Three-dimensional Maxwell lattices — periodic spring networks with equal numbers of constraints and degrees of freedom — can host topological Weyl lines: one-dimensional manifolds in the bulk Brillouin zone on which extra zero-frequency phonon modes appear, carrying integer winding-number charge and forcing the number of surface zero modes to jump when the surface-parallel wave vector crosses their projection. Stenull, Kane, and Lubensky [Phys. Rev. Lett. 117, 068001 (2016)] established this physics explicitly in a generalized pyrochlore lattice (GPL), analyzing only one deformation family and two conformations, and left two linked directions open: experimentally realizing and detecting the predicted Weyl lines and the wave-vector-resolved flips of surface zero modes in a fabricated 3D metamaterial, and systematically characterizing the presence, number, charge, and connectivity of Weyl lines across the broader class of 3D Maxwell lattices. Later work has fabricated 3D-printed GPL-based isostatic metamaterials and indirectly detected Weyl-line phases through boundary-stiffness measurements (Tang et al., 2024), but no direct dynamical identification of the Weyl-line singularities or of the surface-mode flips exists, and no classification, including a treatment of the full GPL parameter space, has been given.

Background

A Maxwell lattice (also called an isostatic lattice) is a periodic network of point masses connected by central-force springs in which the number of constraints equals the number of degrees of freedom, dN=nBdN = n_B, where NN is the number of sites per unit cell, dd the spatial dimension, and nBn_B the number of bonds. Such lattices sit exactly at the threshold of mechanical stability and are described by the Calladine–Maxwell counting relation n0(q)−s(q)=dN−nBn_0(q) - s(q) = dN - n_B at each wave vector qq in the Brillouin zone (BZ, the set of distinct wave vectors allowed by the periodic lattice), where n0(q)n_0(q) counts zero modes (ZMs) — collective displacements that stretch no bonds and cost no energy — and s(q)s(q) counts states of self stress (SSSs) — bond tension patterns that exert no net force on any site. Equivalently, the compatibility matrix C(q)C(q), mapping site displacements to bond extensions, has null vectors (ZMs) whose count jumps wherever det⁡C(q)\det C(q) vanishes. Kane and Lubensky showed in two dimensions that ZMs and SSSs on open surfaces are controlled by a bulk topological invariant, the mechanical polarization, and Lubensky, Kane, Mao, Souslov, and Sun reviewed the resulting 'topological mechanics' framework for critically coordinated lattices; the 2D prototype is the generalized kagome lattice (GKL), a deformed version of the kagome lattice (the 2D Maxwell-coordinated network of corner-sharing triangles) whose floppy edge modes can be switched from one edge to the other by lattice distortion. Rocklin, Chen, Falk, Vitelli, and Lubensky then showed that 2D deformed square Maxwell lattices generically host Weyl points: isolated qq-space points at which a zero mode appears, carrying an integer winding number of det⁡C(q)\det C(q) around any loop encircling it, so that the number of ZMs changes by that integer when the loop is crossed — the mechanical analog of a Weyl semimetal node. Sussman, Stenull, and Lubensky found such Weyl points to be ubiquitous in 2D models of jammed matter. In three dimensions, the analogous singularities become Weyl lines: closed curves in the 3D BZ on which det⁡C(q)=0\det C(q) = 0, each carrying winding-number charge that changes the ZM count in reciprocal-space planes crossing it. Stenull, Kane, and Lubensky constructed the first explicit 3D example, the generalized pyrochlore lattice (GPL), obtained by deforming the pyrochlore lattice (the 3D network of corner-sharing tetrahedra, itself built from stacked kagome planes). Deforming straight lines of bonds ('filaments') into zigzag ones removes the SSSs that fill the BZ in the undeformed lattice and leaves topologically charged Weyl lines traversing the BZ. Their key prediction is a bulk–surface correspondence: for a surface perpendicular to reciprocal lattice vector GG, the number of ZMs localized on that surface at surface-parallel wave vector q⊥q_{\perp} is fixed by a winding number that jumps when q⊥q_{\perp} crosses the projection of a Weyl line, so that the surface zero modes flip abruptly from one edge to the opposite edge as a function of surface wavenumber. The 2016 study was deliberately restricted to one deformation family (their Eq. (1), parameters z,x1,x2,x3z, x_1, x_2, x_3) and two conformations X1X_1 and X−1X_{-1}; the authors noted that moving the kagome-plane basis sites 1–3 out of their planes splits the two-fold degenerate zero-mode line along the (1,1,1)(1,1,1) direction n^\hat{n} into two oppositely charged Weyl lines with separation proportional to the out-of-plane displacement, that their results 'hint' that Weyl singularities are common in Maxwell lattices while the GKL unit cell lacks the degrees of freedom to host them, and that 3D printing should make a GPL-like metamaterial realizable in the laboratory. Related but distinct lines of work should not be confused with this core: finite-frequency Weyl points and nodal lines in phononic crystals and mechanical metacrystals (e.g. Xie et al. 2019; Xiong et al. 2018) arise from symmetry-enforced band degeneracies in over-coordinated elastic media, not from Maxwell-lattice zero modes; and topological polarization experiments (Bilal et al. 2017; Baardink et al. 2018) concern gapped polar phases with no Weyl singularities.

Problem Statement

Establish the Weyl-line phenomenology of three-dimensional Maxwell lattices beyond the two conformations of the single deformation family analyzed by Stenull, Kane, and Lubensky (2016), on two jointly linked fronts. (1) Experimental detection: fabricate a 3D isostatic (Maxwell) metamaterial whose unit cell realizes the generalized-pyrochlore lattice (GPL) deformation family of Stenull, Kane, and Lubensky Eq. (1), or an isostatic lattice that provably hosts charged Weyl lines, and dynamically identify the bulk Weyl-line singularities and their surface-mode consequence — the abrupt flips of surface zero modes from one edge to the opposite edge as the surface-parallel wave vector crosses the projected Weyl lines — rather than only indirect static signatures such as boundary-stiffness contrast. (2) Systematic characterization: determine, for the general class of 3D Maxwell lattices and for the full generalized-pyrochlore parameter space (including geometries with basis sites displaced out of the kagome planes, where the degenerate zero-mode line along the (1,1,1)(1,1,1) direction splits into two oppositely charged Weyl lines), the conditions under which Weyl lines occur — necessary and sufficient geometric constraints — together with rules for their number, topological charge, and connectivity, thereby settling whether Weyl singularities are generic in 3D Maxwell lattices and why lattices such as the GKL lack them. Both parts serve one objective, the one articulated in the source paper's outlook: confirming and generalizing the predicted Weyl lines and their bulk–surface correspondence for 3D isostatic lattices, with part (1) providing the experimental confirmation and part (2) the predictive theory that part (1) would test. The two fronts are jointly posed: part (1) realizes a specimen of the admissible class and directly detects the bulk Weyl lines and the surface-mode flips, and part (2) supplies, over the stated class, necessary and sufficient geometric conditions for the occurrence of Weyl lines together with rules for their number, charge, and connectivity.

References

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