Is there any way to check whether in a Monte Carlo simulation using Ising model is stuck in any (false) local minima of energy or not, particularly in 3D system ?

  • $\begingroup$ I have edited the question to be more specific. $\endgroup$
    – cosmicraga
    Commented Apr 3, 2013 at 15:36
  • $\begingroup$ There's probably still a misunderstanding. There are only two true ground states for the Ising model (at zero magnetic field): all spins up or all spins down. All other states must be either false vacua or not even local minima of the energy. Whether another local minimum is 'close by' depends on how you update your configuration. $\endgroup$
    – Vibert
    Commented Apr 3, 2013 at 21:00

1 Answer 1


For nearest-neighbor interactions in 1D and 2D, the free energy of the system can be computed analytically. We can then check that this free energy is at its global minimum for a certain state. In 3D, we do not know the free energy analytically, so we have to resort to some kind of simulation (Monte-Carlo probably). If you reach a final state of your simulation, you can always give it a 'kick' and check that it comes back to the same state. This can't rule out the possibility of very deep local minima, but it does increase your confidence that you have found the ground state.

For situations where the Ising model DOES get trapped in a local minimum, check out the work of Sidney Redner at Boston University. The gist is that if you quench the system, it can get 'stuck' in local minima and the dynamics are surprisingly non-trivial (in 2D and 3D, the 1D system always goes to the ground state).

  • $\begingroup$ Based on the edit the OP made to the question, my answer is no longer specific enough. I don't think I have the expertise to answer the more detailed question. $\endgroup$ Commented Apr 3, 2013 at 15:38
  • $\begingroup$ Your answer is very helpful. Lets wait to see whether something more gets added by others or not. $\endgroup$
    – cosmicraga
    Commented Apr 3, 2013 at 15:41
  • $\begingroup$ Do you have any idea where I can find out more about the work of Sidney Redner? $\endgroup$
    – user86840
    Commented Jul 27, 2015 at 11:29

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