# Phase transition in a 2D classical Heisenberg model Monte Carlo simulation

I have performed a Monte Carlo simulation of the classical Heisenberg model for a simple 2D square lattice. But I obtain some strange results. In fact, there is clearly a phase transition and a magnetized phase for low temperatures.

I attach some plots of the results.   I know that the symmetry can be broken only for $$T=0$$. I've checked multiple times the code I wrote and all seems ok. I have run some simulation of the 3D Heisenberg model and all went fine. I have also estimated some critical exponents for the 3D model with good accuracy so I don't think my code is bugged. What could be the reason of this behavior?

I have already read Phase transition in 2D Heisenberg model and I use the correct way to sample point on the sphere.

EDIT:

For each sample, I compute and save the average magnetization per spin $$\vec m_i$$ and the system energy $$E_i$$

In the analysis, I simply compute the mean of the magnitude $$\langle | \vec m | \rangle$$ for the first plot.

In the second one, the heat capacity per spin is estimated by

$$c_v = \frac{1}{L^2} \frac{\langle E^2 \rangle - \langle E \rangle^2}{T^2}$$

• Should be the 2D Heisenberg model without phase transitions because of Mermin-Wagner theorem? – Andrea Maiani Nov 5 '18 at 17:39
• Oh sorry, I somehow read Ising model. You are right, MW forbids symmetry breaking in the Heisenberg model. – Nephente Nov 5 '18 at 17:42
• Can you expand on how you're calculating the mean magnetization and the heat capacity? Is this the classical or quantum Heisenberg model? – Jahan Claes Nov 5 '18 at 18:19
• I added the formula I used. They are the simple ones to compute thermodynamic quantity by the fluctuations. The model is the classical Heisenberg model. – Andrea Maiani Nov 5 '18 at 18:46

This model (the classical Heisenberg model, with 3-component spins, on a 2D $$L\times L$$ square lattice) is one of the challenging ones. I believe that the current thinking is that there is not a true phase transition, but instead it exhibits pseudocritical behaviour: the correlation length becomes extremely large, but not infinite. The latest publication that I found is by Y Tomita, Phys Rev E, 90, 032109 (2014), and there a finite-size-scaling analysis was employed on systems up to $$1024\times1024$$. Also, the Monte Carlo simulations employed special techniques (Swendsen-Wang cluster updating). I'm afraid that I couldn't find an open-access version of the paper cited above.