Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices.

Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potential for a new class of topological metamaterials. One important feature of hyperbolic lattices is that their translation group is nonabelian, permitting high-dimensional irreducible representations (irreps), in contrast to abelian translation groups in Euclidean lattices. Here we introduce a general framework to construct wave eigenstates of high-dimensional irreps of infinite hyperbolic lattices, thereby generalizing Bloch's theorem, and discuss its implications on unusual mode counting and degeneracy, as well as bulk-edge correspondence in hyperbolic lattices. We apply this method to a mechanical hyperbolic lattice, and characterize its band structure and zero modes of high-dimensional irreps.

Shape and size changes of adherent elastic epithelia.

Epithelial tissues play a fundamental role in various morphogenetic events during development and early embryogenesis. Although epithelial monolayers are often modeled as two-dimensional (2D) elastic surfaces, they distinguish themselves from conventional thin elastic plates in three important ways- the presence of an apical-basal polarity, spatial variability of cellular thickness, and their nonequilibrium active nature. Here, we develop a minimal continuum model of a planar epithelial tissue as an active elastic material that incorporates all these features. We start from a full three-dimensional (3D) description of the tissue and derive an effective 2D model that captures, through the curvature of the apical surface, both the apical-basal asymmetry and the spatial geometry of the tissue. Crucially, variations of active stresses across the apical-basal axis lead to active torques that can drive curvature transitions. By identifying four distinct sources of activity, we find that bulk active stresses arising from actomyosin contractility and growth compete with boundary active tensions due to localized actomyosin cables and lamellipodial activity to generate the various states spanning the morphospace of a planar epithelium. Our treatment hence unifies 3D shape deformations through the coupled mechanics of apical curvature change and in-plane expansion/contraction of substrate-adhered tissues. Finally, we discuss the implications of our results for some biologically relevant processes such as tissue folding at the onset of lumen formation.

Topology and ground-state degeneracy of tetrahedral smectic vesicles.

Chemical design of block copolymers makes it possible to create polymer vesicles with tunable microscopic structure. Here we focus on a model of a vesicle made of smectic liquid-crystalline block copolymers at zero temperature. The vesicle assumes a faceted tetrahedral shape and the smectic layers arrange in a stack of parallel straight lines with topological defects localized at the vertices. We counted the number of allowed states at [Formula: see text]. For any fixed shape, we found a two-dimensional countable degeneracy in the smectic pattern depending on the tilt angle between the smectic layers and the edge of the tetrahedral shell. For most values of the tilt angle, the smectic layers contain spiral topological defects. The system can spontaneously break chiral symmetry when the layers organize into spiral patterns, composed of a bound pair of +1/2 disclinations. Finally, we suggest possible applications of tetrahedral smectic vesicles in the context of functionalizing defects and the possible consequences of the spiral structures for the rigidity of the vesicle.

Frustrated self-assembly of non-Euclidean crystals of nanoparticles.

Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.