# Transverse field Ising model quantum phase transition

I am looking at the quantum phase transition of the transverse field Ising model.

Let: $$$$H = -J \sum_{x=1}^{N-1} \sigma_x^3\sigma_{x+1}^3 - B \sum_{x=1}^N \sigma_x^1$$$$

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?

• It is indeed known that the transition at $B/J=1$ is second-order, which means there is singularity (discontinuity) in the second derivative of the energy with respect to the tuning parameter. Your first order derivative does not look discontinuous, so I don't know what the problem is. Feb 25, 2022 at 23:28
• @MengCheng My first derivative is discontinuous. It has a hole at $x = 1$. But it is smooth. Does that make sense? Feb 25, 2022 at 23:30
• If it is smooth for both $x>1$ and $x<1$ and has the same limit approaching $x=1$ from both sides, then it should just be a continuous curve and the hole should not be there. Feb 25, 2022 at 23:43

On the divergence near $$\mathbf{(\mathit{B/J}) =0}$$: Since $$B/J$$ has been chosen as the parameter (i.e., $$J$$ has been chosen as the energy-scale), we have to be careful as we set $$B/J$$ to zero.
Since $$(B/J)\rightarrow0$$ could imply $$J\rightarrow\infty$$, or $$B\rightarrow0$$, this can anomalously cause the energy to diverge.