# What is this secondary transition in the simulation of the Ising model?

Here, the horizontal axis is the strength of the ambient magnetic field. The Hamiltonian I used is $$H = -h\sum_i \sigma_i - J\sum_{\langle i \, j \rangle}\sigma_i\sigma_j.$$ The horizontal axis is $$h$$, and the vertical axis is the average of the spin values $$\sigma_i\in\{\pm1\}$$. $$J$$ is set to $$1$$ for simplicity. For low temperatures ($$\beta$$ is $$3$$, $$5$$, $$7$$ for blue, red, green respectively), apart from the expected phase transition at $$h=0$$, there is two other 'steps' appearing on both sides of the plot. The asymptotic value seems to be about 0.65, not very reasonable for any explanations I can think of: fluctuations from the last round of Markov spin-flipping (only 100 is candidate to be flipped each round, out of 10000), the external field has quenched most flipping (the average should have been $$\pm 1$$, then), etc.

Is this phenomenon caused by computational flaws, or is it actually appearing in the physic model? Thanks in advance!

• What is the used value for the coupling constant $J$? – WarreG Jan 18 '19 at 14:14
• @WarreG edited. $J=1$. – Trebor Jan 18 '19 at 14:18
• +1, this is a very good question. Have you tried varying the time for which you run the simulation? It could be that this is a transition in kinetic rates, not just the equilibrium state. – Nathaniel Jan 18 '19 at 14:22
• Though, at low temperatures you should have average spin values of +1 or -1, so there is something strange here in any case. You may have a bug. Have you tried watching your simulation graphically? – Nathaniel Jan 18 '19 at 14:26
• Not sure if it has any influence but, what is the dimensionality of your system? How much neighbors does each spin have? – WarreG Jan 18 '19 at 14:26

• And convergence of MC algorithms is quite slow, O($\sqrt{n}$) ($n$ is number of samples)! The life of a computationalist... good results are not easy/cheap/fast results. – tpg2114 Jan 18 '19 at 14:34