Topological mechanics without the topology: A universal, model-free approach to mechanical response

Over two millennia ago, Archimedes recorded rules for balancing the forces on a lever and boasted that this principle, along with a lever and a place to stand, could move the world. In a sense he was right: New mechanical devices have taken us to new places, from rockets that have let a few human beings walk on the moon to scalpels that have excised disease and given many human beings new life. Increasingly, devices can manipulate solid objects at the atomic scale. The challenge in all cases is directing force and motion as desired, and thereby shaping the world in useful and beneficial ways. Guzman et al. (1) introduce crucial new tools and language for quantifying these fundamental mechanical properties. This gives us a new way of telling whether we have the right lever and the right place to stand.

At first glance, this has little to do with the revolution in quantum mechanics and materials synthesis that has given rise to diverse new electronic states. Many such states arose within the Landau theory of phase transitions, as when spontaneous breaking of symmetry gives rise to superconductivity. However, a new paradigm that eventually came to be known as topological insulators (2) arose beginning with the Integer Quantum Hall Effect (3) in 1980. Topological properties are those which remain constant when a system undergoes certain deformations: just as it is impossible to remove a knot from a closed loop without cutting the loop itself. Because of this property, the states are quite robust and precise, to the point that condensed matter, long regarded as a dirty and sample-dependent discipline, was able to provide the von Klitzing constant, which now provides a precise measurement of the fine structure constant of electrical interactions and serves as the basis for resistance in metric units. This effect arises via bulk-boundary correspondence, in which an insulating system’s bulk structure gives rise to a topological number which ensures the existence of conducting electronic modes on its edge (2).

In (4), Kane and Lubensky showed that mechanical systems with equal numbers of degrees of freedom and constraints (one such constraint could be that a bond between two elements remains of fixed length) were governed by equations that could be mapped onto topological electronic systems. In the mechanical context, this meant that a system of rigid elements joined at hinges might be rigid in the bulk, such that deformations necessarily deformed rigid pieces and cost energy, but flexible on the boundaries or at interfaces, such that deformations required only rotations at free hinges. The topological origin of this property ensures its robustness against disorder. This basic phenomenon presents tantalizing promises for new ways of directing force, such as structures that can shift effortlessly between soft and malleable and stiff and sturdy. These critically coordinate lattices, also referred to as Maxwell lattices and closely related to isostatic structures, have in recent years served as the basis for important theoretical and experimental advances (5–7).

However, achieving such tantalizing possibilities requires meeting a number of challenges, ranging from mind-numbingly abstract mathematical analysis to brutally practical experimental synthesis. The simplest story of topological mechanics involves a system that is one-dimensional but allows elements to rotate, that exists in space but cannot undergo translation, that is infinitely long but has a boundary, and that is devoid of disorder or nonlinearity. Once this highly specific system is achieved, periodicity permits a Fourier transform onto the Brillouin Zone, but this one-dimensional Brillouin Zone must be embedded in a complex plane, which contains the boundary modes (7).

In a well-crafted experiment on a $5\times 5\times 5$ unit cell 3D-printed model system (8), it was found that when a topologically soft mode was placed on one surface, it reduced the stiffness of that surface by less than a factor of 2 relative to the opposite surface. Part of the difficulty arises in that a soft hinge is not a free hinge, setting a minimum surface stiffness. Moreover, finite-size effects can be brutal: a surface indentation with a characteristic width of ten (more generally, $n$) unit cells requires a depth of one hundred ($n^2$) unit cells to decay (9), blurring the difference between bulk and boundary in realistic experimental settings.

To bridge this gap and move the field forward, it is vital to develop new, broad, and accessible language that allows different communities to assess topological mechanical systems and compare them fairly to alternative mechanical schemes. This is the elegant contribution of Guzman et al. (1), building on the collaboration’s earlier work (10). The authors note that displacements of point particles and extensions of bonds between them comprise distinct linear spaces, though the displacements determine the extensions (similarly, the tensions in bonds determine the forces on the sites). As has been observed previously (11), in a topologically polarized system even in the bulk applying a force can lead to extensions on only one side of this source. This would seem to violate Maxwell Betti reciprocity (12) but it is indeed permitted because of the distinction between site displacement and bond elongation.

Guzman et al. (1) introduce new expressions for the polarization of a structure as a vector describing the relative position of displacements and extensions in response to a source term. These further develop the notion of Kane–Lubensky systems as being polarized in a manner analogous to electromagnetic systems. Here, site displacements and bond extensions play the role of opposite charges. For an arbitrary system undergoing an arbitrary deformation, one can ask then about the “charge” distribution based on the magnitudes of site displacements and bond extensions. One can similarly calculate the net polarization vector of the entire structure (these are real structures in real space, without any periodic boundary conditions to confuse the notion of position).

This approach liberates the system from the particularities of the topological model. As Guzman et al. (1) show, a domain wall predicted to have a floppy mode (in which the system can be deformed with little or no restoring force) protected by a topological invariant derived from infinite-system reciprocal-space calculations does indeed have such a mode. However, this is detected via a discontinuity in the polarization that can readily be obtained via a relatively mechanical and straightforward calculation without any invocation of topological considerations at all. In this way, it is well suited to a new generation of mechanically polarized structures that contain disorder or structural complexity.

Guzman et al. have developed and applied a new concept for complex mechanical structures.

It may well be possible to translate long-desired mechanical properties, such as the ability to convey deformation toward a target (e.g., energy harvesting, activation), or away from one (e.g., protection, cloaking) into this new language of polarization, or into a generalization thereof. Numerical schemes have already been developed to effectively generate mechanical networks with desired properties (13, 14), and new generations of machine learning may yield even more innovative approaches. Polarization textures may prove an effective metric to use as an objective function, achieving interesting and functional structures in the absence of any underlying theory, topological or otherwise. This raises the possibility of achieving closer connection to the general field of flexible mechanical metamaterials (15), particularly if the approach can be extended to nonlinear deformations.

Even within mechanical metamaterials, there are significant other topological modes, especially those appearing at finite frequency (see, e.g., ref. 16). Schemes exist, similar to the chiral symmetry that Guzman et al. (1) exploit, to classify such topological mechanical metamaterials (17), and any such class may yet admit some analogous polarization or order parameter. Intriguingly, topological interface modes have already been detected in systems of mechanically coupled gyroscopes that are amorphous, ruling out an orthodox reciprocal-space origin (18). Looking further afield, these polarization techniques could find application in the original (i.e., electronic) classes of topological insulators, in active fluids (19) or even in the study of ocean waves (20).

The notion that topological properties may be detected (or even generated) via local polarization may seem strange. In full mathematical rigor, one cannot locally measure a topological invariant such as whether a surface has a hole in it–whatever part of the surface one cannot see might contain myriad holes. However, in practice, one can often guess how many holes exist in an object (e.g., one hole in a donut) from a partial view. Similarly, in 2017 quantum researchers used machine learning to detect topological order based on partial system information (21). Similar techniques might permit mechanical polarizationGuzman et al. have developed and applied a new concept for complex mechanical structures. to detect novel forms of topological order.

Guzman et al. (1) have developed and applied a new concept for complex mechanical structures. This mechanical polarization sheds new light on the intimate connection between the displacements and extensions of elements of solid structures. In particular, it manages to liberate the field of topological mechanics from highly abstract reciprocal-space methods, permitting new and tighter connections between highly idealized structures and broader, more readily achievable existing structures. This model-free method of characterizing mechanical response promises both to permit new connections between different communities and to advance the field, opening new fields to place our Archimedean levers.