Diagnosing weakly first-order phase transitions by coupling to order parameters

We thank the referees for their overall positive assessment of our paper and for recommending it for publication in SciPost Physics after minor revision. We are also grateful for the detailed referees comments, which we will clarify below, point by point.

Response to Report 1:

The method of the authors assumes that the Hamiltonian is already fine-tuned to the critical point (as clearly stated in the paper). However, in many practical cases, the exact location of the transition is not necessary known a priori. Hence, I was wondering what happens if the method is applied slightly away from the critical point, to either side of the transition. My expectation would be: in the symmetry broken phase, $\langle O \rangle$ would approach a finite value as $\lambda \to 0$, such that a continuous transition might show similar signatures as a first-order transition - do you agree in that respect? If my intuition is right, I think it might be beneficial for future application to caution the reader and discuss this in the paper or at least an appendix.

Indeed our method requires a prior value for the critical point, although we found that locating it (e.g. by using binder ratio crossings as we did for the rectangular J-Q model) to the required precision was rather easy in this case. It is true that one may worry about being detuned from the transition and falsely claiming a first-order transition, when really it is just the ordered phase. To address this we have included such a study for the Q-state Potts model in the Appendix C with a new figure 4, which helps to show the critical and first-order behavior in a larger parameter space near the transition. One notices that the running exponents shows a qualitatively different behaviour in the parameter plane: a fanlike structure in the continuous case, and a horizontal rod or bar-like structure in the first order case. We believe that more generally it is a good idea to consider several values of the transition as a function of the field before diagnosing a first order transition, although we note that such a subtlety is not present in the J-Q case, since we have ordered phases on both sides.

Do you see a way to apply the method to transitions from/to topological phases? It seems difficult if there is no local order parameter, or am I missing something here?

We have not explored an application to topological phase transition, where there is no local order parameter on either side of the transition. It is likely that our ideas are not applicable for such transitions.

Discuss what happens if the method is applied slightly away from the critical point, see report. 
 Done as detailed above. Add "University of Waterloo" and/or URI to Ref. 59. Thanks for pointing this out, we have updated the corresponding reference.

Response to Report 2:

I would not call it a weakness, but I am not sure how this method can be employed in cases when the critical point is not known exactly. And I could not find this point addressed by the authors, though I think it would be beneficial to do so.

Indeed our method requires a prior value for the critical point, although we found that locating it (e.g. by using binder ratio crossings as we did for the rectangular J-Q model) to the required precision was rather easy in this case. It is true that one may worry about being detuned from the transition and falsely claiming a first-order transition, when really it is just the ordered phase. To address this we have included such a study for the Q-state Potts model in the Appendix C with a new figure 4, which helps to show the critical and first-order behavior in a larger parameter space near the transition. One notices that the running exponents shows a qualitatively different behaviour in the parameter plane: a fanlike structure in the continuous case, and a horizontal rod or bar-like structure in the first order case. We believe that more generally it is a good idea to consider several values of the transition as a function of the field before diagnosing a first order transition, although we note that such a subtlety is not present in the J-Q case, since we have ordered phases on both sides.

Response to Report 3:

It is argued that using external fields is more efficient than the symmetry-preserving method. It might be helpful to show corresponding (inconclusive) data from the symmetric approach if possible. For the Q-state Potts model and the O(3) transition, probably both methods would work equally well. The case of the J-Q models seems more intricate. While the data in Fig. 3 is rather suggestive in favor of a first order transition it might still be possible for the running exponent to converge to a nonzero value.

We remark that while the literature on symmetry preserving studies of these models is extensive, it is true that we do not provide a side-by-side comparison of both the symmetric and perturbed approaches. However, we find such a comparison to be beyond the scope of that which we hope to accomplish here. In the symmetric J-Q setup, it is already well known that the critical exponents show a drift with system size, which have been interpreted as unconventional corrections to scaling. It is our view that the perturbed setup produces a visibly more striking effect that is more easily interpretable as a first order transition, at smaller system sizes than the symmetric simulations.

We agree that it is impossible to exclude nonzero convergence of the running exponents, but this argument is always present in any extrapolation scheme, and it is based on the appearance of a distinct scaling form emerging at inaccessible system sizes.
We do not detect any deviation away from a steady drift of the exponents toward zero on the (rather large) system sizes that we have studied. Given the similarity with the Potts model, where the first order nature is known analytically, we believe a saturation or upturn at much smaller perturbation scales would constitute new, unexpected physics. While we can not rule it out, there is also no mechanism known to us which could give rise to such a saturation or upturn. So the simplest explanation of the observed behaviour remains a first order transition.

The simulations used previously determined values for transition points. If these were not very precise and actually lying in the symmetry broken phase of a continuous transition, wouldn't the running exponent look as if the transition was first-order?

Indeed our method requires a prior value for the critical point, although we found that locating it (e.g. by using binder ratio crossings as we did for the rectangular J-Q model) to the required precision was rather easy in this case. It is true that one may worry about being detuned from the transition and falsely claiming a first-order transition, when really it is just the ordered phase. To address this we have included such a study for the Q-state Potts model in the Appendix C with a new figure 4, which helps to show the critical and first-order behavior in a larger parameter space near the transition. One notices that the running exponents shows a qualitatively different behaviour in the parameter plane: a fanlike structure in the continuous case, and a horizontal rod or bar-like structure in the first order case. We believe that more generally it is a good idea to consider several values of the transition as a function of the field before diagnosing a first order transition, although we note that such a subtlety is not present in the J-Q case, since we have ordered phases on both sides.

It is argued that double limits with respect to system size and the external field are no major concern. Is this still true for Monte Carlo and tensor network state simulations with limited accuracy? Typically tensor network state simulations for the relevant two-dimensional models are limited to rather small bond dimensions and biased by choosing a certain elementary cell of tensors. It seems that the resulting errors would pose a challenge for the proper double limit.

It is hard for us to predict the behaviour or our method when used with Monte Carlo results of poor accuracy. Most likely results are not conclusive and one should improve the accuracy of the Monte Carlo results. Regarding the tensor network simulations, there is a recent proof of principle application of the method proposed in the current manscript in section III.C of the paper: J. Hasik et al., Phys. Rev. B 106, 125154 (2022). In there the authors studied a different microscopic implementation of the O(3) quantum phase transition and compared the iPEPS results to finite size QMC results. Indeed one has to carefully check that the results are converged in bond dimension D for a given value of the perturbation, but for intermediate to large values of the perturbation the scheme worked well in that application. Comparing these converged values of the perturbation with Fig. 3 of our manuscript, an iPEPS study of the JQ model might be within reach.

the use of external fields is basically a textbook approach for the study of phase transitions that has been applied before

We agree (and also state in the manuscript) that the approach as such is text book material. We stress however that it has been overlooked that this approach has an unappreciated strength in detecting weakly first order transition, and we use our finding to rule the transition in the J-Q model to be first order.

the distinction between continuous and weakly first-order is of theoretical but relatively limited experimental or technological interest

Of course the technological prospects of our work is quite limited indeed, but Neel-VBS transitions on the Shastry Sutherland lattice are currently being discussed in the context of experiments on Strontium Copper Borate (SCBO).

the addressed models are rather well-known

We find this adds to the importance of our work, since the nature of this well studied transition was still disputed in the community before our work.

Minor adjustments were made to formatting and wording throughout the document.

We added emphasis on locating transition point of the rectangular J-Q model in section 5 on page 10.

We added appendix C “Detuning from phase transition” to address a point made by alle 3 referees. This section includes a new figure, Fig. 4, which illustrates the difference in behaviour for continuous and first order phase transitions upon detuning from the phase transition in the case of the Q-state Potts model.