I am looking at the quantum phase transition of the transverse field Ising model.
Let: \begin{equation} H = -J \sum_{x=1}^{N-1} \sigma_x^3\sigma_{x+1}^3 - B \sum_{x=1}^N \sigma_x^1 \end{equation}
Once I plot the ground state energy as a function of $B/J$ it looks something like this:
The first derivative around $B/J = 1$:
And second derivative:
Third derivative:
I am a bit confused. Is this meant to be a first/second/third order phase transition?
The first derivative is smooth, but it is discontinuous at $B/J = 1$. Does this mean that it is a first order phase transition?
If it is not, then the second derivative still converges to the same point at $B/J = 1$, but it is now $\infty$ does it mean it is a second order phase transition?
Otherwise, the third derivative is definitely divergent. Is it then a third order phase transition?
I think that this phase transition should be second order (?), but my first derivative being discontinuous disagrees. Does this mean that my ground state is wrong?