Topological to magnetically ordered quantum phase transition in antiferromagnetic spin ladders with long-range interactions

We thank Referee 3 for carefully reading the manuscript and his/her constructive criticism. We hereby address his/her main concerns, hoping that it can now be accepted as is.

Referee; "the writing is poor and not very accessible..."

Authors: We agree that some parts of the manuscript require revisions, but we believe it should be perfectly accessible to a specialized audience. We already made considerable corrections in the new version, and re-written entire paragraphs, as detailed in the previous resubmission and we describe below. In addition, we point out that, to the best of our knowledge, this is the first time this model has been studied in the literature. It provides an unprecedented one-dimensional example of a transition from a gapped topological phase to one with spontaneous symmetry breaking, possibly of second order. We are very much convinced that our work may appeal to both broad, and specialized audiences and may provide the test ground for new field theories.

Ref: In the introduction, the authors should provide more details and background to make the paper more accessible to non-specialists. For instance, they should specify what "in a RPA-like sense" means; the meaning of "strong rung coupling limit"; and the definition of J_rung.

Authors: We thank the Referee for pointing out this omission. We re-wrote this paragraph and we hope that the new version clarifies this point:

"One could in principle envision such interactions emerging from a proximity coupling with a higher dimensional antiferromagnet or other ladders in a perturbative sense. The only free parameter in the problem is the exponent $\alpha$; for large $\alpha$ we expect the ground state to be in the same phase as the conventional Heisenberg ladder and the physics is well understood: the correlation length is short, of a few lattice spaces, and the gap is of the order of the coupling $J$ [9–20]. This ground state is adiabatically connected to the trivial limit of the conventional Heisenberg ladder corresponding to anisotropic couplings along the legs and rungs $J_{rung} \gg J_{leg}$. In this ''strong rung coupling limit'' the ground state is a product of rung dimers, the single-triplet gap is of order $\mathcal{O}(J_{rung})$ and excitations are rung triplets that can propagate coherently along the ladder." ... "On the other hand, in the model described by Eq.(1) we anticipate that the all-to-all unfrustrating interactions will yield a ground state with long-range AFM (N\'eel) order and gapless excitations for relatively small $\alpha$. These expectations are based on previous studies of Hamiltonian (1) in 1D chains [29–34], where a transition between a gapless spin-liquid and a gapless ordered phase was revealed. "

Notice that we have also added a figure (new Fig. 1) describing the geometry of the interactions.

Ref: On page 4, the paragraph starting with "In Fig. 1..." should be rewritten. It is the first paragraph presenting the numerical results, and it is written in a very obscure way.

Authors: We agree. This is the new version:

"In Fig. 2 we show both $B$ and $R$ as a function of $\alpha$ for various system sizes. We observe that $B$ monotonically increases for decreasing $\alpha$. This behavior indicates the onset of long-range N\'eel order for small values of $\alpha$. On the other hand, $R$, which is always negative by definition, is growing in absolute value as $\alpha$ decreases. For larger values of $\alpha$, in the gapped phase, $R$ tends to $0$ with increasing system size, implying that $O>0$. In the N\'eel phase (small $\alpha$) $R$ converges to a finite value, indicating that $O=0$. The ''steepness'' of the $B$ and $R$ curves increases with increasing system size. This observation is consistent with the finite-size behavior one would expect from a phase transition [43]. "

Ref: On page 6, in the first paragraph of part 3, the statement that the ground state entanglement entropy obeys a volume law due to long-range interactions is stunning. This would be an extraordinary observation. I suspect it is not true, and the entanglement entropy obeys in fact a log(L) scaling, as found in all sudies of long-range models. This should be clarified.

Authors: We point out that we never claimed to see a volume law. We believe that we never said anything incorrect. What we implied is that, in the presence of long range interactions one may naively expect a volume law behavior. In fact, we observe that "Surprisingly, the entanglement entropy does not grow dramatically in the gapless phase and across the transition." We have rephrased the text in a way that may sound less confusing:

"What can make this problem particularly challenging is the possibility of a volume law entanglement law due to the presence of all-to-all interactions. However, in the gapped phase, the correlation length remains finite and the entanglement remains under control. Surprisingly, the entanglement entropy does not grow dramatically in the gapless phase and across the transition. This may appear to be a general feature of one-dimensional models with long-range interactions as has been observed in quantum spin chains, which display a $log(L)$ behavior [33, 44–46]."

We added these three references that we considered relevant. Notice that the only rigorous results for the scaling of the entanglement entropy in systems with long range interactions is presented in the new reference Gong2017, which is valid for exponent alpha > 2. The other results are empirical and come from numerical simulations, some out of equilibrium, and some in ferromagnetic models.

Ref: On the same page 6. What polynomial fit is done for the gap to extrapolate finite-size data? In Fig.4 (inset), it seems that the gap is linear in 1/L. But this contradicts the scaling 1/Lz, which would imply a vertical slope in 0. This should be clarified, and possibly corrected.

Authors: We clarified that we used a second order polynomial (this information was in the figure caption but not the body of the manuscript).

Concerning the scaling, we emphasize that in reality the behavior should be 1/L^z with z<1. This is pretty obvious in the spectral function and is a general feature of one-dimensional systems with long-range interactions. The apparent linear behavior for large L is an artifact of this extrapolation. As a matter of fact, our extrapolation of z in the (newly renumbered) Figure 6 clearly shows both facts: z <1, and a power law extrapolation is ill behaved due to the small dataset. Similarly, a fit using Eq.(8) is ill behaved and does not yield physical results.

In short: a polynomial fit is not accurate and cannot describe the proper scaling for large L, where the sub-linear corrections become dominant. Consider a f(x)=sqrt(x), for instance: if one has data for x > 5, one would never see the sub-linear behavior and would completely miss the correct scaling near the origin. In that case, it is tempting to carry our a quadratic fit. In our case we are subject to the same dilemma: we cannot observe the sub-linear scaling in the dataset because we cannot simulate sufficiently large ladders. That's why the extrapolation of the gap should be taken with a grain of salt and is not very accurate. That is also why we had to proceed to other strategies in order to properly account for these corrections, as we discuss in great detail in the paragraph starting right above Eq.(8).

Ref: Why is the extrapolation of z(L) to L→∞ not showed? A plot should be done to illustrate this extrapolation and therby complement Fig. 4 (right panel)

Authors: We answered this in the previous point. In short: it is pointless because the extrapolation to the thermodynamic limit is not possible due to the huge variability based on the type of extrapolation used. In the newly labeled Fig. 6 we demonstrate this issue explicitly by showing that the power-law and polynomial extrapolations give wildly different results when extrapolating the dynamic exponent. On the other hand, the critical point value seems much less sensitive to the extrapolation method used and both give consistent results that are within error bars.

Summary of changes:

• We corrected several typos.
• We re-wrote two paragraphs in the introduction and section 3.
• We added three references 44,45,46 in this version of the manuscript.

We thank the referees for their valuable comments and, in particular, for recommending our manuscript for publication. We have addressed their concerns in our new version and in the following we proceed to answer Report 2 in detail:

REFEREE:

The main concern I have is on the nature of the transition between the two phases. The authors claim it is a second order phase transition by computing the Binder cumulant B and an indicator R related to the string order parameter. They then locate the transition in α by looking at the point where the curves which represent B and R cross for different system sizes. By looking at Fig. 1 while this crossing is evident for R, it is not clear that it happens also for B.

RESPONSE:

While we agree that in Fig. 1 it is a bit hard to see the crossing points, results for B and R from our large scale QMC calculations are shown in more detail in Fig. 2. These crossing points are obtained from a much more refined set of data compared to the points presented in Fig. 1.

REFEREE:

Moreover, the ground state gap (Fig. 4) does not seem to close in the vicinity of the transition. I think the authors should exclude the possibility that the change of the phase when α decreases is a simple crossover and not a phase transition.

RESPONSE:

We already discuss this issue in the manuscript, namely that finite-size corrections to the gap are strongly sub-linear, making an accurate extrapolation of the gap very problematic. A way to see this is by supposing that the scaling at large L (small 1/L) goes as sqrt(1/L). If L is not large enough, we would not be able to resolve the sublinear behavior, which is precisely the problem that affects our extrapolation here. We try to get around this issue by estimating the dynamic exponent. Similar to the plots for $B$ and $R$, the extrapolation of the "finite length" dynamic exponent shows crossing points such that on one side of the transition the extrapolation tends to a constant and on the other side it is tending to 0. This behavior indicates a transition from a gapped phase to a gapless phase with a finite staggered magnetization. Using the crossing points we can estimate the position of the critical point, which agrees with the QMC results for the order parameter. The issue of this critical point being of the Landau type, or a deconfined critical point, is left open as a possibility that deserves further study.

REFEREE:

I have also few remarks

1. There's a space missing in the abstract (6th line)
2. I would add a figure in order to explain the interactions of Hamiltonian (1).
3. page 4, line 4 affect -> effect
4. page 4, paragraph Considering a functional... The symbol Ol is repeated twice
5. In the captions of fig 2 and 3 the symbol chi is not typed correctly.
6. Page 6, paragraph As discussed..., "the limit alpha" -> "in the limit alpha"

RESPONSE:

We have updated the manuscript to fix these issues, including a new figure. The changes are reflected in arXiv v3.