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 $$ \mathcal{H} = J\sum _{i,j} s_i s_j + B \sum s_i. $$
The 1D Ising model shows no phase transitions as temperature changes: there are no divergences to look out for.
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 ($T_c$)?
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 ($T_c$)?