Mechanical metamaterials achieve their extraordinary properties through their intricate architectures. While amorphous designs can minimize directional bias relative to their regular counterparts, their vast configuration space poses a significant challenge for conventional design strategies. Here, we introduce a physics-constrained, energy-based model framework to navigate this complexity with machine learning. We formulate a multiobjective energy function that encodes desired macroscopic properties—specifically, a target negative Poisson ratio and approximate isotropy—into the network’s topology. The design problem is then recasted as finding the ground state of this energy landscape, while operating within a mechanically stable or physics-constrained configuration space. We employ Boltzmann annealing as a physically consistent inference algorithm to identify the optimal low-energy configurations. The structures discovered and fabricated via 3D printing exhibit a highly negative Poisson ratio together with near-equal responses along principal axes (i.e., isotropy along both $x$ and $y$ axes). Remarkably, these optimized structures also reveal highly desirable emergent properties, including exceptional performance in specific energy absorption, impact resistance, and fracture toughness, significantly outperforming regular lattice counterparts. This work establishes a robust and interpretable machine-learning framework for the design of high-performance amorphous metamaterials.

Knits and crochets are mechanical metamaterials with a long history that can typically be produced from a single yarn. Despite the simplicity of the manufacturing process, they exhibit a wide range of structural configurations with diverse mechanical properties and application potentials. Although there has been recent growing interest in textile-based metamaterials, a rigorous topological characterization of what makes a structure knittable has been lacking. Here, we introduce a general criterion based on topological constraints that distinguish knits and crochets from other textile structures. We demonstrate how the introduction of topological defects and their propagation makes this classification practical. Our approach highlights a fundamental link between manufacturing processes and structural fragility. Within this framework, we show how the rationalization of defect propagation unlocks the design of fabrics with controllable damage resistance.

We report the discovery of super-resonance—a new regime of resonant behavior in which a mode’s out-of-phase response persists far beyond its classical bandwidth. This effect emerges from a coiled phononic structure composed of a locally resonant elastic metamaterial and architected to support multiple internal energy pathways. These pathways converge at an effectively single spatial location, enabling extended modal dominance and significantly broadening the frequency range over which a resonant phase is sustained. We demonstrate by direct numerical simulations the implications of this mechanism in the context of flow instability control, where current approaches are inherently constrained by the characteristically narrow spectral bandwidth of conventional resonances. Using a super-resonant phononic subsurface structure interfacing with a channel flow, we show passive simultaneous suppression of four unstable flow perturbations across a frequency range more than 5 times wider than that is attainable with a standard resonance in an equivalent uncoiled structure. By enabling broadband control of flow instabilities, super-resonance overcomes a long-standing limitation in laminar flow control strategies. More broadly, it introduces a powerful new tool for phase-engineered wave-matter interaction. The ability to preserve out-of-phase modal response across wide spectral ranges establishes a fundamental advance in the physics of resonance, with far-reaching implications for suppressing fully developed turbulent flows and beyond.

Despite being governed by the familiar laws of Hookean mechanics, elastic shells patterned with an internal structure (i.e., metashells) exhibit a wealth of unusual mechanical properties with no counterparts in unstructured materials. Here I show that much of this behavior can be captured by a real-valued analog of the inhomogeneous Schrödinger equation, with the lateral pressure experienced by the internal structure in the role of the wave function. In the fine structure limit—i.e., when the length scale associated with the internal structure is much smaller than the local radius of curvature—this approach reveals the existence of localized states, in which elastic deformations are prevented to diffuse away from their origin, thereby allowing the internal structure to smoothly adapt to the intrinsic geometry of the metashell. Leveraging on an analogy with scattering states in nearly free electrons, I further show that periodic metashells, obtained from the repetition of the same structural unit periodically in space, support elastic Bloch waves, corresponding to stationary periodic configurations of the internal structure and characterized by a geometry-dependent band structure. When applied to crystalline monolayers, this approach provides a generalization of the elastic theory of interacting topological defect to compressible systems.

Hyperuniform structures are spatial patterns whose fluctuations disappear on long length scales, making them effectively homogeneous when observed from afar. Mathematically, this means that their spectral density $\tilde{\rho }(\mathbf{k})$ approaches zero for low wave number $|\mathbf{k}|$. Crystalline lattices are hyperuniform, as are certain quasicrystals, maximally random jammed packings of spheres, and electrons in the fractional quantum Hall state. Stealthy hyperuniformity is an even stronger constraint on the spectral density: It requires that $\tilde{\rho }(\mathbf{k})$ is strictly zero in a finite range of wave vectors around $\mathbf{k}=\mathbf{0}$, called the stealthy regime, or exclusion region. Since the degree of scattering by disorder is, to leading order, proportional to $\tilde{\rho }(\mathbf{k})$, waves propagating through such structures may do so without scattering for sufficiently long wavelengths and short distances. Here, we measure scattering by disorder in photonic crystal slabs with stealthy-hyperuniform disorder by measuring the linewidths of the photonic bands. We observe the transition between the stealthy and nonstealthy regimes, marked by a sharp increase in linewidth. We also observe the effects of multiple scattering in the stealthy regime, which implies diminishing transparency. Moreover, we show that residual single scattering in the stealthy regime arises from an intrinsically non-Hermitian effect: Propagating light has a complex effective mass due to radiative loss out of the slab.

A remarkable feat of active matter physics is that systems as diverse as collections of self-propelled particles, nematics mixed with molecular motors, and interacting robots can all be described by symmetry-based continuum theories. These descriptions rely on reducing complex effects of individual motors to a few key active parameters, which increase with activity. Here we observe a striking anomaly in the continuum description of nonreciprocal active solids, a ubiquitous class of active materials. Using a combination of metamaterial experiments and coarse-graining theory we find that as microscopic activity increases, macroscale active response can vanish: more is less. In this highly active regime, nonaffine and localized modes prevail and destroy the large-scale signature of microscopic activity. These modes exist in any dilute periodic structure and emerge in random lattices below a percolation transition. Our results unveil a counterintuitive facet of active matter, offering new principles for engineering materials far from equilibrium.

We present an experimental protocol for the fabrication and characterization of scalable microarchitected elastic waveguides. Using silicon microfabrication techniques, we develop free-standing 2D truss-based architected waveguides with a maximum diameter of 80 mm, unit cells size of $100\mathrm{\mu }\mathrm{m}$, and minimum beam width of $5\mathrm{\mu }\mathrm{m}$, thus achieving scale separation. To characterize elastic wave propagation, we introduce a custom-built scanning optical pump-probe experiment that enables contactless excitation of elastic wave modes and full spatiotemporal reconstruction of wave propagation across hundreds of unit cells with subunit cell resolution. Results on periodic architectures show excellent agreement with finite element simulations and equivalent experimental data at larger length scales. Motivated by scalable computational inverse design, we fabricate a specific example of a spatially graded waveguide and demonstrate its ability to guide elastic waves along an arbitrary predesigned path.

Spin waves (magnons) in two-dimensional (2D) materials have received increasing interest due to their unique states and potential for tunability. However, many interesting features of these systems, including Dirac points and topological states, occur at high frequencies, where experimental probes are limited. Here, we study a crystal formed by patterning a hexagonal array of holes in a perpendicularly magnetized thin film. Through simulation, we find that the magnonic band structure imitates that of graphene, but additionally has some kagomelike character and includes a few flat bands. Surprisingly, its nature can be understood using a nine-band tight-binding Hamiltonian. This clear analogy to 2D materials enables band-gap engineering in 2D, topological magnons along 1D phase boundaries, and spectrally isolated modes at 0D point defects. Interestingly, the 1D phase boundaries allow access to the valley degree of freedom through a magnonic analog of the quantum valley Hall insulator. These approaches can be extended to other magnonic systems, but are potentially more general due to the simplicity of the model, which resembles existing results from electron, phonon, photon, and cold-atom systems. This finding brings the physics of spin waves in 2D materials to more experimentally accessible scales, augments it, and outlines a few principles for controlling magnonic states.

Origami principles are used to create strong, lightweight structures with complex mechanical response. However, identifying the fundamental physical principles that determine a sheet’s behavior remains a challenge. We introduce a new analytic theory to account for the low-energy, spatially varying deformations of parallelogram-face origami, a broad class that includes widely studied sheets. We identify topological classes—previously predicted in quantum systems—with sharply distinct mechanical properties, offering a mechanism for control over origami’s effective stiffness and the smoothness of its mechanical response to external loads. The key control parameter is the Poisson’s ratio; when it is negative, as in the Miura ori, sheets have conventional, smooth mechanical response amenable to continuum-based approaches. In contrast, positive Poisson’s ratio, as in the eggbox ori, generates a topological transition to lines of doubly degenerate zero modes that lead to dramatically softer structures with uneven, complex patterns of spatial response. These patterns interact in complicated ways with origami boundary conditions and source terms, leading to rich physical phenomena in experimentally accessible systems. Indeed, these rich topological phenomena demonstrate that the complex response of some origami sheets is both intrinsic and topological in origin and cannot be captured via continuum-based approaches. Doubly degenerate modes, such as the topological ones identified here, are related to non-Abelian physics and coherent computing paradigms. By exploring such consequences and extending the theory to the broader class of origami with nonparallelogram faces, this work sheds light on intriguing new areas of physics.

In physical networks trained using supervised learning, physical parameters are adjusted to produce desired responses to inputs. An example is an electrical contrastive local learning network of nodes connected by edges that adjust their conductances during training. When an edge conductance changes, it upsets the current balance of every node. In response, physics adjusts the node voltages to minimize the dissipated power. Learning in these systems is therefore a coupled double-optimization process, in which the network descends both a cost landscape in the high-dimensional space of edge conductances and a physical landscape—the power dissipation—in the high-dimensional space of node voltages. Because of this coupling, the physical landscape of a trained network contains information about the learned task. Here, we derive a structure-function relation for trained tunable networks and demonstrate that all the physical information relevant to the trained input-output relation can be captured by a tuning susceptibility, an experimentally measurable quantity. We supplement our theoretical results with simulations to show that the tuning susceptibility is correlated with functional importance and that we can extract physical insight into how the system performs the task from the conductances of highly susceptible edges. Our analysis is general and can be applied directly to mechanical networks, such as networks trained for protein-inspired function such as allostery.

Breathing solitons consist of a fast beating wave within a compact envelope of stable shape and velocity. They can propagate and carry information and energy in a variety of contexts such as plasmas, optical fibers, and cold atoms, but propagating breathers have remained elusive when energy conservation is broken. Here, we report on the observation of breathing, unidirectional, arbitrarily long-lived solitons in nonreciprocal, nonconservative active metamaterials. Combining precision desktop experiments, numerical simulations, and perturbation theory on generalizations of the sine-Gordon and nonlinear Schrödinger equations, we demonstrate that unidirectional breathers generically emerge in weakly nonlinear nonreciprocal materials, and that their dynamics are governed by an unstable fixed point. Crucially, breathing solitons can persist for arbitrarily long times provided that (i) this fixed point displays a bifurcation when a delicate balance between energy injection and dissipation is struck and (ii) the initial conditions allow the dynamics to reach this bifurcation point. Importantly, discrete effects tend to stabilize these nonreciprocal breathers over a wider range of initial conditions. Our work establishes nonreciprocity as a promising avenue to generate stable nonlinear unidirectional waves and could be generalized beyond metamaterials to optics, soft matter, and superconducting circuits.

Networks of interconnected materials permeate throughout nature, biology, and technology due to exceptional mechanical performance. Despite the importance of failure resistance in network design and utility, no existing physical model effectively links strand mechanics and connectivity to predict bulk fracture. Here, we reveal a scaling law that bridges these levels to predict the intrinsic fracture energy of diverse stretchable networks. Simulations and experiments demonstrate its remarkable applicability to a breadth of strand constitutive behaviors, topologies, dimensionalities, and length scales. We show that local strand rupture and nonlocal energy release contribute synergistically to the measured intrinsic fracture energy in networks. These effects coordinate such that the intrinsic fracture energy scales independent of the energy to rupture a strand; it instead depends on the strand rupture force, breaking length, and connectivity. Our scaling law establishes a physical basis for fracture of homogeneous networks with uniform strand mechanics and lattice connectivity throughout. The scaling also extends generally for fabricating tough materials from homogeneous networks across multiple length scales.

Following the discovery of moiré-driven superconductivity and density waves in twisted-graphene multilayers, twistronics has spurred a surge of interest in tailored broken symmetries through angular rotations enabling new properties, from electronics to photonics and phononics. Analogously, in monoclinic polar crystals a nontrivial angle between nondegenerate dipolar phonon resonances can naturally emerge due to asymmetries in their crystal lattice, and its variations are associated with intriguing polaritonic phenomena, including axial dispersion, i.e., the rotation of the optical axis with frequency, and microscopic shear effects that result in an asymmetric distribution of material loss. So far, these phenomena have been restricted to specific midinfrared frequencies difficult to access with conventional laser sources and fundamentally limited by the degree of asymmetry and by the strength of light-matter interactions available in natural crystals. Here, we leverage the twistronics concept to demonstrate maximal axial dispersion and loss redistribution of hyperbolic waves in elastic metasurfaces, achieved by tailoring the angle between coupled metasurface pairs featuring tailored anisotropy. We show extreme control over elastic wave dispersion and absorption via the twist angle and leverage the resulting phenomena to demonstrate enhanced propagation distance, in-plane reflection-free negative refraction and diffraction-free defect detection. Our work welds the concepts of twistronics, non-Hermiticity, and extreme anisotropy, demonstrating the powerful opportunities enabled by metasurfaces for tunable, highly directional surface-acoustic-wave propagation of great interest for a wide range of applications spanning from seismic mitigation to on-chip phononics and wireless communication systems, hence paving the way toward their translation into emerging photonic and polaritonic metasurface technologies.

The non-Hermitian skin effect dramatically reshapes the energy bands of non-Hermitian systems, meaning that the usual Bloch band theory is fundamentally inadequate as their characterization. The non-Bloch band theory, in which the concept of Brillouin zone is generalized, has been widely applied to investigate non-Hermitian systems in one spatial dimension. However, its generalization to higher dimensions has been challenging. Here, we develop a formulation of the non-Hermitian skin effect and non-Bloch band theory in arbitrary spatial dimensions, which is based on a natural geometrical object known as the amoeba. Our theory provides a general framework for studying non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian bands, including the energy spectrum, eigenstates profiles, and the generalized Brillouin zone, can be efficiently obtained from this approach.

Metamaterials, from optics to radio frequencies and acoustics, have attracted significant attention over the last few decades, with promising applications in a wide range of technological areas. However, it has been recognized that their performance is often hindered by ubiquitous material loss and nonlocal phenomena. A canonical problem consists in imaging through metamaterial superlenses, which hold the promise of superresolution but which are, in practice, limited by material loss as we attempt to image deeply subwavelength details. Active metamaterials have been explored to compensate for loss; however, material gain introduces other obstacles, e.g., instabilities, nonlinearity, and noise. Here, we demonstrate that the temporal excitation of passive metamaterials using signals oscillating at complex frequencies can effectively compensate material loss, leading to resolution enhancement when applied to metamaterial superlenses. More broadly, our results demonstrate that virtual gain stemming from tailored forms of excitation can tackle the impact of loss in metamaterials, opening promising avenues for a broad range of applications from acoustic to photonic technologies.

Imaging is of great importance in everyday life and various fields of science and technology. Conventional imaging is achieved by bending light rays originating from an object with a lens. Such ray bending requires space-variant structures, inevitably introducing a geometric center to the lens. To overcome the limitations arising from the conventional imaging mechanism, we consider imaging elements that employ a different mechanism, which we call reciprocal lenses. This type of imaging element relies on lateral ray shifting, enabled by momentum-space-variant phase modulations in periodic structures. As such, it has the distinct advantage of not requiring alignment with a geometric center. Moreover, upright real images can be produced directly with a single reciprocal lens as the directions of rays are not changed. We realize an ultrathin reciprocal lens based on a photonic crystal slab. We characterize the lateral ray shifting behavior of the reciprocal lens and demonstrate imaging. Our work gives an alternative mechanism for imaging and provides a new way to modulate electromagnetic waves.

In $PT$-symmetric systems, the notion of non-Abelian frame charges enables multiband topological characterization of degeneracies by examining the eigenvector-frame rotations. Interestingly, some features of these frame charges can be viewed as an analog of electric charges confined in conducting wires, only that they flow in momentum space along nodal lines. But different from electric charges, these frame charges are not integers, and non-Abelian signatures emerge when braiding between adjacent band nodal lines occurs, which flips the direction of the flow. In photonic systems, the photonic $\mathrm{\Gamma }$ point serves as the source or sink of such a frame charge flow due to a hidden braiding induced by the often-ignored electrostatic mode at zero frequency. This source naturally generates non-Abelian topological signatures in $PT$-symmetric photonic systems that can be identified even in ordinary dielectric media. We use biaxial photonic crystals as examples to show how complex nodal line configurations can be explained as the topological consequences of the frame charge flow from the $\mathrm{\Gamma }$ point to the Brillouin zone boundaries. Our results reveal that non-Abelian band topology is intrinsically manifested in general photonic systems, making them easily accessible platforms for exploring non-Abelian physics.

Cladding layers of waveguides prevent interchannel interference yet are unfavorable for the integration of photonic circuits. Here, we report the realization of ultracompact waveguide arrays, bends, and circuits with essentially zero interchannel separation. This supercompactness is achieved via arrays of waveguides with shifted spatial dispersions, where each waveguide functions as both the transmission channel and an effective “cladding layer” of its neighboring waveguides, and has been experimentally realized in low-loss all-dielectric photonic crystals. We show that the zero-spacing transmission array possesses the remarkable features of negligible crosstalk, high-efficiency sharp bending, and ultracompact photonic routing where the light can traverse the entire physical space. This finding opens a new avenue for extreme space utilization efficiency in waveguide physics and integrated photonics.

From enzyme binding to robot grasping, the function of many mechanical systems depends upon large, coordinated motions of their components. Such motions arise from a network of physical interactions in the form of links (edges) that transmit forces between constituent elements (nodes) and have been fruitfully modeled in known networks. However, the principled design of precise motions in novel networks is made difficult by the number and nonlinearity of interactions. Here, we formulate a simple but powerful framework for designing fully nonlinear motions using concepts from dynamical systems theory. We demonstrate that a small network unit acts as a one-dimensional map between the distances across pairs of nodes, and we represent the act of combining units as an iteration of this map. By tying the map’s attractors and their stability to the shape and folding sequence in a network of combined units, we program the precise coordinated motion between arbitrarily complex macroscopic shapes, the exact folding sequence between the shapes, and exotic network behaviors such as a mechanical and gate and a period-doubling route to chaos. Further, we construct a unit with a 3-cycle that combines to form a lattice with any positive integer period as a result of Sharkovskii’s theorem. Finally, we construct physical networks and analyze the effect of bond elasticity to demonstrate the framework’s potential and versatility. The precise design of shape change and folding sequence makes this framework ideal as a starting minimal model for many applications, such as robotics, providing a promising direction for future work in metamaterials.

Quantum emitters in the presence of an electromagnetic reservoir with varying density of states, or structure, can undergo a rich set of dynamical behavior. In particular, the reservoir can be tailored to have a memory of past interactions with emitters, in contrast to memoryless Markovian dynamics of typical open systems. In this article, we investigate the non-Markovian dynamics of a superconducting qubit strongly coupled to a superconducting waveguide engineered to have both a sharp spectral variation in its transmission properties and a slowing of light by a factor of 650. Tuning the qubit into the spectral vicinity of the passband of this slow-light waveguide reservoir, we observe a 400-fold change in the emission rate of the qubit, along with oscillatory energy relaxation of the qubit resulting from the beating of bound and radiative dressed qubit-photon states. Furthermore, upon addition of a reflective boundary to one end of the waveguide, we observe revivals in the qubit population on a timescale 30 times longer than the inverse of the qubit’s emission rate, corresponding to the round-trip travel time of an emitted photon. By in situ tuning of the qubit-waveguide interaction strength, we also probe a crossover between Markovian and non-Markovian qubit emission dynamics in the presence of feedback from waveguide reflections. With this superconducting circuit platform, future studies of multiqubit interactions via highly structured reservoirs and the generation of multiphoton highly entangled states are possible.