Soft topological modes protected by symmetry in rigid mechanical metamaterials

Topological mechanics can realize soft modes in mechanical metamaterials in which the number of degrees of freedom for particle motion is finely balanced by the constraints provided by interparticle interactions. However, solid objects are generally hyperstatic (or overconstrained). Here, we show how symmetries may be applied to generate topological soft modes even in overconstrained, rigid systems. To do so, we consider non-Hermitian topology based on non-square matrices, and design a hyperstatic material in which low-energy modes protected by topology and symmetry appear at interfaces. Our approach presents a novel way of generating softness in robust scale-free architectures suitable for miniaturization to the nanoscale.

Drive-specific adaptation in disordered mechanical networks of bistable springs

Systems with many stable configurations abound in nature, both in living and inanimate matter. Their inherent nonlinearity and sensitivity to small perturbations make them challenging to study, particularly in the presence of external driving, which can alter the relative stability of different attractors. Under such circumstances, one may ask whether any clear relationship holds between the specific pattern of external driving and the particular attractor states selected by a driven multistable system. To gain insight into this question, we numerically study driven disordered mechanical networks of bistable springs which possess a vast number of stable configurations arising from the two stable rest lengths of each spring, thereby capturing the essential physical properties of a broad class of multistable systems. We find that the attractor states of driven disordered multistable mechanical networks are fine-tuned with respect to the pattern of external forcing to have low work absorption from it. Furthermore, we find that these drive-specific attractor states are even more stable than expected for a given level of work absorption. Our results suggest that the driven exploration of the vast configuration space of these systems is biased towards states with exceptional relationship to the driving environment, and could therefore be used to `discover states with desired response properties in systems with a vast landscape of diverse configurations.

Helicity in Superfluids: existence and the classical limit

In addition to mass, energy, and momentum, classical dissipationless flows conserve helicity, a measure of the topology of the flow. Helicity has far-reaching consequences for classical flows from Newtonian fluids to plasmas. Since superfluids flow without dissipation, a fundamental question is whether such a conserved quantity exists for superfluid flows. We address the existence of a superfluid helicity using an analytical approach based on the the symmetry underlying classical helicity conservation: the particle relabeling symmetry. Furthermore, we use numerical simulations to study whether bundles of superfluid vortices which approximate the structure of a classical vortex, recover the conservation of classical helicity and find dynamics consistent with classical vortices in a viscous fluid.

When do knots in light stay knotted?

An initially knotted light field will stay knotted if it satisfies a set of nonlinear, geometric constraints, i.e. the null conditions, for all space-time. However, the question of when an initially null light field stays null has remained challenging to answer. By establishing a mapping between Maxwells equations and transport along the flow of a pressureless Euler fluid, we show that an initially analytic null light field stays null if and only if the flow of the initial Poynting field is shear-free, giving a design rule for the construction of persistently knotted light fields. Furthermore we outline methods for constructing initially knotted null light fields, and initially null, shear-free light fields, and give sufficient conditions for the magnetic (or electric) field lines of a null light field to lie tangent to surfaces. Our results pave the way for the design of persistently knotted light fields and the study of their field line structure.

Weaving knotted vector fields with tunable helicity

We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot and its generalizations. As finite-energy physical fields they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.

Tying knots in light fields

We construct a new family of null solutions to Maxwells equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is both geodesic and shear-free, preserves the topology of the knots and links. Our approach combines the Bateman and spinor formalisms for the construction of null fields with complex polynomials on $mathbb{S}^3$. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.