Soft Annealing: A New Approach to Difficult Computational Problems

I propose a new method to study computationally difficult problems. I consider a new system, larger than the one I want to simulate. The original system is recovered by imposing constraints on the large system. I simulate the large system with the hard constraints replaced by soft constraints. I illustrate the method in the case of a ferromagnetic Ising model and in the case of a three-dimensional spin-glass model. I show that in both models the phases of the soft problem have the same properties as the phases of the original model and that the softened model belongs to the same universality class as the original one. I show that correlation times are much shorter in the larger soft constrained system and that it is computationally advantageous to study it instead of the original system. This method is quite general and can be applied to many other systems.

Figure 1

Binder ratio $r$ plotted as a function of $\beta =1/T$. Circles represent data for linear size $L=16$, triangles for $L=14$, $\times$'s for $L=12$, and stars for $L=8$.

Figure 2

Binder ratio $r$ plotted versus the scaling variable $s$ (see the text). Circles represent data for linear size $L=16$, squares for $L=14$, $\times$'s for $L=12$, and stars for $L=8$.

Figure 3

Average probability distribution $P(q)$ of the overlaps $q$ for $\beta =1.05$ and sizes $L=14$ (stars) and $L=8$ (crosses).

Figure 4

Spin-spin autocorrelations $C^h(t)$ and $C^s(t)$ at ${\beta }_c^h$ and ${\beta }_c^s$ (see the text) as a function of the number of the lattice Monte Carlo sweeps.