Non-trivial Surfaces Research Articles

Null states and time evolution in a toy model of black hole dynamics

Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states eS in a gravitational system much smaller than the number of states eSp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{S_{\ extrm{p}}} $$\\end{document} predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves N = O(eS) of the possible eSp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{S_{\ extrm{p}}} $$\\end{document} perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for N ≪ eS (in which case no quantum extremal surfaces can arise). In particular, even for small N, the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension D ≪ eS. We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.

Yang-Baxter deformed wedge holography

Yang-Baxter deformed wedge holography

The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method

Abstract We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $C^{k+1}$ surfaces, with degree $k$ (Lagrangian, parametrically curved) approximation of the surface, for any $k \geqslant 1$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $C^{2,1}$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.

Topological properties of noncentrosymmetric superconductors TIr2B2 ( T=Nb,Ta )

A recent experiment reported two new non-centrosymmetric superconductors NbIr$_{2}$B$_{2}$ and TaIr$_{2}$B$_{2}$ with respective superconducting transition temperatures of 7.2 K and 5.2 K and further suggested their superconductivity to be unconventional [K. G\'ornicka \textit{et al}., Adv. Funct. Mater. 2007960 (2020)]. Here, based on first-principles calculations and symmetry analysis, we propose that $T$Ir$_{2}$B$_{2}$ ($T$=Nb, Ta) are topological Weyl metals in the normal state. In the absence of spin-orbit coupling (SOC), we find that NbIr$_{2}$B$_{2}$ has 12 Weyl points, and TaIr$_{2}$B$_{2}$ has 4 Weyl points, i.e. the minimum number under time-reversal symmetry; meanwhile, both of them have a nodal net composed of three nodal lines. In the presence of SOC, a nodal loop on the mirror plane evolves into two hourglass Weyl rings, along with the Weyl points, which are dictated by the nonsymmorphic glide mirror symmetry. Besides the rings, NbIr$_{2}$B$_{2}$ and TaIr$_{2}$B$_{2}$ have 16 and 20 pairs of Weyl points, respectively. The surface Fermi arcs are explicitly demonstrated. On the (110) surface of TaIr$_{2}$B$_{2}$, we find extremely long surface Fermi arcs ($\sim$0.6 ${\text{\AA}}^{-1}$) located 1.4 meV below the Fermi level, which should be readily probed in experiment. Combined with the intrinsic superconductivity and the nontrivial bulk Fermi surfaces, $T$Ir$_{2}$B$_{2}$ may thus provide a very promising platform to explore the three-dimensional topological superconductivity.

Topological d+s wave superconductors in a multiorbital quadratic band touching system

Realization of topological superconductors is one of the most important goals in studies of topological phases in quantum materials. In this work, we theoretically propose a novel way to attain topological superconductors with non-trivial Fermi surfaces of Bogoliubov quasiparticles. Considering the interacting Luttinger model with $j\!=\!3/2$ electrons, we investigate the dominant superconducting channels for a multi-orbital quadratic band-touching system with finite chemical potential, which breaks the particle-hole symmetry in the normal state. Notably, while the system generally favors d-wave pairing, the absence of the particle-hole symmetry necessarily induces parasitic s-wave pairing. Based on the Landau theory with $SO(3)$ symmetry, we demonstrate that two kinds of topological superconductors are energetically favored; uniaxial nematic phase with parasitic $s$ wave pairing ($d_{(3z^2-r^2)}\!+\!s$) and time-reversal-symmetry broken phase with parasitic $s$ wave pairing ($d_{(3z^2-r^2,xy)}\!+\!id_{x^2-y^2}\!+\!s$). These superconductors contain either nodal lines or Fermi pockets of gapless Bogoliubov quasiparticles and moreover exhibit topological winding numbers $\pm2$, leading to non-trivial surface states such as drumhead-like surface states or Fermi arcs. We discuss applications of our theory to relevant families of materials, especially half-heusler compound YPtBi, and suggest possible future experiments.

Freeform irradiance tailoring for light fields.

We show how to correct an optical surface to transform an arbitrary incident light field into a desired irradiance pattern on a projection surface. Beam dilation errors and optical surface corrections are derived from the pullbacks of the actual and desired irradiances. Étendue effects - the principal obstacle to extended-source tailoring - are factored out by solving a sparse linear system. The method accommodates nontrivial projection surfaces, transport phenomena, and incident wavefronts, including those from multiple extended light sources. Numerical experiments achieve high fidelity and contrast ratios in as little as O(NlogN)-time for a surface represented by N height values.

Instability of rapidly accelerating rupture fronts in nanostrips of monolayer hexagonal boron nitride

A molecular structural mechanics model of monolayer hexagonal boron nitride is constructed by finite element (FE) method, in which BN bonds are equated with Timoshenko beam elements. Edge crack is introduced in nanostrip of FE model. Crack propagates straight and smoothly under pure opening displacement-loading, and crack speed reaches up to a stable value of 8.45 km/s finally at tensile loading rate 3.33 m/s of both upper and bottom boundaries. While crack branching or kinking occurs beyond critical speeds of 8.74 km/s and 8.71 km/s at higher loading rates of 16.67 m/s and 8.33 m/s respectively, with the formation of non-trivial crack surfaces. The above results are also examined by molecular dynamics models of the same sizes and geometry. Simultaneously, the critical energy release rate is equal to 0.136 TPa·Å at a critical tensile strain 8.27% with the occurrences of crack instabilities. Moreover, critical strains of crack initiation 5.75% and branching 8.27% are independent of displacement-loading rates.

Non-analytic behavior of the Loschmidt echo in XXZ spin chains: Exact results

We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of "excited-state" thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the N\'eel state, and we determine the first few non-analytic points.

Distinct Evolutions of Weyl Fermion Quasiparticles and Fermi Arcs with Bulk Band Topology in Weyl Semimetals.

The Weyl semimetal phase is a recently discovered topological quantum state of matter characterized by the presence of topologically protected degeneracies near the Fermi level. These degeneracies are the source of exotic phenomena, including the realization of chiral Weyl fermions as quasiparticles in the bulk and the formation of Fermi arc states on the surfaces. Here, we demonstrate that these two key signatures show distinct evolutions with the bulk band topology by performing angle-resolved photoemission spectroscopy, supported by first-principles calculations, on transition-metal monophosphides. While Weyl fermion quasiparticles exist only when the chemical potential is located between two saddle points of the Weyl cone features, the Fermi arc states extend in a larger energy scale and are robust across the bulk Lifshitz transitions associated with the recombination of two nontrivial Fermi surfaces enclosing one Weyl point into a single trivial Fermi surface enclosing two Weyl points of opposite chirality. Therefore, in some systems (e.g., NbP), topological Fermi arc states are preserved even if Weyl fermion quasiparticles are absent in the bulk. Our findings not only provide insight into the relationship between the exotic physical phenomena and the intrinsic bulk band topology in Weyl semimetals, but also resolve the apparent puzzle of the different magnetotransport properties observed in TaAs, TaP, and NbP, where the Fermi arc states are similar.

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern.

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.

Nontrivial minimal surfaces in a class of Finsler 3-spheres

In this paper, we study the rotationally invariant minimal surfaces in the Bao–Shen's spheres, which are a class of 3-spheres endowed with Randers metrics [Formula: see text] of constant flag curvature K = 1, where [Formula: see text] are Berger metrics, [Formula: see text] are one-forms and k > 1 is an arbitrary real number. We obtain a class of nontrivial minimal surfaces isometrically immersed in the Bao–Shen's spheres, which is the first class of nontrivial minimal surfaces with respect to the Busemann–Hausdorff measure in Finsler spheres. Moreover, we also obtain a new class of explicit minimal surfaces in the classical Berger spheres [Formula: see text], which was expected to get in [F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl.28(5) (2010) 593–607].

A Method for Segment Based Surface Reconstruction from Discrete Inclination Values

Object orientation tracking in local reference frame can serve as a useful tool in fields such as surface shape recognition. Often measurement methods provide incomplete data about object orientation. In this paper a method is proposed for selection of surface segment direction when only inclination levels of each segment are known. Method is applicable in cases when some additional constraints can be applied to the surface, such as limited flexion and segment layout. During surface model approximation a specially adapted exhaustive search algorithm is used to find the most appropriate segment direction angle to satisfy physical constraints of the surface. This method proved to significantly increase approximation precision of non-trivial surfaces. DOI: http://dx.doi.org/10.5755/j01.eee.20.2.6380

Generation of Tactile Maps for Artificial Skin

Prior research has shown that representations of retinal surfaces can be learned from the intrinsic structure of visual sensory data in neural simulations, in robots, as well as by animals. Furthermore, representations of cochlear (frequency) surfaces can be learned from auditory data in neural simulations. Advances in hardware technology have allowed the development of artificial skin for robots, realising a new sensory modality which differs in important respects from vision and audition in its sensorimotor characteristics. This provides an opportunity to further investigate ordered sensory map formation using computational tools. We show that it is possible to learn representations of non-trivial tactile surfaces, which require topologically and geometrically involved three-dimensional embeddings. Our method automatically constructs a somatotopic map corresponding to the configuration of tactile sensors on a rigid body, using only intrinsic properties of the tactile data. The additional complexities involved in processing the tactile modality require the development of a novel multi-dimensional scaling algorithm. This algorithm, ANISOMAP, extends previous methods and outperforms them, producing high-quality reconstructions of tactile surfaces in both simulation and hardware tests. In addition, the reconstruction turns out to be robust to unanticipated hardware failure.

On the GUT scale of F-theory [formula omitted

In F-theory GUTs, threshold corrections from Kaluza-Klein massive modes arising from gauge and matter multiplets play an important role in the determination of the weak mixing angle and the strong gauge coupling of the effective low energy model. In this letter we further explore the induced modifications on the gauge couplings running and the GUT scale. In particular, we focus on the KK-contributions from matter curves and analyse the conditions on the chiral and Higgs matter spectrum which imply a GUT scale consistent with the minimal unification scenario. As an application, we present an explicit computation of these thresholds for matter fields residing on specific non-trivial Riemann surfaces.

Doubling constant mean curvature tori in S3

The Clifford tori in S 3 constitute a one-parameter family of flat, two- dimensional, constant mean curvature (CMC) submanifolds. This paper demon- strates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE meth- ods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid into the neighbourhood of each point of a sub-lattice of the Clifford torus; and then one can show that a constant mean curvature perturbation of this submanifold does exist.

Periodic DFT modeling of bulk and surface properties of MgCl2

MgCl(2) is the preferred support for the industrial Ziegler-Natta catalysts, and is believed to act as a template for the epitactic chemisorption of the active Ti species. As the first step of a thorough computational modeling of these systems, we studied the bulk and surface structure of the ordered alpha and beta phases of MgCl(2) by means of periodic DFT (B3LYP) methods using localized basis sets. The layer structure of both phases was reproduced satisfactorily with the inclusion of a (small) empirical dispersion correction ("DFT-D") as a practical method to describe the attraction between the layers. Surface models were studied on slabs with adequate thickness. It appears that various surfaces exposing 5-coordinated Mg are very similar in energy and are the lowest non-trivial surfaces. Cuts exposing 4-coordinated Mg are significantly less stable; both kinetic and equilibrium models of crystal growth indicate that they should normally not be formed to a significant extent. "Nano-ribbons" of single, flat chains of MgCl(2), sometimes proposed as components of the disordered delta phase, were also evaluated, but are predicted to be unstable to rearrangement. Implications for the role of MgCl(2) as catalyst support are discussed.

Anisotropic scattering in angular-dependent magnetoresistance oscillations of quasi-two-dimensional and quasi-one-dimensional metals: Beyond the relaxation-time approximation

The electrical resistivity for a current moving perpendicular to layers (chains) in quasi-two-dimensional (2D) [quasi-one-dimensional (1D)] metals under an applied magnetic field of varying orientation is studied using Boltzmann transport theory. We consider the simplest nontrivial quasi-2D and quasi-1D Fermi surfaces but allow for an arbitrary elastic collision integral (i.e., a scattering probability with arbitrary dependence on momentum transfer) and obtain an expression for the resistivity which generalizes that previously found using a single relaxation-time approximation. The dependence of the resistivity on the angle between the magnetic field and current changes depending on the momentum dependence of the scattering probability. So, whereas zero-field intralayer transport is sensitive only to the momentum-averaged scattering probability (the transport relaxation rate), the resistivity perpendicular to layers measured in a tilted magnetic field provides detailed information about the momentum-dependence of interlayer scattering. These results help us to clarify the meaning of the relaxation rate determined from fits of angular-dependent magnetoresistance oscillations (AMROs) experimental data to theoretical expressions. Furthermore, we suggest how AMRO might be used to probe the dominant scattering mechanism.

Hyperbolic surfaces in the Grassmannian

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations. On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.

Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories

We study the possibility of extending ghost systems with higher spin to a logarithmic conformal field theory. In particular we are interested in c=-26 which turns out to behave very differently to the already known c=-2 case. The energy momentum tensor cannot be built anymore by a combination of derivatives of generalized symplectic fermion fields. Moreover, the logarithmically extended theory is only consistent when considered on nontrivial Riemann surfaces. This results in a LCFT with some unexpected properties. For instance the Virasoro mode L_0 is diagonal and for certain values of the deformation parameters even the whole global conformal group is non-logarithmic.

A Note on the Null Surface Formulation of GR

The purpose of this work is to study the structure and nature of the singularities of wavefronts in flat space-time. We computed the behavior at the singularities of important objects that take place in the null surface formulation of general relativity. As a secondary result we show that the Minkowski space-time with non-trivial null surfaces is a solution of the null surface approach to general relativity.