How does phase transition occur in finite sized ising model?

Question

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?

Answer 1

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 $\gamma$.