Can first order transitions accidentally have continuous free energy derivatives Obviously if you define a first order phase transition as having a discontinuity in the first derivative of the free energy then the answer is no, but I'm asking about if the following situation would still 'effectively' be a first order phase transition in all ways except for the wording of this definition.
The following diagram shows that as you go from a temperature $t$ above the critical temperature $t^*$ to a temperature $t$ below the critical temperature, a previous local minimum of the free energy 'overtakes' the initial global minimum so the system discontinuously jumps to the part of phase space where the new global minimum is - this (I think) is the 'meat' of a first order phase transition.

In the old local minimum, the precise free energy value $\mathcal{L}_0$ at the minimum point will generically continuously vary as the temperature is changed (or in this case (trivially) continuously stay at $\eta=0$. When the system jumps into the new global minimum, this part of the graph will also have a minimum $\mathcal{L}_0$ value which continuosly varies as you vary temperature - but crucially this 'gradient of $\mathcal{L}_0$' is different in the new global minimum than in the old one - this is what causes the cusp in the $\mathcal{L}_0/T$ graph (see second graph below).

My question is, could we envisage a situation where accidentally/coincidentally this new global minimum has a gradient $\frac{d\mathcal{L}_0}{dt}$ which is the same as the old global minimum, therefore not causing a cusp in the $\mathcal{L}_0/T$ graph but nonetheless seeming like a first order phase transition (in that there is a discontinuous jump in the order parameter $\eta$ at the critical temperature (which is indicative of a first not second order transition according to the graph below)?

(I understand this would be unbelievably unlikely on a non-contrived graph)
 A: I am not 100% sure that I have understood your question, but I believe that the Thouless effect occurring in the one-dimensional Ising model with interactions decaying as $1/r^2$ is something similar to what you have in mind.
Let me provide more detail. Let us consider the Ising model on $\mathbb{Z}$ with formal Hamiltonian
$$
\mathcal{H} = -\sum_{i,j} \frac{\sigma_i\sigma_j}{|j-i|^2}.
$$
The following scenario was first suggested by Thouless and is now known as the Thouless effect:

*

*there is a nonzero spontaneous magnetization at sufficiently low temperatures;

*the spontaneous magnetization exhibits a discontinuous behavior at the transition: it vanishes for all $\beta<\beta_{\rm c}$, but is strictly positive for all $\beta\geq\beta_{\rm c}$;

*the free energy density is infinitely differentiable in $\beta$ at $\beta_{\rm c}$ (in particular, the energy density is continuous at $\beta_{\rm c}$).

As far as I know, only 1. and 2. have been established rigorously (in this paper and this paper, respectively). That a pathological behavior must happen at $\beta_{\rm c}$ is also known rigorously (see here), but that precisely the behavior in 3. happens remains open, as far as I know.
It is also known that 1. and 2. together with the validity of correlation inequalities in this model imply the divergence of the susceptibility at $\beta_{\rm c}$ (see the discussion here, for instance). The behavior is thus dramatically different from what is seen in standard first-order transitions. Since such transitions exhibit characteristics of both first and second-order phase transitions, some people call them of mixed order. (This might help you find more on this topic, if you're interested.)
