Symmetries of cosmological Cauchy horizons

We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic toT ³ we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT ³ and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.     

Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings

We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to generate three-dimensional Euclidean (E³) networks. The technique involves projection of edges of tilings of the hyperbolic plane (H²) onto three-periodic minimal surfaces, embedded in E³. Given the extraordinary wealth of symmetries commensurate with H², we can generate networks in E³ that are difficult to construct otherwise. In particular, we form four-, five- and seven-connected (E³) nets containing three- and five-rings, viz. (3, 7), (5, 4) and (5, 5) tilings in H². Received 14 January 2002 / Received in final form 12 August 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: stephen.hyde@anu.edu.au     

Nonlinear holomorphic supersymmetry on Riemann surfaces

We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature metrics. The cases of the sphere and Lobachevski plane are elaborated in detail. The partial algebraization of the spectrum of the corresponding Hamiltonians is proved by the reduction to one-dimensional quasi-exactly solvable families. It is found that these families possess the “duality” transformations, which form a discrete group of symmetries of the corresponding 1D potentials and partially relate the spectra of different 2D systems. The algebraic structure of the systems on the sphere and hyperbolic plane is explored in the context of the Onsager algebra associated with the nonlinear holomorphic supersymmetry. Inspired by this analysis, a general algebraic method for obtaining the covariant form of integrals of motion of the quantum systems in external fields is proposed.     

The Rayleigh hypothesis in scattering off photonic crystal interfaces

We examine the Rayleigh hypothesis in the context of scattering of light off photonic crystal interfaces. First, the hypothesis - which was initially suggested for scattering of waves off rough surfaces between homogeneous media - is rephrased to apply to photonic crystal interfaces. Next, an exact and explicit functional form is presented that maps plane photonic crystal surfaces to periodic rough surfaces in free space, so that known criteria for the validity of the Rayleigh hypothesis for scattering at rough surfaces can be applied directly to scattering at the photonic crystal surfaces. The same map also allows the scattering problem to be solved exactly.     

From 2D hyperbolic forests to 3D Euclidean entangled thickets

A method is developed to construct and analyse a wide class of graphs embedded in Euclidean 3D space, including multiply-connected and entangled examples. The graphs are derived via embeddings of infinite families of trees (forests) in the hyperbolic plane, and subsequent folding into triply periodic minimal surfaces, including the P, D, gyroid and H surfaces. Some of these graphs are natural generalisations of bicontinuous topologies to bi-, tri-, quadra- and octa-continuous forms. Interwoven layer graphs and periodic sets of finite clusters also emerge from the algorithm. Many of the graphs are chiral. The generated graphs are compared with some organo-metallic molecular crystals with multiple frameworks and molecular mesophases found in copolymer melts. Received 10 December 1999     

Elastic surface waves in crystals. Part 1: review of the physics

We present a review of wave propagation at the surface of anisotropic media (crystal symmetries). The physics for media of cubic and hexagonal symmetries has been extensively studied based on analytical and semi-analytical methods. However, some controversies regarding surfaces waves and the use of different notations for the same modes require a review of the research done and a clarification of the terminology. In a companion paper we obtain the full-wave solution for the wave propagation at the surface of media with arbitrary symmetry (including cubic and hexagonal symmetries) using two spectral numerical modeling algorithms.     

Beltrami Gauge Symmetry and 2D Induced Gravity

We introduce the Beltrami gauge symmetry related to the Beltrami parametrization of the metrics and the Beltrami equations. We explore its role in 2d induced gravity and show that Polynkov's. SL(R) and KPZ's residual symmetries are subsymmetries of the Beltrami gnuge symmetry in the light-cone gauge. We also find that the 2d induced gravity with or without matter can be reformulated as Beltrami-Liouville field theories on Riemann surfaces of higher genus.     

Optical nonreciprocity in magnetic structures related to high-Tc superconductors

Rotation of the plane of polarization of reflected light (Kerr effect) is a direct manifestation of broken time-reversal symmetry and is generally associated with the appearance of a ferromagnetic moment. Here I identify magnetic structures that may arise within the unit cell of cuprate superconductors that generate polarization rotation despite the absence of a net moment. For these magnetic symmetries the Kerr effect is mediated by magnetoelectric coupling, which can arise when antiferromagnetic order breaks inversion symmetry. The structures identified are candidates for a time-reversal breaking phase in the pseudogap regime of the cuprates.     

A note on the two symmetry-preserving covering maps of the gyroid minimal surface

Our study of the gyroid minimal surface has revealed that there are two distinct covering maps from the hyperbolic plane onto the surface that respect its intrinsic symmetries. We show that if a decoration of is chiral, the projection of this pattern via the two covering maps gives rise to distinct structures in .     

Anisotropic photonic crystals: Generalized plane wave method and dispersion symmetry properties

This article presents a generalized vector plane wave expansion method, applicable to isotropic and anisotropic periodic dielectric media of arbitrary geometry and dimension. The influence of anisotropic material orientation on the symmetry properties of photonic crystal dispersion surface is discussed. It is shown that the overall Brillouin zone symmetry is formed by the intersection of the photonic crystal lattice symmetry and the symmetry determined by the anisotropic material orientation. This work explains how to define the irreducible Brillouin zone of a two-dimensional anisotropic photonic crystal and demonstrates that doing it correctly allows one to avoid erroneous results, when calculating band gap diagrams of anisotropic photonic crystals. With the help of the methods presented, the possibility of controlling the band gaps of anisotropic photonic crystals by means of external electric field is shown.     

Occurrence of rotation domains in heteroepitaxy

Heteroepitaxy can involve materials with a misfit of crystal structure. Rotation domains in the epilayer are a fundamental consequence. We derive a general expression for their (minimum) number which is determined by the mismatch of the rotational symmetries of the substrate and epilayer. In the case of a mismatch of rotational symmetry, the number of rotation domains of material A on material B is different from that of B on A. A larger number of rotation domains can occur due to domain structure or nearly fulfilled additional symmetries of the substrate surface.     

Spectral density of surface states for TiO and VO: (001) surface

We present a calculation of the spectral density of states for crystals of TiO and VO with (001) cleaved surfaces. The transfer matrix formalism is employed, with a hamiltonian including the O 2p and the metal 3d orbitals. Slater-Koster parameters obtained by Mattheiss (1972) from a bulk calculation are used. The results are obtained at three special points in the 2D Brillouin zone for three different (001) planes: surface plane, plane immediately below it and bulk plane. In the neighborhood of the point M there is an intrinsic surface state and a surface resonance due to the hybridization between ligand and metal orbitals. The surface state has mixed e_g and p_z symmetry along the Σ symmetry line and lies above the Fermi level. No surface relaxation or reconstruction is considered.     

Discrete gauge symmetry anomalies

It has been recently argued that quantum gravity effects strongly violate all non-gauge symmetries. This would suggest that all low energy discrete symmetries should be gauge symmetries, either continuous or discrete. Acceptable continuous gauge symmetries are constrained by the condition they should be anomaly free. We show here that any discrete gauge symmetry should also obey certain “discrete anomaly cancellation” conditions. These conditions strongly constrains the massles fermion content of the theory and follow from the “parent” cancellation of the usual continuous gauge anomalies. They have interesting applications in model building. As an example we consider the constraints on the Z_N “generalized matter parities” of the supersymmetric standard model. We show that only a few (including the standard R-parity) are “discrete anomaly free” unless the fermion content of the minimal supersymmetric standard model is enlarged.     

Symmetry properties of periodic orbits extracted from scattering data

Discrete symmetries of a system are reflected in the properties of the shortest periodic orbits. By applying a recent method to extract these from the scaling of the fractal structure in scattering functions, we show how the symmetries can be extracted from scattering data simultaneously with the periods and the Lyapunov exponents. We pay particular attention to the change of scattering data under a small symmetry breaking.     

Quasiperiodicity in irrational interfaces

Recent work on quasiperiodicity in irrational grain boundaries is reviewed and generalized to heterophase interfaces. The concept of local isomorphism is shown to be very important at irrational interfaces, replacing translational invariance at periodic interfaces. Symmetry related variants of irrational interfaces are enumerated and line defects separating energetically degenerate domains of irrational interfaces are characterised by products of space group operations, one from each crystal. The symmetry operations relating the variants belong to a six dimensional crystal and the variants are locally isomorphic. A framework is described for discussing the symmetries and long range order at interfaces regardless of whether they are rational or irrational.     

Band crossing and surface states in solids

A symmetry criterion for the existence of surface states in model calculations in solids is derived. The criterion applies to surfaces that coincide with symmetry planes of the ideal crystal. A detailed analysis is carried out for one-dimensional model calculations and it is shown how band crossing can be interpreted in the framework of the crystal symmetry.     

Models for a Finite Universe

A method of constructing 3-dimensional hyperbolic manifolds is described. Because of their high degree of symmetry, these may be suitable models for a finite universe. Because their group of symmetries is different from any in the list of manifolds given in Hodgeson and Weeks (available at ftp://ftp.northnet.org/pub/weeks/snappea/closedcensus), it is claimed that these are new.     

Periodic-cylinder vesicle with minimal energy

We give some details about the periodic cylindrical solution found by Zhang and Ou-Yang in [1996 Phys.Rev.E 53 4206] for the general shape equation of vesicle.Three different kinds of periodic cylindrical surfaces and a special closed cylindrical surface are obtained.Using the elliptic functions contained in mathematic,we find that this periodic shape has the minimal total energy for one period when the period-amplitude ratio β 1.477,and point out that it is a discontinuous deformation between plane and this periodic shape.Our results also are suitable for DNA and multi-walled carbon nanotubes(MWNTs).     

Comparing light-induced colloidal quasicrystals with different rotational symmetries

Quasicrystals are aperiodic structures with long-range orientational order. Unlike crystals, quasicrystals can, in principle, possess any non-crystallographic rotational symmetry. However, only a few of these rotational symmetries have been observed. By using Monte Carlo simulations of colloidal particles in laser interference patterns with quasicrystalline symmetry, we compare the onset of quasicrystalline order for different rotational symmetries in two dimensions. We find that quasicrystals with 5-, 8-, 10-, and 12-fold rotational axes can be induced with lower laser intensities than quasicrystals with other non-crystallographic rotational symmetries. We relate this finding to the number of local symmetry centers in the respective interference patterns.