Search Results (29 total)

Crystalline assemblages of identical subunits packed together and elastically bent in the form of a torus have been found in the past ten years in a variety of systems of surprisingly different nature, such as viral capsids, self-assembled monolayers, and carbon nanomaterials. In this Rapid Communication we analyze the structural properties of toroidal crystals and provide a unified description based on the elastic theory of defects in curved geometries. We find ground states characterized by the presence of fivefold disclinations on the exterior of the torus and sevenfold disclinations in the interior. The number of excess disclinations is controlled primarily by the aspect ratio of the torus, suggesting a mechanism for creating toroidal templates with precisely controlled valency via functionalization of the defect sites.

We study the organization of topological defects in a system of nematogens confined to the two-dimensional sphere (S²). We first perform Monte Carlo simulations of a fluid system of hard rods (spherocylinders) living in the tangent plane of S². The sphere is adiabatically compressed until we reach a jammed nematic state with maximum packing density. The nematic state exhibits four +1/2 disclinations arrayed on a great circle. This arises from the high elastic anisotropy of the system in which splay (K₁) is far softer than bending (K₃). We also introduce and study a lattice nematic model on S² with tunable elastic constants and map out the preferred defect locations as a function of elastic anisotropy. We find a one-parameter family of degenerate ground states in the extreme splay-dominated limit K₃/K₁→∞. Thus the global defect geometry is controllable by tuning the relative splay to bend modulus.

We investigate crystalline order on a two-dimensional paraboloid of revolution by assembling a single layer of millimeter-sized soap bubbles on the surface of a rotating liquid, thus extending the classic work of Bragg and Nye on planar soap bubble rafts. Topological constraints require crystalline configurations to contain a certain minimum number of topological defects such as disclinations or grain boundary scars whose structure is analyzed as a function of the aspect ratio of the paraboloid. We find the defect structure to agree with theoretical predictions and propose a mechanism for scar nucleation in the presence of large Gaussian curvature.

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57∕4−10sqrt[2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9∕8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1∕4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.

We investigate the zero-temperature structure of a crystalline monolayer constrained to lie on a two-dimensional Riemannian manifold with variable Gaussian curvature and boundary. A full analytical treatment is presented for the case of a paraboloid of revolution. Using the geometrical theory of topological defects in a continuum elastic background, we find that the presence of a variable Gaussian curvature, combined with the additional constraint of a boundary, gives rise to a rich variety of phenomena beyond that known for spherical crystals. We also provide a numerical analysis of a system of classical particles interacting via a Coulomb potential on the surface of a paraboloid.

Point defects are ubiquitous in two-dimensional crystals and play a fundamental role in determining their mechanical and thermodynamical properties. When crystals are formed on a curved background, finite-length grain boundaries (scars) are generally needed to stabilize the crystal. We provide a continuum elasticity analysis of defect dynamics in curved crystals. By exploiting the fact that any point defect can be obtained as an appropriate combination of disclinations, we provide an analytical determination of the elastic spring constants of dislocations within scars and compare them with existing experimental measurements from optical microscopy. We further show that vacancies and interstitials, which are stable defects in flat crystals, are generally unstable in curved geometries. This observation explains why vacancies or interstitials are never found in equilibrium spherical crystals. We finish with some further implications for experiments and future theoretical work.

The original Thomson problem of “spherical crystallography” seeks the ground state of electron shells interacting via the Coulomb potential; however one can also study crystalline ground states of particles interacting with other potentials. We focus here on long-range power-law interactions of the form 1∕r^γ (0<γ<2), with the classic Thomson problem given by γ=1. At large R∕a, where R is the sphere radius and a is the particle spacing, the problem can be reformulated as a continuum elastic model that depends on the Young’s modulus of particles packed in the plane and the universal (independent of the pair potential) geometrical interactions between disclination defects. The energy of the continuum model can be expressed as an expansion in powers of the total number of particles, M∼(R∕a)², with coefficients explicitly related to both geometric and potential-dependent terms. For icosahedral configurations of 12 five-fold disclinations, the first nontrivial coefficient of the expansion agrees with explicit numerical evaluation for discrete particle arrangements to four significant digits; the discrepancy in the fifth digit arises from a contribution to the energy that is sensitive to the particular icosadeltahedral configuration and that is neglected in the continuum calculation. In the limit of a very large number of particles, an instability toward grain boundaries can be understood in terms of a “Debye-Huckel” solution, where dislocations have continuous Burgers’ vector “charges.” Discrete dislocations in grain boundaries for intermediate particle numbers are discussed as well.

Toroidal templates such as vesicles with hexatic bond orientational order are discussed. The total energy including disclination charges is explicitly computed for hexatic order embedded in a toroidal geometry. Related results apply for tilt or nematic order on the torus in the one Frank constant approximation. Although there is no topological necessity for defects in the ground state, we find that excess disclination defects are nevertheless energetically favored for fat torii or moderate vesicle sizes. Some experimental consequences are discussed.

We attack the generalized Thomson problem, i.e., determining the ground state energy and configuration of many particles interacting via an arbitrary repulsive pairwise potential on a sphere via a continuum mapping onto a universal long range interaction between angular disclination defects parametrized by the elastic (Young) modulus Y of the underlying lattice and the core energy E_core of an isolated disclination. Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r^γ (0<γ<2) to four significant figures. The generality of our approach is illustrated by a study of grain boundary proliferation for tilted crystalline order and square lattices on the sphere.

The formation of vortex loops (global cosmic strings) in an O(2) linear sigma model in three spatial dimensions is analyzed numerically. For over-damped Langevin dynamics we find that defect production is suppressed by an interaction between correlated domains that reduces the effective spatial variation of the phase of the order field. The degree of suppression is sensitive to the quench rate. A detailed description of the numerical methods used to analyze the model is also reported.

We determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaquettes, to be σ  =  -0.37(6), in statistical agreement with the Poisson ratio of phantom fixed-connectivity membranes σ  =  -0.32(4). Together with the equality of critical exponents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish that physical fixed-connectivity membranes provide a wide class of auxetic (negative Poisson ratio) materials with significant potential applications in materials science.

We propose and analyze an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting disclinations is mapped to a Laplacian sine-Gordon Hamiltonian suitable for numerical simulations. We discuss general features of the ground state and thereafter specialize to the spherical case. The ground state is analyzed as a function of the ratio of the defect core energy to the Young’s modulus. We argue that the core energy contribution becomes less and less important in the limit R≫a, where R is the radius of the sphere and a is the particle spacing. For large core energies there are 12 disclinations forming an icosahedron. For intermediate core energies unusual finite-length grain boundaries are preferred. The complicated regime of small core energies, appropriate to the limit R/a→∞, is also addressed. Finally we discuss the application of our results to the classic Thomson problem of finding the ground state of electrons distributed on a two sphere.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we compute the critical exponents using an ɛ expansion about the upper critical embedding dimension for general internal dimension D and embedding dimension d. We then exhibit various methods for reliably extrapolating to the physical point (D=2,d=3). Our most accurate estimates are ν=0.62 for the Flory exponent and ζ=0.80 for the roughness exponent.

We study the formation of domains in a continuous phase transition with a finite-temperature quench. The model treated is the Φ⁴ theory in two spatial dimensions with global O(2) symmetry. We investigate this using real-time thermal field theory, following Boyanovsky and collaborators, and find that domain sizes appear to be smaller than those produced in an instantaneous quench in the tree-level approximation. We also propose that a more physical picture emerges by examining the two-point functions which do not involve any cutoff on the short wavelength Goldstone modes.

We study the tubular phase of self-avoiding anisotropic membranes. We discuss the renormalizability of the model Hamiltonian describing this phase, and from a renormalization group equation derive some general scaling relations for the exponents of the model. We show how particular choices of renormalization factors reproduce the Gaussian result, the Flory theory, and the Gaussian variational treatment of the problem. We then study the perturbative renormalization to one loop in the self-avoiding parameter using dimensional regularization and an ε expansion about the upper critical dimension, and determine the critical exponents to first order in ε.

We provide the first numerical evidence for the existence of a tubular phase, predicted by Radzihovsky and Toner (RT), for anisotropic tethered membranes without self-avoidance. Incorporating anisotropy into the bending rigidity of a simple model of a tethered membrane with free boundary conditions, we show that the model indeed has two phase transitions corresponding to the flat-to-tubular and tubular-to-crumpled transitions. For the tubular phase we measure the Flory exponent ν_F and the roughness exponent ζ. We find ν_F  =  0.305(14) and ζ  =  0.895(60), which are in reasonable agreement with the theoretical predictions of RT; ν_F  =  1/4 and ζ  =  1.

It is shown that a black hole of fixed mass can carry arbitrary axionic charge. The unique static black-hole solution is found to have vanishing axion field strength but nonvanishing potential. The axion charge cannot be detected by point particles, but can be detected by strings in a process analogous to the Aharonov-Bohm effect. It is argued that the existence of axion charge may play a significant role in the late stages of black-hole evaporation.

We argue that the Kaaumlhler geometry of the loops on space-time describes bosonic string theory. The Kaaumlhler potential is the dynamical (field) variable of closed-bosonic-string theory. The equation of motion for this field is that a generalized Ricci tensor vanishes. Loops on flat space constitute a solution only in 26 dimensions.

The spontaneous generation of parity- (P) and time-reversal- (T) violating masses in (2+1)-dimensional QED is studied in the large-N limit, where N is the number of two-component complex fermions. Energy considerations of various symmetry-breaking patterns indicate that P and T are not spontaneously broken, even though masses which individually violate these symmetries are dynamically generated.

A detailed analysis is given of chiral-symmetry breaking in the large-flavor (N) limit of quantum electrodynamics in (2+1) dimensions. Analytical and numerical solutions of the homogeneous Dyson-Schwinger equation for the fermion self-energy combined with a computation of the effective potential for the fermion bilinear show that it is energetically preferable for the theory to dynamically generate a mass for fermions. The magnitude of the mass is roughly exponentially suppressed in N from the fundamental dimensionful scale α≡N e² of the gauge coupling constant, but the scale at which the self-mass begins to damp rapidly appears to be of order α, so that there is no spontaneous breaking of an approximate scale invariance that the underlying theory possesses at momentum small compared to α. Higher-order 1/N corrections are analyzed and it is shown that the 1/N expansion can be used consistently to demonstrate chiral-symmetry breaking. Open issues and possible improvements of the analysis are given and some avenues for future investigation suggested.

We argue that the massive modes of the superstring can play an important role in the last stages of black-hole evaporation. If the Bekenstein-Hawking entropy is the true statistical entropy of an evaporating black hole, it becomes probable for a black hole to disappear by making a transition to an excited string state. This excited string state can then decay to massless radiation, avoiding the naked singularity of the semiclassical picture. We also construct the energy-volume phase diagram separating the three phases: pure radiation, black hole and radiation, and massive string modes and radiation.

Chiral-symmetry breaking in (2+1)-dimensional QED is studied in the many-flavor limit. Analytical and numerical solutions of the Dyson-Schwinger equation are found. Substitution of the symmetry-breaking solution in the composite-operator effective potential indicates that it is favored over the symmetric solution. Improvements and possible extensions of the analysis are discussed.