Integrability of a non-integrable quantum spin model at critical point

Question

Is it right, that non-integrable quantum spin models in one dimension become integrable at their critical points? Or do they stay nonintegrable at the critical point also? Are there any examples known? In the field of 2d classical models, the three-state Potts model is not in general integrable, but this model is integrable at the critical point.

Answer 1

I have constructed a counterexample; that is, I have constructed a nonintegrable spin chain that remains nonintegrable at the critical point. Furthermore, my answer seemingly contradicts the one by Exhaustive, as my model is described by (i.e. flows under the renormalization group to) a conformal fixed point at the critical point, as also shown below. I haven't found any references on my model yet, but it's likely it has been studied before. I will show strong evidence for a phase transition, for the phase transition being in the Ising universality class, and that the critical point is nonintegrable below.

First, we are familiar with the integrable transverse Ising model: $$\sum_i \sigma^x_i \sigma^x_{i+1} + h \sigma^z_i $$ This model has a duality transformation to the model $\sum_i h \sigma^x_i \sigma^x_{i+1} + \sigma^z_i $ up to details about boundary conditions; assuming there is only one critical point, this predicts a critical point of $h=1$. The model is integrable because it can be mapped via a Jordan-Wigner transformation to a moderately simple quadratic fermionic Hamiltonian.

Let us use the insight of duality above to craft a nonintegrable model that undergoes an Ising transition at a known critical field $h$, but has more complicated interactions to break integrability.

$$\sum_i (\sigma^x_i \sigma^x_{i+1} + \Delta \sigma^x_i \sigma^x_{i+1}\sigma^x_{i+2}\sigma^x_{i+3}) + h (\sigma^z_i+\Delta \sigma^z_i \sigma^z_{i+2})$$

Though nonintegrable, the duality means the critical point will remain at $h=1$. I'll consider small $\Delta=1/4$ to be safe to avoid changing the type or existence of transitions. All my calculations below are for chains in periodic boundary conditions, though I vary the length of the chain according to my patience for numerics.

First, let's check that there's a transition between a ferromagnetic phase at $h<1$ and a paramagnetic phase at $h>1$ by checking the degeneracy of the ground state. Below, I plot the energy gap between the ground state and the first excited state for $L=14$:

Indeed, we see the gap opens up at $h=1$ as anticipated.

Next, let's verify that the point at $h=1$ is described by the Ising CFT with central charge $c=1/2$ by computing the entanglement entropy of a length $l$ portion of the ground state:

The orange curve is a fit to the function $$S(l) = \frac{c}{3} \ln(\frac{L}{\pi} \sin(\frac{l \pi}{L}))+A$$ where $c$ and $A$ are treated as unknowns. This function is a prediction of conformal field theory for a CFT with a central charge of $c$. The best fit yields $c=.53$, which, at this relatively small system size, is a good sign that the true $c=1/2$, corresponding to the Ising universality class. Thus the highly probable guesses that the model shares the same kind of phase transition as the transverse Ising model and that the critical point flows to a CFT with $c=1/2$ gain some considerable evidence.

Next, let us verify that the model is nonintegrable. To do so, I will look at the level statistics within symmetry sectors. For $h \neq 1$, I look at the symmetry sector $k=0, I=1, F=1$, by which I mean $0$ momentum, an eigenvalue of $1$ under spatial inversion of the chain about the center, and an eigenvalue of $1$ under Ising symmetry. For $h=1$ there is an additional discrete symmetry by virtue of the duality transformation within the symmetry sector above, so I look at the more fine-grained sector $k=0, I=1, F=1, D=1$, where $D=1$ refers to an eigenvalue of $1$ under the duality transformation.

More technically, at the critical point, I implemented an operator form for $D$ inspired by Gec's answer to one of my questions: in Gec's notation, I created an operator that maps a state $|s_1,\ldots,s_N\rangle$ to $|(s_N s_1) (s_1 s_2),(s_2 s_3)\ldots,(s_{N-1} s_{N})\rangle$. This is generically noninvertible, but it is invertible within a given Ising symmetry sector. In particular, using the projector $P$ to stand for the rectangular projector into the $k=0, I=1, F=1$ sector, and $R$ for a $\pi/2$ rotation of spins about the y-axis, I found that (the factor of 1/2 is just a vagary of my $D$)

$$1/2 (PRD) H (PRD)^\dagger = P H P^\dagger $$ and more critically that $$[PRDP^\dagger, PHP^\dagger]=0$$ which means $PRDP^\dagger$ and $PHP^\dagger$ can be simultaneously diagonalized, as they are both normal and in fact both Hermitian (obvious for $PHP^\dagger$, and I checked numerically for $PRDP^\dagger$, which might mean $RD$ has a simpler representation than I think). I explicitly diagonalized $PRDP^\dagger$ to construct the $k=0, I=1, F=1, D=1$ sector; there may be shortcuts, but those aren't so necessary for the problem at hand.

With these symmetry sectors in hand, I checked the level statistics of the cases of $h=1/2$, $h=1$, and $h=3$. I am plotting a histogram of the oft-used level ratio statistic discussed in Atas et al., with the red curve below corresponding to the GOE prediction and the orange curve the integrable model prediction. As seen below, the models at these points are all nonintegrable; the model is nonintegrable on either side of the transition, and it is also nonintegrable at the transition.

$h=1/2$, in the ferromagnetic phase:

$h=1$, at the critical point:

$h=3$, solidly in the paramagnetic phase:

To summarize, I have created (it's probably been studied before, though, and I just don't know about it!) a nonintegrable model that has a transverse-Ising type phase transition at a known parameter value $h=1$, numerically confirmed and characterized the critical point, and checked that the model is nonintegrable in both phases as well as at the critical point itself. The critical point itself had an extra discrete symmetry stemming from the duality transformation, but once that was handled, the level statistics at the critical point showed repulsion and hence strongly point to nonintegrability.

Answer 2

If the non-integrable quantum spin chain at the critical points can be described as a conformal field theory (not always the case), we can say that the model is "integrable''. Because CFT can be seen as an "integrable'' theory since it can be solved exactly and Yang-Baxter relation is satisfied naturally. If the critical points cannot be described by CFT, there is no general guarantee whether the underlying field theory is integrable or not.