# How does phase transition occur in finite sized ising model?

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?

• You do single spin flip metropolis? – Norbert Schuch Aug 25 '19 at 19:25
• Yes i do single flip metropolis. – Sudipta Nayak Aug 27 '19 at 6:40
• 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. – Norbert Schuch Aug 27 '19 at 6:42

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 $$\gamma$$.
• 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 $T = T_c$. – Hans Moleman Aug 25 '19 at 20:33