properties of alloys are compared as computed from 'direct' and 'inverse' procedures. The direct procedure involves Monte Carlo simulations of a set of local density approximation (LDA)-derived pair and multibody interactions {v_f}, generating short-range order (SRO), ground states, order-disorder transition temperatures, and structural energy differences. The inverse procedure involves 'inverting' the SRO generated from {v_j} via inverse-Monte Carlo to obtain a set of pair only interactions {ṽ_f}. The physical properties generated from {ṽ_f} are then compared with those from {v_f}. We find the following: (i) Inversion of the SRO is possible (even when {v_f} contains multibody interactions but {ṽ_f} does not), (ii) Nevertheless, the resulting problem interactions {ṽ_f} agree with the input interactions {ṽ_f} only when the problem is dominated by pair interactions. Otherwise, {ṽ_f} are very different from {v_f}. (iii) The same SRO pattern can be produced by drastically different sets {v_f}. Thus, the effective interactions deduced from inverting SRO are not unique, (iv) Inverting SRO always misses configuration-independent (but composition-dependent) energies such as the volume deformation energy G(x); consequently, the ensuing {ṽ_f} cannot be used to describe formation enthalpies or two-phase regions of the phase diagram, which depend on G(x).