Putting a spin on metamaterials: Mechanical incompatibility as magnetic frustration

Dear Editor,

We would like to thank you for handling our manuscript and to express our gratitude for recruiting devoted referees to seriously review it. We are happy to see the very positive comment posted on our paper, and are glad to read that both invited referees in principle recommend the publication of our manuscript in SciPost Physics. We are grateful for their comments and suggestions that have significantly helped to improve the clarity of our manuscript. In the following, we provide point-to-point responses to all of the comments made by the two referees, and detail the modifications that we made to the enclosed revised version of the manuscript, where we highlighted these changes in red. We also used this opportunity of revising the manuscript in order to introduce some further minor edits that do not modify the content of our work.

Sincerely,

Ben Pisanty, Erdal C. Oğuz, Cristiano Nisoli, and Yair Shokef

Response to Anonymous Comment on 2021-02-02

Referee:

What was hard to believe being possible just a decade ago (and with old textbooks saying that this was in fact impossible), is now realized in artificial materials called metamaterials, a class of man-made materials where properties are pre-designed to be very different from those in their natural/classical counterparts. This article describes such mechanical metamaterials. It also defines different types of topological defects, again extending/redefining the concept well beyond the classic one. What is more, however, is that the new effects arising from mechanical incompatibility can be paralleled with and mapped onto the behavior of more standard magnetic systems with intrinsic frustration. This theoretical work is inspired recent experimental realization of mechanical metamaterials, but it also describes new avenues for realizing such unusual materials in systems like liquid crystal elastomers, potentially also combined with director patterning and spatially resolved polymerization (e.g. based on two-photon polymerization could be one way to go). The paper is well written, the schematics and graphs are clear. I can see this work potentially opening the doors to the realizations of various unusual 3D spin ice systems for fundamental research (and they will join all kinds of colloidal, skyrmionic and many other model systems of spin ice pursued recently!), as well as, potentially, pre-programmed mechanical actuators and artificial muscles with complex mechanical responses defined by topological defect networks within the meta materials, as well as through engineering their responses to external stimuli. I have seen some preliminary results of this work at a KITP workshop in Santa Barbara - nice to see that this research is now developed to a very complete and in-depth study that, hopefully, will inspire more developments

Authors:

We thank the referee for his or her very positive evaluation of our work, of its presentation in the paper and of its potential impact on multiple fields in physics.

Response to Anonymous Report 1 on 2021-03-09 Invited Report

Referee:

Strengths

1. The framework presented in this work is general and promising.

Weaknesses

1. Very descriptive paper with a lack of concrete calculations or data to back up their claims. Most quantitative results are to be found in the appendix.

2. Some key concepts are not discussed enough or need clarification.

Report

In this paper, the authors exploit an analogy between frustrated spin systems and the mechanical deformations of metamaterials to shed light on the behavior of the topological defects, and the role they play in the mechanical response of the material. They map the deformations of a metamaterial with a classical system made of Ising spins with FM and AF interactions, resulting in a local topological constraint reminiscent of spin-ice systems (although with a zero-point configurational entropy which is not extensive). Frustration in the spin representation is associated to mechanical incompatibility, providing a useful framework to study deformations of metamaterials. I find the overall work interesting, well presented and worth publication. However, there are several aspects that I think should be clarified before publication.

Authors:

We thank the referee for his or her comments and for finding our work promising, interesting, well presented, and worth publication. We agree that our originally submitted manuscript was quite descriptive, as its main text has primarily focused on presenting the main results, and much of the details, calculations, and explanations appeared only in the appendices. Following the referee’s suggestions, we have extended our manuscript by incorporating details, discussions, and calculations as listed in our point-to-point responses below. All changes made to the manuscript are highlighted in red in the enclosed revised version.

Referee:

1. The authors should explain better how to control the location of the defects by actuating the metamaterial from the boundaries. I believe this is a key point of the paper which remains somehow loose in the main text. The authors could explain more concretely how the material reacts to a perturbation and analyse in more depth a precise situation. Besides, much of the connection between defects and mechanical response is to be found in the appendix B. The latter is not appropriately mentioned in the main text: the appendix is only cited as a place where to look for details of a coarse-grained model, but, to me, it sheds light on the mechanisms controlling the response of the system. I found the results presented in the main text, basically the definition of the mapping deformation<->spins, quite slim and not enough to understand that 'By actuating the metamaterial along two opposing boundaries... we can control the location of the incompatible region'. I overall find it quite unpleasant to go back and forth to the appendix to find the information I need to understand the paper.

Authors:

We believe that one of our key results is the ability to control the location of the stresses by appropriate displacement of the boundary in the presence of topological defects. Importantly, we do not control the location of the defects by actuating the metamaterial from the boundaries. We first design a particular metamaterial with predetermined positions of the topological defects. For this given metamaterial, we investigate its elastic response to specific boundary displacements, where we can control the location of the stresses, while defects do not change their positions. To clarify the relation between defects and mechanics, we first extended the description of the mechanical model in the main text (pages 3-4). Following the referee’s suggestion, we have incorporated substantial parts of Appendix B (Mechanical response model) into the main text: We have added Eq. 2 (energy expression), Eq. 3 (interaction constants), and Fig. 5 (interaction constants and coarse-grained displacements of our model) in the revised manuscript. Once our mechanical model is presented, we explain in greater detail how to actuate the boundaries for a given defected structure to concentrate the stresses in the one or the other part of the system (page 4, right column).

Referee:

2. Related to the latter point: a study of the dynamics of topological defects when actuating the system would be useful to understand the response of the system from the viewpoint of defects. This is a general criticism, but I think the paper is mostly focused on the description of the mapping but does not provide much quantitative results to sustain their claims. At this level, the paper is largely descriptive, and it is only when looking at the appendix that one realizes that some calculations have been carried out..

Authors:

We appreciate the comments of the referee on the dynamics of topological defects upon actuation of the system from the boundaries. The temporal and positional evolution of defects in mechanical metamaterials upon mechanical forcing could indeed be an interesting direction to pursue in future studies. However, unlike in, e.g., liquid crystals or other condensed matter systems, the defects in our system are not mobile, as they are quenched after the mechanical metamaterial is fabricated. In principle, we can design materials with desired location and number of defects. But, once a metamaterial is designed, defects cannot move anymore. Applying a mechanical load from the boundaries can only change the location of stresses and deformations, but not the location of the defects.

Moreover, it is noteworthy that we do not have dynamics in our model. We apply a static mechanical load and compute the static elastic response of the material to this load. This having said, even if we were to change dynamically the boundary forcing (for example an oscillating boundary displacement field), the defects in the metamaterials presented in this manuscript would still not move unless we would allow the internal structure of the underlying material to change during the boundary actuation.

Referee:

3. The spin assignment could be clarified further. For instance, showing the two sublattices A and B. I found it hard to follow how the mapping between a deformation configuration and a spin works in practice, and it is important to be very clear about that from the very beginning, since the rest of the work is based on it.

Authors:

We agree that the spin assignment in the 2D case could have been clarified further. In the revised manuscript, we split the original Figure 1 into two figures, namely Figure 1 and Figure 2: In the new Figure 1 we show the sublattices A and B, based on which we define +1 and -1 spins, and ferromagnetic and antiferromagnetic interactions among those spins. Figure 2 is now solely devoted to presenting the relation of the compatibility with the number of (anti)ferromagnetic bonds. Note that such a spin assignment is not always possible, and what is important (and general) is the assignment of ferromagnetic and antiferromagnetic bonds. Specifically, in the 2D system that we present, a spin assignment is possible, while for the 3D system that we analyze in the manuscript, such an assignment does not exist. For this reason, we decided to focus on the ferromagnetic and antiferromagnetic bond distributions, which are the more fundamental and general objects.

Response to Anonymous Report 2 on 2021-04-09 Invited Report

Referee:

Strengths

An analogy between mechanical metamaterials and frustrated spin systems is developed in this manuscript. The authors present a novel strategy for controlling defects by designing frustrated mechanical metamaterials. This paper is intriguing and timely. The ideas described are interesting and promising.

Weaknesses

Not really a weakness, but I would like to mention an area point of potential improvement. The authors made an effort to make their work understandable without mathematics. This effort is commendable, but the manuscript could be improved by also including a formal version of the arguments currently developed only verbally. Doing so would allow the authors to make some of their results and statements sharper. It would also help the readers less familiar with both mechanical metamaterials and/or gauge theory to understand the results and their connections with existing literature.

Report

Overall, I recommend publication of this work. In the light of the comments above, I would encourage the authors to benefit from the fact that SciPost allows mathematical formula even in the main text to give more technical details. This would make the paper stronger and more accessible. The changes I list under "requested changes" are not necessary changes but hopefully helpful suggestions on presentation and some technical points.

Authors:

We are delighted to read that the referee finds our manuscript intriguing and timely, and the ideas described interesting and promising. We thank the referee for his or her comments, which majorly concerned the compact presentation of our findings in the manuscript, with the technical details being mainly in the appendices. Following the referee’s suggestions, we have rewritten parts of the manuscript, where we have provided a more formal description of our findings, and inserted equations and additional figures wherever necessary. All changes made to the manuscript are highlighted in red in the enclosed revised version, and detailed in the following point-to-point responses to each of the referee’s comments.

Referee:

Requested changes

1) The manuscript alludes to Wilson loops. While this is a well-known object, it would be useful to remind the reader of their definition. More importantly, this would give an opportunity to explain how these show up in the context of mechanical metamaterials, how they are computed, etc. (Usually, Wilson loops are holonomies of a connection. What is the connection here? etc.)

Authors:

Loops in the underlying bond distribution of the mechanical metamaterial are analogous to Wilson loops, known as holonomies of a connection. In the context of mechanical metamaterials, the connection emerges through the introduction of the spin gauge, and it is defined as the product of ferromagnetic (+1) and antiferromagnetic (-1) interactions along a line. As demonstrated by Villain in Ref. [28], a connection along a closed loop in a bond distribution accommodates information about frustration through the parity of the antiferromagnetic interactions. Fradkin et al. [27] further showed that this parity is gauge invariant, i.e. a holonomy of the connection. We thank the referee for pointing out the opportunity to explicitly explain the relation between Wilson loops and our mechanical metamaterials. On page 3 of the revised manuscript we now briefly discuss the analogy between Wilson loops and the loops comprised of bonds in our metamaterials, providing thereby the definition of a Wilson loop, and explaining how it relates to mechanical compatibility. Moreover, in Figure 4(a,b), we now highlight closed loops of different connections with different colors for illustrative purposes. Finally, for readers less familiar with the work by Villain, we have added Appendix B (Topological defects), which explains the notions of mechanical defects using the familiar language of topology, and in which we prove that the parity of antiferromagnetic bonds is shared between homotopic loops.

Referee:

2) Consider the sentence "Note that in this system compatible configurations exhibit holographic order in the soft mode maintained by the displacements of each pair of opposing facets." This is a very captivating sentence, but the physics behind is not completely clear. How is the "holographic order" defined? Why is it holographic? What is the basis for these statements?

Authors:

In both the 2D and the 3D metamaterials presented here, if the metamaterial is compatible, the displacements of surface building blocks entirely determine displacements in the bulk of the structure. To be precise, if one deforms one boundary, this deformation propagates in the soft mode through the interior of the structure to the opposing boundary in an alternating manner. This is because each pair of opposing facets of each building block deforms in opposite directions. This unusual relation between the surface and the bulk behavior is referred to as ‘holographic order’ both in the present manuscript and in previous work on the 3D systems, Ref. [10].
In the revised manuscript, we illustrate the holographic order in Figure 3(a) using a lattice of hexagonal building blocks, where the displacements along the (2L-1) directions a, b, and c of the yellow surface hexagons dictates the displacements in the whole interior of the lattice. We have further added a set of equations on page 3 (Eq. 1) describing the displacements along the principal axes in the hexagonal metamaterial shown in Fig. 3(a).

Referee:

3) The discussion on mechanical defects is also very interesting. It is not very clear to me how the defects are controlled from the boundary in Fig. 2. A more detailed discussion of the control parameters and what control they allow would be welcome. I suggest to show explicitly the statement: "Locally rotating building blocks changes the number of defects by an even amount, suggesting that in our metamaterials the parity of the defects is the topologically protected quality".

Authors:

The defects are not controlled from the boundary. We design a particular metamaterial with a predetermined number and locations of defects. And, once a metamaterial is designed and fabricated, defects are basically “frozen”. What we can control from the boundary is, however, the location of stresses and strains in the presence of defects. The stress and strain steering is achieved by the specific choice of displacements on the boundaries, as indicated by the red arrows at the right and left boundaries in what is now Figure 4 (c-f). We have added two paragraphs to page 4, left column, to discuss in greater detail how we can control the stresses and deformations in the presence of defects and from the boundary.

Lastly, we have extended our discussion on the parity of defects. In the revised manuscript, Appendix B contains now a detailed description of the topological aspects of the defects. Specifically, in the new Figure 10, we show that locally rotating a single hexagon can only alter the number of defects by an even number, and as such, it does not change the defect parity. We further show that this claim is independent of the soft deformation mode describing the building block and that it is a result of the Z_2 topological charge of the defects. In a system with topological charges in Z_2, the defect parity is the topologically stable quantity.

Referee:

On this point, I would like to ask: shouldn't it still be possible to write down a topological charge, that would have value in Z_2? If not, what prevents one from doing that? Defects with Z_2 charges can indeed occur. This is the case in 3D nematics or superfluid 3-He, in which \pi_1 is Z_2 (see for instance (5.12) in V.B.3 in Mermin's review on the topological theory of defects). The review of Alexander, Chen, Matsumoto, and Kamien in RMP gives a nice picture of how to transform a +1/2 disclination line into a -1/2 disclination line in 3D (see Fig. 4). Can the authors harness ideas from these sources to make their point more precise?

Authors:

We thank the referee for suggesting to include a more in-depth discussion of the mechanical defects in our system using the familiar language and notions of topological defects. In our system, frustration is deduced from the parity of antiferromagnetic bonds, which in turn is induced by the parity of the defects. As the referee points out, these relations indeed imply that the defects can be assigned a topological charge in Z_2. In the revised manuscript, we have added Appendix B, in which we extend our discussion on the topological aspects of the mechanical defects. We show that the parity of the defects is topologically stable and that the defect charge has a Z_2 symmetry. We further show that a similar Z_2 symmetry applies also to defect lines in three-dimensional systems, where the parity of antiferromagnetic bonds along a loop equals to the parity of the winding number around defect lines.

Referee:

4) The statement: "If the arrangement of the hexagons leads to a compatible structure, the ground state of the corresponding unfrustrated Ising model describes the deformations of the soft mode. However, if the system is incompatible, the lowest energy configurations of the corresponding Ising system do not necessarily describe its elastic deformations." is intriguing. It would be very nice to have an explicit map written down, so that the readers can see for themselves when it works and when it fails from the equations.

Authors:

The analogy to spin systems introduced in our manuscript is specifically aimed towards describing and understanding (local) mechanical incompatibilities as (topological) defects. Using this characterization, we uncover the possibilities of unique mechanics such as stress and strain steering. However, the specific details of the mechanical response do not rely on the spin analogy. Instead, it is derived from a vast set of linear equations, which encodes the orientations of all the building blocks in the structure (~3L^2 equations for LxL hexagonal metamaterials, and ~3L^3 equations for LxLxL metacubes). Within this framework, it is unintuitive to compare the solutions of different structures, which are described by different equations.

Instead, it is useful to consider a fundamental difference between the spin and the deformation degrees of freedom. Spin degrees of freedom are discrete, which results in a high energetic cost locally attributed to specific frustrated interaction bonds, whereas (mechanical) deformation degrees of freedom are continuous, which allows for alleviating the stresses and lowering the energy through smearing. Furthermore, frustrated spin systems are often degenerate whereas the mechanical solution to externally applied boundary conditions is unique. For these reasons, the mechanical and spin solutions generally differ from one another, except for the very special case in which the mechanical solution completely coincides with the lowest energy deformation mode, which in turn was used to define the (unfrustrated) spin gauge.

In the revised manuscript, we have added a paragraph on page 4 that explains our statement based on the aforementioned difference between spin and deformation degrees of freedom.

Referee:

5) Minor issues:

• Fig. 2: please add "percentile of the stored energy" below the color bar

Authors:

We thank the referee for identifying this issue. We have added this text to what is now Figure 4 as well as to Figures 7 and 16 that display the corresponding results in 3D.

Referee:

• I don't understand the use of italic in the paper.

Authors:

We removed the use of italic in the manuscript.

Referee:

• What is the meaning of "-" in "(y, z)− plane"?

Authors:

We removed the “-” in such expressions.

Referee:

• Fig. 10: typo "Ref. [1]" instead of "Ref. [10]" in the legend

Authors:

We corrected the reference number in the legend.

Referee:

6) The paper "Topological Elasticity of Nonorientable Ribbons" https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.041058 by Bartolo and Carpentier might interest the authors and perhaps be a useful reference.

Authors:

We thank the referee for introducing us to this intriguing paper. Indeed, there is a striking similarity between elastic non-orientable ribbons and a (single) defect in our metamaterials. Furthermore, the orientability of ribbons, as well as the topological charge of defects in our metamaterials, are both characterized by a Z_2 gauge field. As a result, in both systems, it is possible to capitalize on this topological structure to produce exotic mechanics, such as the steering of stresses and strains, which can also result in nonadditive elasticity, as demonstrated by Bartolo and Carpentier. However, there are also a few significant differences between these two classes of systems. In non-orientable ribbons, the topological signature is evident in the (shear) deformation field already in a stress-free configuration, whereas in our metamaterials it only becomes evident upon applied external constraints or the introduction of the spin gauge. Furthermore, the mechanical response proves slightly different: in our system, the region of vanishing stresses unintuitively coincides with the region of maximal deformations. This is a result of the floppy mode describing the building blocks, which can be regarded as non-local constitutive strain-stress relations. In addition, another interesting difference between these two types of systems is that with non-orientable ribbons one can theoretically acquire a greater charge by winding several times, whereas in our system winding around several defects still preserves a Z_2 charge.

We found this comparison of great interest, and in the revised manuscript we have added a discussion on the connection between the two cases at the end of Section IV.