Anti-Holomorphic Modes in Vortex Lattices

The paper addresses a problem of current interest in the field concerning the emergence of edge modes in chiral vortex systems. It has the advantage of complementing related previous works that present continuum hydrodynamic theories by directly analysing the form of these edge modes in a specific point vortex system. This helps uncover how the predictions of the the continuum theory in the linearised approximation emerge from the underlying discrete model. A specific example is presented and excellent agreement is obtained between the numerical solutions and analytical results that are derived.

More analysis of the results could be presented in places to strengthen some of the conclusions and reinforce some of points being made. There are also a few minor points that need further clarification.

The paper addresses a problem of current interest that is related to the emergence of edge waves in a chiral vortex system. The approach is based on a linearised analysis of a rotating point vortex system where the base state corresponds to Tkachenko vortex solutions. Results are presented that recover a continuum theory from the point vortex model in the linearised approximation.

Two methods are used to recover the continuum equation and are shown to lead to the same result. The resulting linearised equation is then solved under further approximations by considering length scales sufficiently large to the intervortex separation and assuming that the perturbed vortices maintain circular paths. From this, an analytical expression for the radial dependence of the edge modes is recovered and is compared against numerical solutions of the modes representing the solutions of the continuum theory.

The paper ends with some interesting observations concerning the topological properties of these solutions.

I generally found the paper very well written and for the most part addresses the problem in a relatively clear manner despite the technical aspects of some of the derivations presented. However, I have some comments that I would like the authors to consider before recommending publication.

Main points:

1. As pointed out in the paper, edge waves in vortex systems have been discussed in hydrodynamic theories of vortex systems (e.g. [25],[26],[34]). However, these works recover fully non-linear equations (such as the Benjamin-Davis-Ono equation) that describe the edge waves. It would be helpful to the reader to place the current results in context with previously established results from the literature to better understand in what limiting regimes the two sets of results (those presented elsewhere and those derived in the present work) relate to one another.

2. Is there any scope to extend some of the analysis in order to compare solutions of the fully non-linear point vortex model to the solutions presented of the linearised theory to see to what extent there is agreement between the two. This would help clarify how small the amplitude of the perturbation needs to be and on what time-scales breakdown may occur, considerations that would be useful to make particularly if one seeks to realize these modes in a real physical system.

3. For a system with a smaller number of vortices how does the continuum theory breakdown? Would it be possible to include a case with a relatively smaller number of vortices to give some indication of how big the system needs to be to clearly observe these edge modes? Indeed, one of the advantages of the results presented in this work over previous works recovering hydrodynamic theories is that there is a clear connection made between the point vortex system and the emergent edge modes. It would be nice to establish whether the predictions for these edge modes persist for systems with a relatively small number of vortices which would then be within experimental reach, for example in some superfluid systems.

Minor points:

4. Above Eq.(50), it is assumed that the perturbed vortex trajectories can be assumed to be circular. Although the assumption sounds plausible, how can this be better justified?

5. It is stated that the solutions obtained in Eq. (62) correspond to neglecting the term involving the second derivative with respect to z. What is the physical effect of this term on the solutions obtained for higher anti-holomorphic modes where this term would become more important?

6. Phrasing of sentence following Eq. (63) is a little unclear.

7. It is stated that the red curve (rather than red line) shown in Fig. 2 is obtained from the continuous theory and does now contain the edge modes. I was unsure what equation was used to recover this curve so it would be useful to state this explicitly.

1) The paper tackles an interesting and long-standing question and provides new insights into the nature of edge modes in incompressible superfluid vortex crystals.
2) The coarse-grained continuum theory and numerics give consistent results according to the authors.

1) Some aspects of the paper are unclear, see below.

I find the paper very interesting. However, before I can provide my recommendation regarding publication, I would like the authors to address the following comments:

-In Introduction the authors state:"In the laboratory frame these modes are zero-frequency, coupled with the overall rotation of the lattice leads to a time dependent deformation structure rotating in the same direction as the lattice, but at a slower frequency." I do not understand this sentence and encourage the authors to clarify it. Why the frequency is slower?

-In the end of "Introduction" the authors suggest that their edge modes might be related to the Kohn mode discussed previously in Refs [17,42] in vortex crystals. As far as I understand, the Kohn theorem is valid also in infinite systems without any boundary, so why do the authors see any connection to their edge excitations?

- In Fig 2, what is meant by "(totalling 331)"?

- After Eq. (22) what are g_2 and g_3?

- If for the triangular lattice $\alpha=0$, why the function $f(z)$ is introduced at all before Eq. (26)?

- This might be simple, but I do not understand why Eq. (48) is valid.

- In the section "Anti-holomorphic modes" the authors start from the ansatz (53). How come that the expansion coefficients depend on $|z|^2$? Does not that contradict the recursion relation (56)?

- In Eq. (63) it appears there is a typo s->i.

- Could the authors provide some details on the linearized solution of the microscopic model discussed in "Numerics"?

-This is beyond the scope of the paper, but can the bulk-boundary correspondence from the end of the paper be extended to the compressible superfluids, where the Tkachenko mode is quadratic?

Excitations on the edge of the vortex matter are an important problem. It holds keys to fundamental problems of melting quantum crystals and understanding edge states in the fractional quantum Hall effect. The paper poses and addresses a timely problem.

This being said, I think the paper needs some improvements/ clarifications. Below I list a few points whose clarification will help me to develop an opinion recommendation about the paper.

1) The main coarse grain approximation is the old and well-known problem of `contour dynamics’. see e.g.D. G. Dritschel, The repeated filamentation of two-dimensional vorticity interfaces J. Fluid Mech, 194, 511 (1988);
I understand that it corresponds to the first two terms in the rho of (20).

It is known that contour dynamics develop filament instabilities. I believe the authors must include some discussion on this issue.

2) I understand that the last term in (20) is related to the lattice structure. This term would be absent if the vortices form a liquid rather than crystal. Is it correct?
Is this term sensitive to Tkachenko's moduli of the lattice? For example, how does this term depend on the lattice, say, square or triangular?

3) Does this term stabilize the development of filaments?

4) In the case of the liquid (the first two terms in the rho of 20) the linear edge modes had been extensively studied in papers by Crowdy. A discussion of a relationship with these works will be helpful;

5) It will be desirable to formulate the equation for the edge mode as an equation in terms of the real coordinate along the edge. And when it is done what is the relation between the last term in 20 and the
dispersive term of the Benjamin-Ono equation discussed in [34].

See above