The most straightforward way of carrying out Monte Carlo in the Ising (and similar) models is by using
local updates, which consider changes to only one lattice site (spin) at a time. While this is the easiest method both in concept and in practice, it does have its drawbacks. Chief among these is the occurrence of “critical slowing down,” which becomes a problem in the neighborhood of critical points. Because critical phenomena involve the development of long-range correlations—the divergence of correlation lengths is a key marker of criticality—local updates struggle to effectively capture such behavior. Each local update is statistically independent of previous ones, so that it takes a very long time for large clusters to form. One
0.0 0.3 0.6 0.9 1.2 1.5 βJ −2.0 −1.8 −1.6 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 E /J N L= 6 L= 12 L= 24 L= 48 L= 96 High-T plaq High-T hex Low-T
Figure 6.6: Energy per lattice site of the three-body model, where the thermodynamic limit is approached with increasing darkness of curves. Also shown: approximations for single plaquette at high temperature (dotted pink), hexagon at high temperature (dashed red), and hexagon at low temperature (dashed blue).
must wait for the update to chance upon the boundary of the growing cluster, but the probability of the boundary being landed on randomly tends to zero in the thermodynamic limit.3 Since large-scale changes slow down significantly, samples modified by local updates are highly correlated, and an increasing number of updates must be carried out before samples become statistically independent [137].
To overcome the difficulties of critical slowing down associated with local updates, cluster update algo- rithms have been developed, which update many lattices sites at once. The most efficient cluster algorithm developed for the Ising model to date is the Wolff algorithm [138], which builds clusters whose size tends to grow with decreasing temperatures: given that one lattice site is in the cluster, the probability for each of its neighboring sites to be added to the cluster is p= 1−e−2βJ. The Wolff algorithm is rejection-free,4 so that proposed updates arealways accepted while still satisfying detailed balance.
As described in Ref. [139], the Wolff algorithm is built by recursively considering the probability of adding (or not adding) same-spin neighbors of lattice sites to the cluster. Since the Ising model has only two possible spin values, it is straightforward to compare the statistical weights of a cluster before and after flipping it. By following similar reasoning, we may construct a cluster algorithm for the three-body model
3Ind-dimensional space, the probability of randomly choosing a boundary location (with sizeLd−1) out of the whole volume (with sizeLd) scales asL−1; the thermodynamic limit sendsL→ ∞.
on the triangular lattice as follows.
First, randomly choose a plaquette that is in an interacting configuration. Its neighbors are those plaquettes that share an edge (two vertices); if they are in the same phase orientation, they are added to the cluster with probabilityp(not yet specified), and so on for each of their neighbors. Once the cluster is constructed, it is collectively “flipped” to a new orientation with some probabilityP (to be determined). This “flip” is carried out by performing a permutation on the values of all of the lattice sites belonging to plaquettes in the cluster.
Now, we must determine the probabilitiespandP. To do so, collect the cluster plaquettes’ neighboring plaquettes that were not added to the cluster, whether by non-compatibility of orientation or by failure to meet the probabilityp. This set of plaquettes is the same before and after the flip (“set A”). Find the vertices that are members of both set A and of the cluster, then collect those plaquettes to which these vertices belong that are neither in the cluster nor in set A. These plaquettes form “set B”, also the same before and after a flip. Define xas the number of interacting plaquettes in set B.5 Then, if we choose p= 1−e−βJ, the probabilityP of flipping the cluster from its initial configuration to a new one is given by
P(i→f) = minn1, eβ(xf−xi)o, (6.16)
wherexiandxfare values ofxbefore and after the flip, respectively. This algorithm is thus not rejection-free,
but it still may prove more efficient than the local update scheme.
Carrying out this cluster update, the energy (not shown) appears very similar to that obtained with the local update, shown in Fig. 6.6. A phase transition still appears to occur, but with a higher critical temperature, appearing closer toβJ≈0.6. Why the disagreement? It turns out that this cluster algorithm does not actually obey detailed balance; the critical step lies at the very beginning of the procedure.
As laid out clearly by Ref. [139], the probability of carrying out a Monte Carlo update consists of not only the probability of accepting vs. rejecting, but also of the probability ofproposing any given change. In formulating proposals, we must not introduce any undue bias to a particular region of configuration space (this is the requirement of ergodicity). Notice that in the first step of this algorithm, however, we chose a plaquette that wasalready in an interacting configuration. As a result, although the steps that follow may be in accord with detailed balance, we have restricted ourselves to updating only those regions which are immediate neighbors of interacting plaquettes; noninteracting plaquettes are never chosen as the starting point for an update. Ordered phases are thus preferentially chosen for updates at the exclusion of disordered
5In most cases,xis equal to the number of interacting plaquettes that are intersected by the boundary of the cluster and have only one vertex inside the cluster. While helpful for visualization, this is not always the case.
phases. The Wolff algorithm does not suffer from this deficiency, as any randomly chosen site can serve as the beginning of a cluster. To date, we have not discovered a suitable remedy for this bias and leave it for future work.