# Why does the Wolff algorithm slow down in a 4-body Ising model?

In the paper that introduced "Self-learning MC" (an ML-inspired MC technique, as I understand) the authors consider a many-body Ising model as an example to show the efficiency of their algorithm. The model looks like this: $$$$H= -J \sum_{\langle i,j\rangle} S_i S_j - K \sum_{ijkl\in p}S_iS_jS_k S_l \tag{1} ,$$$$ where the second term runs for all plaquettes $$p$$ on the square lattice.

In the paper, I found an explanation that says "For $$K=0$$, this model reduces to the standard Ising model which can be simulated efficiently by the Wolff method. However, for $$K\neq0$$, no simple and efficient global update method is known." The Wolff algorithm can be applied to this model, and they indeed compare the "naive Wolff algorithm" with the new self-learning MC. It seems that while the Wolff algorithm does speed up MC compared to local updates, it still has a very similar asymptotic suffering in relaxation (Fig 3 in the paper).

My question is this: Is there an intuitive way to understand WHY this slowing down happens in the Wolff algorithm? To me, it seems like it might work just fine as in the 2-body Ising model, because it's not that different (at least in terms of implementation).

• Discussion about this question, including the answer to the above comment, is taking place in this chat room. – Nike Dattani Mar 25 at 2:18