Many Phases of Generalized 3D Instanton Crystals

We thank the referee for report and for the useful comments. Our response:

1/ This was not explained well in the first version of the manuscript and we thank the referee for pointing this out. Indeed also four dimensional crystals are possible. However at low density, one expects that the solitons are confined to a plane in the holographic direction so that the only relevant dimensions are the spatial ones. As the density increases, we expect a "popcorn" transition from three dimensional to four dimensional crystals, similar to the one to two dimensional transition considered, for example, in Ref. [15]. Studying the four dimensional crystals and the transition to them would be an interesting (but extremely challenging) topic for future research. There is a parametrically large region of densities where the three dimensional crystals can be embedded in the Witten-Sakai-Sugimoto setup. We have clarified the setup by adding comments on the 3D to 4D transition in the introduction on page 3 before Eq. (1) and in the beginning of Sec. 2.

2/ Indeed due to the long range sensitivity of the two-body interaction, we add an additional external force in our simulations. In the probably most interesting cases (i.e. the non-Abelian crystals) the external force is however well-defined: there is only a single choice which makes sense. This choice is the external force due to an unoriented continuum of instantons outside the cavity of the simulation. We stress that this is not an ad-hoc force but added because of the setup that we want to consider: in principle we want to consider an infinite sample, but due to limited resources, we can only simulate a relatively small bubble. The external force is then the mean field force due to all those instantons which were left out of the simulation due to this limitation. For the non-Abelian (and also for the global ferromagnetic) crystals this external force also leads to crystals having constant instanton density (approximated over a large enough volume), and we see no other reasonable choice for the force in this case.

For the oriented (spherical antiferromagnetic or ferromagnetic) phases, the situation is a bit more complicated. This is because the force which leads to constant instanton density is not the same as that obtained by averaging over an (anti)ferromagnetic continuum of instantons outside the cavity of simulations. Here we have chosen the external force such that the averaged instanton density is constant. This makes it possible to carry out the IR analysis of section 4. However we have also tried simulating with other choices of the eternal forces (these attempts are not documented in detail in the article for brevity), and did not see any significant changes in the results for the crystal structures locally. Globally, there were some changes, as the density of the instanton varies are one easily develops even empty regions in the middle of the cavity.

Notice that one can also do simulation without external force, even though this is presumably less well physically motivated. In this case one some of the solitons first cluster at the surface of the simulation volume, and then one obtains crystal formation in the middle. Even though we have not studied such configurations in detail, we expect that the crystal structures would be the same even in this case as with the simulations with the additional external force.

In summary, we do not expect that any of our main results are sensitive to the choices for the external force.

We have improved the discussion of the external force in the introduction (page 4, the first item (i) of the list and discussion before it), in the first paragraph of Sec. 3, and at the end of Sec 3.4. We now point out more clearly that the external force for the non-Abelian case appears to be essentially unique, and have added references to Appendix A.2, where a detailed discussion of the forces is given.

For the authors, Matti Jarvinen

We thank the referee for careful reading of the manuscript and for the useful comments, and apologize for the slow response.

We will clarify the discussion of the numerical details in the second version of the manuscript.


Our response to the specific issues:


(1) The two body potential is motivated by the analysis in the Witten-Sakai-Sugimoto model. In this context there is indeed a parametrically large range where our expression for the potential is valid. This requires that the separation $d$ between the instantons satisfies $ 1/(M\sqrt{\lambda}) \ll d \ll 1/M$ where $\lambda$ is the coupling constant and $M$ is the scale of strong interactions. The lower limit arises from the instanton size and the upper limit from neglecting curvature corrections in the AdS geometry. In the strong coupling limit $\lambda \to \infty$, where the classical dual description is valid, these two scales are therefore well separated. This is discussed in detail in ref. [15] of the manuscript.

We notice that this point is not explained in the current version of the manuscript. We suggest to add an explanation in section 2.2 where the two-body potential is introduced.


(2) We agree that there should be more discussion of the interpretation of the parameters. Basic properties of the potential can be read from the two-body interaction term, eq. (8). The $\alpha$ coefficient multiplies a term where the dependence on orientation is independent of the spatial interactions. For positive (negative) $\alpha$ perpendicular (parallel) spins of nearby neighbors are preferred, and the effect increases with increasing $|\alpha|$. The interaction involving $\beta$ is a bit more complicated since there is nontrivial coupling between the orientations and directions in coordinate space. Picking an instanton with unit orientation $y=1$, positive (negative) $\beta$ means that orientation $y$ of a neighboring instanton which is perpendicular (parallel) to the spatial link between the two instantons is preferred (with the understanding that e.g. the orientation $y=i$ is parallel to the $x$ axis in coordinate space).

There are some scattered comments in sections 3 and 4 about the interactions but we plan to add a more detailed discussion in section 2.2. We will also comment how the final results for the phase diagram reflect the structure of the interaction term in section 5.


(3) The presence of the spherical phases is indeed a nontrivial question. We have carried out various checks (see below) some of which are mentioned in the manuscript. Based on them, we are confident that the spherical configurations are dominant over "regular" crystals in the regions indicated in figure 8. In particular, our analytic findings (which are independent of the boundary effects) agree to a great degree with the simulation results.

However it might happen that the true ground state in the continuum limit is some other, even more complicated structure (which we do not see in our simulations due to finite size/boundary effects) -- this we do not know. But even in with the largest volumes we could with relative ease simulate on a personal computer (about 40000 instantons), we did not see signs of any other structures. In addition we have checked that the spherical antiferromagnetic structure is insensitive to the shape of the cavity where we carry out the simulations. We have tried simulating i.e. in a cubic cavity and still found configurations which are clearly spherical near the center of the simulation result.

The strong tendency to form spherical structures for $\beta>\alpha>0$ seen in the simulations (the simulations converge to the final state more than 10 times faster than for $\alpha>\beta>0$) may mean that such configurations have a large energy gap with respect to regular crystals. This is in part confirmed by the IR analysis of the configurations.

We suggest to add more discussion of these points in section 3.4.


We also thank the referee for pointing out the issue with Preview. However, we did not observe issues in our own tests with this viewer, so we cannot fix the problem. (The 3D content in the figures is not available on Preview, and to our knowledge it only works with Adobe Reader.)


For the authors,
Matti Jarvinen