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In lattice systems such as Ising model or spin glasses, the standard Monte Carlo simulation with Metropolis algorithm works by proposig a single spin flip and then accepting or rejecting the proposal with the probability given by the algorithm itself.

I'd like to know if anything changes when the proposal changes from a single spin flip to a group of spins (choosen randomly and independently at each step, but fixed in number).

In other words, suppose that we suggest a flip of N spins. Then we measure the new energy and then use the standard Metropolis choice for the acceptance.

  1. Does this change the equilibrium probability distribution?
  2. If (1) is yes, does this make the algorithm any faster?
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    $\begingroup$ See Swendsen-Wang and Wolff cluster flip algorithms. $\endgroup$ – JamalS Jun 25 '18 at 16:31
  • $\begingroup$ From what I get from the two algorithms, they are quite different from the one in the question. Should I assume that therefore the Metropolis algorithm can’t be generalized that way? $\endgroup$ – Drebin J. Jun 25 '18 at 18:56
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The short answers to your question are

  1. No, the algorithm you describe does not change the equilibrium distribution, and so it is correct. Of course you must take care to choose the flipping spins independently and randomly, as you say. The crucial point is that the probability of selecting the reverse trial move is the same as the probability of selecting the forward trial move, so as to satisfy detailed balance. But it is a valid example of a Metropolis method.
  2. I guess you mean "if (1) is no... ", not "if (1) is yes... ". I.e., if the method is valid .... Generally, the crude multi spin flip will not be faster: the total energy change which appears in the Metropolis formula will be larger, in most cases, and trial moves will be rejected more often. So we tend not to do this.

The two cluster move algorithms mentioned in the comment of JamalS are smart ways of constructing clusters of spins for a trial flip, which will be faster in some circumstances, especially near the critical point. They are more complicated than the method you are discussing: the clusters are connected sets of neighbouring spins, constructed in a probabilistic way.

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