Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet

Reply to referees:

We are grateful to both referees for taking the time to carefully study the manuscript, and give responses below to their specific recommendations and comments.

The first point to make is that we have substantially reorganised the manuscript in response to the referees comments. The motivation for this was to try and find a balance between the comment of referee 2 who considered the original manuscript to be overlong, and the comments of both referees praising the thoroughness and systematic nature of the presentation. In particular, a large amount of information has been moved into appendices. The main text is now focused almost exclusively on the dipolar model, and as such has been reduced from 53 pages to 24 (we note that this is in the relatively sparse scipost style - in a denser double column format the main text would be roughly 15 pages). We hope that this will make it more accessible, in particular to scientists performing experiments on artificial frustrated systems. Studies of related toy models, which we think provide useful physical insights into various aspects of the dipolar model, as well as triangular-lattice Ising antiferromagnets in general, are now presented as appendices. These are referred to in the main text whenever we think the study of a toy model can lead to a deeper understanding, but we have nevertheless tried to design the main text so that it can be read without it being necessary to refer to the appendices.

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Response to referee 1
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“1. I would like the authors to comment more explicitly in the manuscript about the peculiar behavior of the stripe order across the phase transition. In the limit where defects are forbidden, the stripe order apparently disappears discontinuously across the transition, as in Figure 8(a), even for a continuous transition. Does this feature survive in the full model? It seems to be possible only because the staggered magnetization is a nonlocal quantity when written in terms of the strings.”
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This is an important point. As the referee states, the stripe order disappears discontinuously if the density of defect triangles (triangles with three spins aligned) is exactly zero. This is exactly as would be expected for a Kasteleyn transition, which is driven by the appearance of non-local defects (strings) that wind the system.

In a strict sense this behaviour doesn’t survive in the dipolar TLIAF, since the density of defect triangles is never exactly zero at the transition. This means that close to the transition, the critical behaviour is in the 2D Ising universality class and is driven by the appearance of local defects.

However, a point that we try to make in the manuscript is that the temperature window for Ising criticality is exponentially suppressed at low temperature, and so in practice the Ising temperature window can be negligible (see for example analysis of the transition at \delta=0.05 in the new Fig. 8b). When this is the case, the transition appears to all intents and purposes to be of the Kasteleyn type, with a discontinuous jump in the stripe order parameter, but a continuous increase in the string density.

We have rewritten the manuscript to try and make this point clearer - see in particular Section 5.1.

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“2. I have some concerns about the interpretation of the results of perturbation theory on p. 33. First of all, is it not possible to continue the expansion in Eq. (74) beyond first-order, and see whether higher-order terms are indeed small? Separately, is there any reason to believe that first-order perturbation theory will be accurate for other quantities, besides the critical temperature? The latter is special, in that it depends only on single-string properties, and one might worry that other quantities will not be so accurate at this order.”
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The perturbation theory involves an expansion in the small parameter 1-z_2(T), and this parameter is small for T>>J_2 (i.e. it is a perturbation expansion around the non-interacting limit).

At large dJ the critical temperature is many multiples of J_2, and so the perturbation expansion works well.

Approaching the tricritical point the transition temperature decreases and it is necessary to question whether the perturbation theory remains valid. Monte Carlo simulations reveal a tricritical temperature of T_tri~9J_2, and this translates to a perturbative parameter 1-z_2(T_tri)~0.2. This seems just about small enough for perturbation theory to be reasonable.

As an example the exact 2nd-order critical temperature is given by solving the equation:

z(T_K) = 1/((2-y)(1-2y+y^2))

where y=1-z_2(T_K). It can be checked that a first-order expansion of this equation gives a value of z correct to within about 15% for T_K=9J_2, and that the error is predominantly captured by the 2nd order term in the expansion.

Any other physical quantity that can simply be expanded in 1-z_2 should be computable to reasonable accuracy within (first-order) perturbation theory. The obvious exception is the spin-spin correlation function, for which it is necessary to take a matrix determinant that amplifies any small errors in the matrix elements (i.e. there is not a simple expansion in 1-z_2).

It is worth pointing out that the point of the perturbation expansion is not to provide accurate estimates of physical quantities - for this we use Monte Carlo simulation. Rather the point is to try and get a more physical understanding of the phase diagram by studying the effect of 2nd neighbour interactions on the spectrum \epsilon_k (which does have a simple expansion in terms of 1-z_2) and through it the nature of the string-string interactions and how to write down a phenomenological fermionic model.

We have rewritten the manuscript to try and make this clearer.

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“3. Can Eq. (81) be extended to include the effect of defect triangles considered in Section 5.2? This would allow the crossover between PT and Ising critical behaviors to be quantified. I believe it is also possible to write down scaling forms for this crossover, analogous to Eq. (83), including the Boltzmann weight of defect triangles as an additional relevant scaling variable [see PRB 87, 064414 (2013)].”
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We think this is a good suggestion, and have used scaling to analyse the crossover behaviour for both the dipolar model (section 5.1) and the nearest-neighbour TLIAF (Appendix D.4).

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“4. In Section 5.4, a free-fermion model is constructed that has a flat dispersion, which reproduces the behavior of the string density across the first-order transition. It is not completely clear how this should be interpreted. I agree that this is the only way to get a discontinuity in the string density from noninteracting fermions, but does it actually provide a useful description of the transition? In particular, does it make any nontrivial predictions for properties near the first-order transition?”
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The motivation for discussing this was to try and understand how the first-order transition can be understood within the same framework as the second-order and tricritical transitions.

We think the way to understand the flattening of the dispersion is as the effect of strong fermion-fermion interactions within a system that doesn’t allow for fermion pair creation or annihilation (i.e. in the approximation that there are no defect triangles), and so retains a sharp quasiparticle dispersion. This makes it clear that the first-order transition is driven by fermion/string interactions.

While we think that it is intellectually satisfying to explain all the transitions within the same framework, we agree with the referee that writing down a phenomenological model with these features built in by hand doesn’t make any useful predictions. As a consequence we have considerably shortened the discussion.

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“1-If the points in Figure 1(b) are from MC, this should be stated in the caption.”
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They are and this has been done.

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“2-Do the nano-magnets referred to on page 4 have their moments oriented perpendicular to the plane? This should be clarified—the physics is presumably quite different if the moments are in the plane.”
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Yes, the moments are perpendicular to the plane, and we have now made this clear.

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“3-In Section 3.1, is it possible to give some typical acceptance ratios for the MC algorithm?”
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We have added a paragraph to the methods section that reads:

“For the isotropic, dipolar TLIAF just above to the phase transition, an acceptance probability of about 0.035 was found for the worm updates, which dropped to about 0.004 on crossing the transition, before continuing to decrease. By running a dense set of temperatures, parallel tempering steps were accepted with a probability of at least 0.8 across the transition, providing a considerable aid to equilibration.”

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“4-In Section 3.3, it would be worth showing an example energy or winding-number histogram to illustrate the change to a continuous transition.”
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This has been included as the new Fig. 6b

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“5-Toeplitz is spelled incorrectly on p. 18.”
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This has been corrected.

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“6-After Eq. (94), I think the value quoted for ζ is actually ν.”
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The referee is correct and we have corrected the manuscript.

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“7-In Figure 25, error bars should be added to help judge the quality of the data collapse.”
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Error bars were already present, but smaller than the point size. We have added a comment to the caption to make this clear.

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Response to referee 2
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“To my taste, it is disappointing that the height model [H. W. J. Bloete and H. J. Hilhorst, J. Phys. A 15, L631 (1982)] is not one of the mappings that the authors use, since I think that it provides the clearest representation of the nearest neighbor model, which stands at the center of the discussion.”

[6] Page 37: the discussion here would be much better if the authors had used a mapping to a height model. Using strings as they do, three symmetry-related phases appear quite different. In a height model, they would be tilted phases with different tilt directions.

[8] Page 51: I disagree with the statement in paragraph 2 of Sec 7: "... the most intuitive way to understand such models is by considering the strong degrees of freedom ...". I think a height representation would be better.
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We agree that the height model is the model of choice for the nearest-neighbour TLIAF, while the string mapping is slightly clumsy in that it requires a special spatial direction to be chosen arbitrarily. However, we have thought about how the height model can be modified in order to explain the behaviour of the dipolar TLIAF (with or without anisotropy), and think that, for the questions of interest in the dipolar system, the string mapping is more intuitive. We have added a comment to this effect at the start of Section 2.2. However, if the referee thinks it would be enlightening for readers, it would be possible to add another appendix discussing the height model description of the phase diagram.

The main points of interest in the dipolar TLIAF are the phase transition and the correlations in the spin-liquid region. For the phase transition the string mapping allows the first/tricritical/second-order nature of the transition to be understood in terms of essentially two quantities, the internal free energy of a string, which controls the transition temperature, and the sign of the interaction between the strings, which controls its nature. An attractive string-string interaction leads to a first-order transition, a repulsive interaction to a second-order transition and when the interactions cancel there is a tricritical point. To us, this is a relatively intuitive picture, that can be understood without necessarily going into the mathematical details. However, it also allows a connection to be made to the Pokrovsky-Talapov analysis of critical behaviour in the vicinity of a Kasteleyn transition, which was performed in terms of 1D fermions (i.e. strings).

The height-model description of the transition requires one to analyse the interplay of an anisotropy parameter appearing in the quadratic term and the cubic interaction, all in the presence of a stabilising quartic interaction (see the Supplementary information to Ref. 11 for an analysis of this model in a closely related situation). While it is of course possible to mathematically analyse the position and nature of the transition within such a height model, we don’t find the picture so intuitive.

In the case of correlations in the spin-liquid region for isotropic dipolar interactions, the power of the string mapping is that it provides a simple picture for the crossover between the domain-wall network state and the string Luttinger liquid, in terms of a change of sign of the string-string interactions. We think this more than compensates for the fact that the domain-wall network region looks simpler/more-symmetric in the height-model picture (regions of different uniform slope) as opposed to the string picture (clumping of strings).

In the spin-liquid region, the string picture is also very natural as soon as the dipolar interactions are anisotropic and so themselves select a special direction. For example for the spin-liquid region just above a second-order transition, there is a natural description in terms of a low density of fluctuating strings, whose density is controlled by the competition between their internal (free) energy and their repulsive interactions.

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[1] Eq 9 omits entropic effects, which are important at low temperature and become clear if one views the defects as vortices in a height model. This should be acknowledged and ideally corrected.

[9] Eq 110 misses the relevant physics, at least for the nearest neighbor model, because it omits the dependence of degeneracy on the number and location of defects.
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We believe that the derivation of the density of defect triangles presented in Appendix A in the original manuscript is consistent with the nearest-neighbour model, but that we were not careful enough in taking the low-temperature limit.

The density of defect triangles in the isotropic, nearest-neighbour TLIAF is relatively simple to calculate in integral form. Expansion in terms of z_def = exp(-E_def/T) results in the density of defect triangles being given by n_def ~ z_def^2+O(z_def^4). This can be seen to match the expansion of n_def from our simple argument.

However, the coefficients in front of the z_def^2, z_def^4 etc. terms depend on N (in both the exact solution to the nearest-neighbour model and our simple calculation), and so care needs to be taken whether the expansion can be truncated. When N->infinity while z_def remains finite, the expansion cannot be truncated, but instead our original way of taking the thermodynamic limit applies.

For the analysis of finite size simulations, we do not take a limit, but instead work with the full expansion.

We have rewritten Appendix A to try and better explain these points.

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[2] It seems surprising if the content in Sec 4.2.2 is not already available in the literature. Have the authors checked this carefully?
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We have found a paper in the literature that uses the related technique of q-deformed Grassmanns to solve the TLIAF, and have added a citation. We still feel that this section is worth having as an Appendix, since the techniques we use are slightly different, and are important for later developments in the paper.

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[3] Page 25: the authors write "for T<<J_A then to all practical purposes xi-> infinity". It would be better to indicate the functional form of the divergence of xi with 1/T.
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The functional form has been added.

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[4] End of page 25 and top of page 26: the issues referred to here were well-known long before Ref 51 - see e.g Schultz, Mattis and Lieb, Rev Mod Phys 36, 856 (1964). Similar comments apply to the last paragraph of Sec 7.
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The discussion referred to has been removed in the new version of the manuscript, since we decided it was not relevant for the problem we are trying to solve.

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[5] End of page 33: an extensive discussion is given of an approximate dispersion relation, but I am unclear what the significance of the dispersion relation is, beyond the Hartree Fock approximation.
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The purpose of discussing the dispersion relation is to get some insight into how it behaves at the crossover between first and second order transitions. This can then be used to guide the phenomenological theory we use to analyse the dipolar model.

Monte Carlo simulations reveal that the transition temperature in this region is approximately T=9J_2, which corresponds to a perturbative expansion parameter of 1-z_2=0.2. This is small enough that the use of first-order perturbation theory seems reasonably well motivated.

The insight from perturbation theory is that the quadratic term in the dispersion disappears at the tricritical point, and one must instead consider the quartic term. This defines what we refer to as a Pokrovsky-Talapov tricritical point, in analogy with the Pokrovsky-Talapov critical point that occurs for higher anisotropy values.

The relevant section has been rewritten to try and make this clearer.

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[7] Eq 94. The argument of the function g_{tri} appears to me to have a misprint in it.
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We believe that the argument was correct but that there was a misprint in the text below the equation. This has been corrected.

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[10] What is the relation of appendix B to the work of Stephenson [Refs 4 and 5]? Is the appendix a rederivation?
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For the isotropic models the appendix is simply a rederivation of Stephenson’s results using the Grassmann language. The point of presenting this is that the analysis carries over trivially to the anisotropic models that are a focus of our manuscript. We have rewritten the first paragraph of the Appendix to try and make this clear.

The manuscript has been substantially reorganised, as set out in the response to the referees.