# First-order phase transition in the Ising model?

I am doing a simulation of the 2D Ising model with a Monte Carlo algorithm. I think that the model should exhibit a second order phase transition at $$\beta=\beta_c$$, but when I try to plot the magnetization for $$0<\beta<1$$, I get the following plot:

Since the the magnetization is discontinuous this is a first order phase transition, right? But I'm pretty sure it should be a second order one. What am i missing?

• looks continuous to me Commented Jan 11, 2022 at 20:07
• So it doesnt approach a step function in the infinite volume, jumping from 0 to 1? Also i see that the susceptibility is peaked at the critical temperature, does thet signal the discontinuity of the second order phase transition? Commented Jan 11, 2022 at 20:14
• If you want to understand what will happen at infinite volume, there is a procedure much better suited than the eyeball method -- namely, a finite size scaling analysis will give you a better indication of how the strength of finite size effects depends on your system size.
– Zack
Commented Jan 11, 2022 at 20:53
• What simulation techniques did you use. Is it single spin-flips? It is known that such algorithms suffer from critical slow-downs near Tc. You can try exploring cluster algorithms to get better (less noisy) results. See en.m.wikipedia.org/wiki/Wolff_algorithm Commented Jan 28, 2022 at 3:06

Magnetization of the 2d Ising model grows too fast in the critical point vicinity. From the exact solution of the Ising model, we know $$M \sim (\beta - \beta_c)^{1/8}.$$ It is hard to catch power law dependence of M on $$\beta$$ with such a small exponent by the approximate stochastic method.
Just as a complement to the previous answers, here is a plot of the spontaneous magnetization for this model, in the same range of $$\beta$$ you are considering, so that you can see how steep the increase is at $$\beta_{\rm c}$$:
The explicit formula is $$m(\beta) = \begin{cases} \bigl( 1 - \sinh(2\beta)^{-4}\bigr)^{1/8} & \text{if \beta>\beta_{\rm c},}\\ 0 & \text{if \beta\leq\beta_{\rm c},} \end{cases}$$ where $$\beta_{\rm c} = \frac12 \operatorname{arsinh}(1) \cong 0.44$$. It was first announced by Onsager and Kaufman in 1949 and derived by Yang in 1952.