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

Question

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 $$ \mathcal{H} = J\sum _{i,j} s_i s_j + B \sum s_i. $$

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

1. 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 ($T_c$)?

2. 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 ($T_c$)?

Answer 1

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.