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:

Ground state energy

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?

  • 1
    $\begingroup$ 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. $\endgroup$
    – Meng Cheng
    Feb 25, 2022 at 23:28
  • $\begingroup$ @MengCheng My first derivative is discontinuous. It has a hole at $x = 1$. But it is smooth. Does that make sense? $\endgroup$ Feb 25, 2022 at 23:30
  • 3
    $\begingroup$ 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. $\endgroup$
    – Meng Cheng
    Feb 25, 2022 at 23:43

1 Answer 1


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.

On the order of the phase transition: The order of a phase transition is (usually) determined by the lowest order of derivative with a singularity. Therefore, by looking at the plots of the derivatives, I would conclude that the transition is second-order.

I hope it helps.


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