# What is the order of the transition for a 2D Ising model?

I have been running around the block trying to find answers for this question, and I keep running into caveats. So, I just want to write down the list of things I want to know: Given that the order parameter is magnetization,

Consider the simple two-spin-state Ising model. Where the Hamiltonian is defined by $$\mathcal{H} = J\sum _{i,j} s_i s_j + B \sum s_i.$$

1. The 1D Ising model shows no phase transitions as temperature changes: there are no divergences to look out for.

2. If we are not present in an external field $$(B=0)$$, will the 2D Ising model will a second-order/continuous phase transition as temperature passes over critical point ($$T_c$$)?

3. In the presence of an external field $$(B\neq 0)$$, will the 2D Ising model show a first-order phase transition as temperature passes over critical point ($$T_c$$)?

• That's too many questions at once. I'd suggest you at least cut #3. (It is not even clear what you mean by "bottleneck".) -- Also: 2D classical? Quantum? As a function of the temperature? Or also of the field? Commented Jan 25, 2022 at 20:41
• @NorbertSchuch, I have added some additions. Hope they help. Commented Jan 25, 2022 at 20:48

For your 1, suppose we are on a 2D square lattice and $$J<0$$ (so ferromagnetic interaction), then yes there is a continuous transition at a critical temperature between ferromagnetic and paramagnetic phases. In fact the model was solved exactly by Onsager, whose main motivation was to prove the existence of a phase transition. The lattice geometry is not crucial: the same thing happens on triangular lattice for example. But the sign of $$J$$ does matter sometimes.
For your 2, the critical point is located at $$T=T_c, B=0$$. For finite $$B$$, there is no singularity in free energy anymore. However, there is indeed a first-order transition as you tune $$B$$ from say positive to negative below the critical temperature crossing $$B=0$$. This first-order line terminates at the critical point.