I was simulating the square lattice Ising model via Metropolis Algorithm and found that at 0 magnetic field, there is spontaneous magnetisation below some temperature.

I have used Periodic Boundary Condition in a 100x100 lattice. Is this an instance of a Phase Transition? I have heard that phase transitions occur in thermodynamic limit. So how does this spontaneous magnetisation occur?

If this is not a phase transition, is it an artifice of the metropolis algorithm and relates to non convergence of this algorithm?

If this is a phase transition how does spontaneous magnetisation occur at all since the probabilities carry the symmetry of the hamiltonian and the partition function is finite allowing the microstates to have boltzmann distribution for all magnetisation values?

  • $\begingroup$ You do single spin flip metropolis? $\endgroup$ Commented Aug 25, 2019 at 19:25
  • $\begingroup$ Yes i do single flip metropolis. $\endgroup$ Commented Aug 27, 2019 at 6:40
  • $\begingroup$ Well, in that case, the two symmetry broken states are essentially disconnected (it takes - at least - exponential to switch between them). So it is not surprising you only see one of the symmetry broken sectors (and thus spontaneous magnetization) in your simulations. $\endgroup$ Commented Aug 27, 2019 at 6:42

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In a strict mathematical sense, you will not observe a phase transition in a finite volume, for the reason you mention. If you measure thermodynamic quantities and their derivatives, when you expect a completely sharp transition, you will instead see a smooth curve that approximates the "correct" behavior if the volume gets larger and larger. There is a theory of finite-size scaling that addresses this quantitatively, and in fact explains how these finite-size effects can be exploited to measure critical exponents effectively.

In practice, on a 100x100 lattice you should have no problem at all to detect a phase transition. If you use a good algorithm (like a cluster algorithm that flips many spins at once) you will find that the susceptibility obtains a maximum $\chi_\text{max}(L)$ at some temperature $T_c(L)$ that slightly depends on the number of spins $L$. By measuring these quantities for different box sizes $L$ you can obtain estimates for the actual critical temperature $T_c$ and the critical exponents $\nu$ and $\gamma$.

  • $\begingroup$ What about average magnetisation though? In a Wolff cluster algorithm simulation, the bond formation probability is p=1-exp[-2*beta*J] which is very high at low temperatures. So in every step we are flipping a large number of spins at every step. So initially if the lattice is 1 dominated, it becomes -1 dominated in next step and this happens alternatively. So the average magnetisation stays close to 0. But we get high magnetisation values in Metropolis algorithm for low temperatures. Is this to do with convergence issues of metropolis algorithm around local minima? $\endgroup$ Commented Aug 25, 2019 at 17:36
  • $\begingroup$ Yes, this is the famous critical slowing down of local spin update algorithms. Still, even if you use Metropolis and wait for a very, very long time, you should also find that the average magnetization is 0 on a finite lattice: it just takes astronomically many steps to go from a "mostly +" to a "mostly -" phase if you're close to $T = T_c$. $\endgroup$ Commented Aug 25, 2019 at 20:33

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