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Elucidating long-range interaction guided organization of matter is a fundamental question in physical systems covering multiple length scales. Here, based on the hexagonal disk model, we analyze the characteristic inhomogeneity created by long-range repulsions, and reveal the intrinsic conformal order in particle packings in mechanical equilibrium. Specifically, we highlight the delicate angle-preserved bending of the lattice to match the inhomogeneity condition. The revealed conformal order is found to be protected by the surrounding topological defects. These results advance our understanding on long-range interacting systems, and open the promising possibilities of using long-range forces to create particle packings not accessible by short-range forces, which may have practical consequences., Comment: 6 pages, 4 figures

Understanding the mechanical instabilities of two-dimensional membranes has strong connection to the subjects of structure instabilities, morphology control and materials failures. In this work, we investigate the plastic mechanism developed in the annular crystalline membrane system for adapting to the shrinking space, which is caused by the controllable gradual expansion of the inner boundary. In the process of plastic deformation, we find the continuous generation of dislocations at the inner boundary, and their collective migration to the outer boundary; this neat dynamic scenario of dislocation current captures the complicated reorganization process of the particles. We also reveal the characteristic vortex structure arising from the interplay of topological defects and the displacement field. These results may find applications in the precise control of structural instabilities in packings of particulate matter and covalently bonded systems., Comment: 8 pages, 4 figures

Understanding the deep connection of microscopic dynamics and statistical regularity yields insights into the foundation of statistical mechanics. In this work, based on the classical three-body system under the Lennard-Jones potential upon disturbance, we illustrated the elusive nonlinear dynamics in terms of the neat frequency-mixing processes, and revealed the emergent statistical regularity in speed distribution along a single particle trajectory. This work demonstrates the promising possibility of classical few-body models for exploring the fundamental questions on the interface of microscopic dynamics and statistical physics., Comment: 7 pages, 4 figures. The SI file is available at: https://ins.sjtu.edu.cn/people/zyao/

Understanding the organization of matter under the long-range electrostatic force is a fundamental problem in multiple fields. In this work, based on the electrically charged tethered membrane model, we reveal regular structures underlying the lowest-energy states of inhomogeneously stretched planar lattices by a combination of numerical simulation and analytical geometric analysis. Specifically, we show the conformal order characterized by the preserved bond angle in the lattice deformation, and reveal the Poincar$\rm{\acute{e}}$-Klein mapping underlying the electrostatics-driven inhomogeneity. The discovery of the Poincar$\rm{\acute{e}}$-Klein mapping, which connects the Poincar$\rm{\acute{e}}$ disk and the Klein disk for the hyperbolic plane, implies the connection of long-range electrostatic force and hyperbolic geometry. We also discuss lattices with patterned charges of opposite signs for modulating in-plane inhomogeneity and even creating 3D shapes, which may have a connection to metamaterials design. This work suggests the geometric analysis as a promising approach for elucidating the organization of matter under the long-range force., Comment: 14 pages, 9 figures

A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic moduli. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism., Comment: 9 pages, 2 figures

Elucidating collective dynamics in crystalline systems is a common scientific question in multiple fields. In this work, by combination of high-precision numerical approach and analytical normal mode analysis, we systematically investigate the dynamical response of two-dimensional Lennard-Jones crystal as a purely classical mechanical system under random disturbance of varying strength, and reveal rich microscopic dynamics. Specifically, we observe highly symmetric velocity field composed of sharply divided coherent and disordered regions, and identify the order-disorder dynamical transition of the velocity field. Under stronger disturbance, we reveal the vacancy-driven shattering of the crystal. This featured disruption mode is fundamentally different from the dislocation-unbinding scenario in two-dimensional melting. We also examine the healing dynamics associated with vacancies of varying size. The results in this work advance our understanding about the formation of collective dynamics and crystal disruption, and may have implications in elucidating relevant non-equilibrium behaviors in a host of crystalline systems., Comment: 7 pages, 4 figures

We report the counter-intuitive instability of charged elastic rings, and the persistence of sinusoidal deformations in the lowest-energy configurations by the combination of high-precision numerical simulations and analytical perturbation calculation. We also study the dynamical evolution of the charged ring under random disturbance, and reveal the modulation of the dominant frequencies by the electrostatic force. The purely mechanical analysis of the classical ring system presented in this work yields insights into the subtlety of long-range forces in the organization and dynamics of matter., Comment: 8 pages, 5 figures

Elucidating the interplay of stress and geometry is a fundamental scientific question arising in multiple fields. In this work, we investigate the geometric frustration of crystalline caps confined on the sphere in both elastic and plastic regimes. Based on the revealed quasi-conformal ordering, we discover the partial, but uniform screening of the substrate curvature by the induced curvature underlying the inhomogeneous lattice. This scenario is fundamentally different from the conventional screening mechanism based on topological defects. In the plastic regime, the yield of highly stressed caps leads to fractures with featured morphologies not found in planar systems. We also demonstrate the strategy of engineering stress and fractures by vacancies. These results advance our general understanding on the organization and adaptivity of geometrically-frustrated crystalline order., Comment: 6 pages, 4 figures

Long-range interacting systems exhibit unusual physical properties not shared by systems with short-range interactions. Understanding the dynamical and statistical effects of long-range interactions yields insights into a host of physical systems in nature and industry. In this work, we investigate the classical microscopic dynamics of screened Coulomb interacting particles confined in the disk, and reveal the featured dynamics and emergent statistical regularities created by the long-range interaction. We highlight the long-range interaction driven fast single-particle and collective dynamics, and the emergent topological defect structure. This work suggests the rich physics arising from the interplay of long-range interaction, topology and dynamics., Comment: 9 pages, 10 figures

Elucidating emergent regularities in intriguing crowd dynamics is a fundamental scientific problem arising in multiple fields. In this work, based on the social force model, we simulate the typical scenario of collective escape towards a single exit and reveal the striking analogy of crowd dynamics and crystallisation. With the outflow of the pedestrians, crystalline order emerges in the compact crowd. In this process, the local misalignment and global rearrangement of pedestrians are well rationalized in terms of the characteristic motions of topological defects in the crystal. Exploiting the notions from the physics of crystallisation further reveals the emergence of multiple fast tracks in the collective escape., Comment: 8 pages, 6 figures

Understanding the intriguing physical effects of long-range interactions is a common theme in a host of physical systems. In this work, based on the classical screened Coulomb interacting ring model, we investigate the dynamical effects of the long-range interaction from the unique perspective of analyzing the dynamical response of the system to disturbance. We reveal the featured dynamics brought by the long-range interaction, including the efficient transformation of the disturbance into the uniform global rotation of the system, the suppression of the intrinsic noise, and the fast relaxation of particle speed., Comment: 6 pages, 5 figures

Packing of particles in confined environments is a common problem in multiple fields. Here, based on the two-dimensional Hertzian particle model, we study the packing of deformable spherical particles under compression, and reveal the crucial role of stress as an ordering field in regulating particle arrangement. Specifically, under increasing compression, the squeezed particles spontaneously organize into regular packings in the sequence of triangular and square lattices, pentagonal tessellation, and the reentrant triangular lattice. The rich ordered patterns and complex structures revealed in this work suggest a fruitful organizational strategy based on the interplay of external stress and intrinsic elastic instability of particle arrays., Comment: 8 pages, 8 figures

Elucidating the interplay of defect and stress at the microscopic level is a fundamental physical problem that has strong connection with materials science. Here, based on the two-dimensional crystal model, we show that the instability mode of vacancies with varying size and morphology conforms to a common scenario. A vacancy under compression is fissioned into a pair of dislocations that glide and vanish at the boundary. This neat process is triggered by the local shear stress around the vacancy. The remarkable fractionalization of vacancies creates rich modes of interaction between vacancies and other topological defects, and provides a new dimension for mechanical engineering of defects in extensive crystalline structures., Comment: 7 pages, 6 figures

Directing individual motions of many constituents to coherent dynamical state is a fundamental challenge in multiple fields. Here, based on the spherical crystal model, we show that topological defects in particle arrays can be a crucial element in regulating collective dynamics. Specifically, we highlight the defect-driven synchronized breathing modes around disclinations and collective oscillations with strong connection to disruption of crystalline order. This work opens the promising possibility of an organizational principle based on topological defects, and may inspire new strategies for harnessing intriguing collective dynamics in extensive nonequilibrium systems., Comment: 5 pages, 4 figures

Generalizing the classical Thomson problem to the quantum regime provides an ideal model to explore the underlying physics regarding electron correlations. In this work, we systematically investigate the combined effects of the geometry of the substrate and the symmetry of the wave function on correlations of geometrically confined electrons. By the numerical configuration interaction method in combination with analytical theory, we construct symmetrized ground-state wave functions; analyze the energetics, correlations, and collective vibration modes of the electrons; and illustrate the routine for the strongly correlated, highly localized electron states with the expansion of the sphere. This work furthers our understanding about electron correlations on confined geometries and shows the promising potential of exploiting confinement geometry to control electron states., Comment: 8 pages, 5 figures

Knotted fields enrich a variety of physical phenomena, ranging from fluid flows, electromagnetic fields, to textures of ordered media. Maxwell's electrostatic equations, whose vacuum solution is mathematically known as a harmonic field, provide an ideal setting to explore the role of domain topology in determining physical fields in confined space. In this work, we show the uniqueness of a harmonic field in knotted tubes, and reduce the construction of a harmonic field to a Neumann boundary value problem. By analyzing the harmonic field in typical knotted tubes, we identify the torsion driven transition from bipolar to vortex patterns. We also analogously extend our discussion to the organization of liquid crystal textures in knotted tubes. These results further our understanding about the general role of topology in shaping a physical field in confined space, and may find applications in the control of physical fields by manipulation of surface topology., Comment: 5 pages, 2 figures

Understanding string dynamics yields insights into the intricate dynamic behaviors of various filamentary thin structures in nature and industry covering multiple length scales. In this work, we investigate the planar dynamics of a flexible string where one end is free and the other end is subject to transverse and longitudinal motions. Under transverse harmonic motion, we reveal the propagating pulse structure in the stress profile over the string, and analyze its role in bringing the system into a chaotic state. For a string where one end is under longitudinal uniform acceleration, we identify the wiggling transition, derive the analytical wiggling solution from the string equations, and present the phase diagram., Comment: 5 pages, 4 figures

Understanding the interplay between ordered structures and substrate curvature is an interesting problem with versatile applications, including functionalization of charged supramolecular surfaces and modern microfluidic technologies. In this work, we investigate the two-dimensional packing structures of charged particles confined on a pinched sphere. By continuously pinching the sphere, we observe cleavage of elongated scars into pleats, proliferation of disclinations, and subsequently, emergence of a depletion zone at the negatively curved waist that is completely void of particles. We systematically study the geometrics and energetics of the depletion zone, and reveal its physical origin as a finite size effect, due to the interplay between Coulomb repulsion and concave geometry of pinched sphere. These results further our understanding of crystallography on curved surfaces, and have implications in design and manipulation of charged, deformable interfaces in various applications., Comment: 6 pages, 6 figures

Various lattice gas automata have been proposed in the past decades to simulate physics and address a host of problems on collective dynamics arising in diverse fields. In this work, we employ the lattice gas model defined on the sphere to investigate the curvature driven dynamic structures and analyze the statistical behaviors in equilibrium. Under the simple propagation and collision rules, we show that the uniform collective movement of the particles on the sphere is geometrically frustrated, leading to several non-equilibrium dynamic structures not found in the planar lattice, such as the emergent bubble and vortex structures. With the accumulation of the collision effect, the system ultimately reaches equilibrium in the sense that the distribution of the coarse-grained speed approaches the two-dimensional Maxwell-Boltzmann distribution despite the population fluctuations in the coarse-grained cells. The emergent regularity in the statistical behavior of the system is rationalized by mapping our system to a generalized random walk model. This work demonstrates the capability of the spherical lattice gas automaton in revealing the lattice-guided dynamic structures and simulating the equilibrium physics. It suggests the promising possibility of using lattice gas automata defined on various curved surfaces to explore geometrically driven non-equilibrium physics., Comment: 8 pages, 7 figures

Understanding geometric frustration of ordered phases in two-dimensional condensed matters on curved surfaces is closely related to a host of scientific problems in condensed matter physics and materials science. Here we show how two-dimensional Lennard-Jones crystal clusters confined on a sphere resolve the geometric frustration and lead to pentagonal vacancy structures. These vacancies, originated from the combination of curvature and physical interaction, are found to be topological defects and they can be further classified into dislocational and disclinational types. We analyze the dual role of these crystallographic defects as both vacancies and topological defects, illustrate their formation mechanism, and present the phase diagram. The revealed dual role of the topological vacancies may find applications in the fabrication of robust nanopores. This work also suggests the promising potential of exploiting the richness in both physical interactions and substrate geometries to create new types of crystallographic defects, which have strong connections with the design of crystalline materials., Comment: 7 pages, 5 figures, Soft Matter (2017)

Stabilizing defects in liquid-crystal systems is crucial for many physical processes and applications ranging from functionalizing liquid-crystal textures to recently reported command of chaotic behaviors of active matters. In this work, we perform analytical calculations to study the curvature driven stability mechanism of defects based on the isotropic nematic disk model that is free of any topological constraint. We show that in a growing spherical disk covering a sphere the accumulation of curvature effect can prevent typical +1 and +1/2 defects from forming boojum textures where the defects are repelled to the boundary of the disk. Our calculations reveal that the movement of the equilibrium position of the +1 defect from the boundary to the center of the spherical disk occurs in a very narrow window of the disk area, exhibiting the first-order phase-transition-like behavior. For the pair of +1/2 defects by splitting a +1 defect, we find the curvature driven alternating repulsive and attractive interactions between the two defects. With the growth of the spherical disk these two defects tend to approach and finally recombine towards a +1 defect texture. The sensitive response of defects to curvature and the curvature driven stability mechanism demonstrated in this work in nematic disk systems may have implications towards versatile control and engineering of liquid crystal textures in various applications., Comment: 11 pages, 5 figures

We investigate the dynamics of an active particle in two-dimensional spherical crystals, which provide an ideal environment to illustrate the interplay of active particle and crystallographic defects. A moving active particle is observed to be surrounded by localized topological defects, becoming a dressed active particle. Such a physical picture characterizes both the lattice distortion around the moving particle and the healing of the distorted lattice in its trajectory. We find that the dynamical behaviors of an active particle in both random and ballistic motions uniformly conform to this featured scenario, whether the particle is initially a defect or not. We further observe that the defect pattern around a dressed ballistic active particle randomly oscillates between two well-defined wing-like defect motifs regardless of its speed. The established physical picture of dressed active particles in this work partially deciphers the complexity of the intriguing nonequilibrium behaviors in active crystals, and opens the promising possibility of introducing the activity to engineer defects, which has strong connections with the design of materials., Comment: 8 pages, 5 figures