Reflection Symmetry Research Articles

Reflection parity and space-time parity photonic conservation laws in parametric nonlinear optics

Conservation laws are some of the most generic and useful concepts in physics. In nonlinear optical parametric processes, conservation of photonic energy, momenta and parity often lead to selection rules, restricting the allowed polarization and frequencies of the emitted radiation. Here we present a scheme to derive conservation laws in optical parametric processes in which many photons are annihilated and a single photon is emitted. We first rederive with it the known nonlinear optical conservation laws, and then utilize it to predict and explore conservations of reflection parity and space-time parity. Conservation of arises from a generalized reflection symmetry of the polarization in a superspace, analogous to the superspace employed in the study of quasicrystals. Conservation of similarly arises from space-time reversal symmetry in superspace. We explore these conservation laws numerically in the context of high-harmonic generation and outline experimental setups where they can be tested. Published by the American Physical Society 2024

Three perspectives on entropy dynamics in a non-Hermitian two-state system

Abstract A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. To begin with, we illustrate the phase portrait of this non-Hermitian model on the Bloch sphere, elucidating the changes in behavior as one moves across the phase transition boundary, as well as the emergent feature of unidirectional state evolution in the spontaneously broken PT-symmetry regime. This is followed by an examination of the purity and entropy dynamics. Here we distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken PT symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its PT-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the PT charge operator C. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.

Optical Singularities in Photonic Microstructures with Rosette Symmetries: A Unified Theoretical Scheme

Optical singularities are topological defects of electromagnetic fields; they include phase singularity in scalar fields, polarization singularity in vector fields, and three-dimensional (3D) singularities such as optical skyrmions. The exploitation of photonic microstructures to generate and manipulate optical singularities has attracted wide research interest in recent years, with many photonic microstructures having been devised to this end. Accompanying these designs, scattered phenomenological theories have been proposed to expound the working mechanisms behind individual designs. In this work, instead of focusing on a specific type of microstructure, we concentrate on the most common geometric features of these microstructures—namely, symmetries—and revisit the process of generating optical singularities in microstructures from a symmetry viewpoint. By systematically employing the projection operator technique in group theory, we develop a widely applicable theoretical scheme to explore optical singularities in microstructures with rosette (i.e., rotational and reflection) symmetries. Our scheme agrees well with previously reported works and further reveals that the eigenmodes of a symmetric microstructure can support multiplexed phase singularities in different components, such as out-of-plane, radial, azimuthal, and left- and right-handed circular components. Based on these phase singularities, more complicated optical singularities may be synthesized, including C points, V points, L lines, Néel- and bubble-type optical skyrmions, and optical lattices, to name a few. We demonstrate that the topological invariants associated with optical singularities are protected by the symmetries of the microstructure. Lastly, based on symmetry arguments, we formulate a so-called symmetry matching condition to clarify the excitation of a specific type of optical singularity. Our work establishes a unified theoretical framework to explore optical singularities in photonic microstructures with symmetries, shedding light on the symmetry origin of multidimensional and multiplexed optical singularities and providing a symmetry perspective for exploring many singularity-related effects in optics and photonics.

Symmetric Piecewise Developable Approximations

AbstractWe propose a novel method for generating symmetric piecewise developable approximations for shapes in approximately global reflectional or rotational symmetry. Given a shape and its symmetry constraint, the algorithm contains two crucial steps: (i) a symmetric deformation to achieve a nearly developable model and (ii) a symmetric segmentation aided by the deformed shape. The key to the deformation step is the use of the symmetric implicit neural representations of the shape and the deformation field. A new mesh extraction from the implicit function is introduced to construct a strictly symmetric mesh for the subsequent segmentation. The symmetry constraint is carefully integrated into the partition to achieve the symmetric piecewise developable approximation. We demonstrate the effectiveness of our algorithm over various meshes.

Absence of bound states for quantum walks and CMV matrices via reflections

We give a criterion based on reflection symmetries in the spirit of Jitomirskaya–Simon to show absence of point spectrum for (split-step) quantum walks and Cantero–Moral–Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue–Morse subshift.

Lagrangian approach to origami vertex analysis: kinematics.

The use of origami in engineering has significantly expanded in recent years, spanning deployable structures across scales, folding robotics and mechanical metamaterials. However, finding foldable paths can be a formidable task as the kinematics are determined by a nonlinear system of equations, often with several degrees of freedom. In this article, we leverage a Lagrangian approach to derive reduced-order compatibility conditions for rigid-facet origami vertices with reflection and rotational symmetries. Then, using the reduced-order conditions, we derive exact, multi-degree of freedom solutions for degree 6 and degree 8 vertices with prescribed symmetries. The exact kinematic solutions allow us to efficiently investigate the topology of allowable kinematics, including the consideration of a self-contact constraint, and then visually interpret the role of geometric design parameters on these admissible fold paths by monitoring the change in the kinematic topology. We then introduce a procedure to construct lower-symmetry kinematic solutions by breaking symmetry of higher-order kinematic solutions in a systematic way that preserves compatibility. The multi-degree of freedom solutions discovered here should assist with building intuition of the kinematic feasibility of higher-degree origami vertices and also facilitate the development of new algorithmic procedures for origami-engineering design.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

The breaking of μ − τ reflection symmetry in a B − L extended 3HDM with (Z2 × Z4)⋊Z2 (II) symmetry

We propose a new neutrino mass matrix texture satisfying the μ−τ reflection symmetry in a B−L model with three Higgs doublets based on non-Abelian discrete symmetry (Z2×Z4)⋊Z2(II). The Dirac and Majorana mass matrices, which is motivated in the framework of the Type I seesaw mechanism in association with (Z2×Z4)⋊Z2(II) symmetry, hint for a realistic lepton mixing pattern where the co-bimaximal mixing pattern is broken by the charged-lepton contribution. The hierarchy of lepton masses is naturally achieved and the constraints on the effective neutrino mass are consistent with the recent experimental limits.

Topologically protected bound modes in non-neighboring hexagonal lattices

In the last decade, the study of wave localization and guiding has gained a renewed interest thanks to the introduction of the concept of topological protection. Originally developed in the field of quantum mechanics, topological protection has successively inspired the search for its analogue in other physical domains, including elasticity. In this context, periodic hexagonal lattices have been proposed as ideal candidates to realize these concepts. In this work, we explore the dynamic behavior of discrete hexagonal lattices with third nearest neighbor connections. First, the formation and transition of multiple Dirac cones above a critical value of the relative stiffness between the first and third nearest neighbor connections is derived. Then, the formation of extremely localized modes (i.e., bound modes) at the interface between regions formed by lattices which are inverted copies of each other is demonstrated. Our results show that this localization derives from a type of interface satisfying a point reflection symmetry for which the considered chain decouples into two identical copies at the high symmetry point of the Brillouin zone. Our findings open novel avenues for topological waveguiding and confinement with potential applications in surface acoustic wave devices.

Double-reflection symmetry of thermal convection for Rayleigh number up to 1010

The Rayleigh–Bénard convection system exhibits certain known symmetries at low Rayleigh numbers that are broken as the Rayleigh number increases. In this study, we investigate the statistical symmetry of Rayleigh–Bénard convection at moderately high Rayleigh numbers through direct numerical simulations. The simulations are conducted for a fluid confined within two-dimensional walls, with an aspect ratio of unity and a fixed Prandtl number. Although elliptical large-scale circulations break both left-right and top-down reflection symmetries, we observe the emergence of a restored double-reflection symmetry. This symmetry is evident in the velocity and temperature fields, as well as in the variations of mean velocity and temperature profiles along the streamwise direction and the characteristics of the kinetic and thermal boundary layers. For Rayleigh numbers ranging between 107 and 1010, our results demonstrate a remarkable data collapse under this double-reflection transformation.

Efficient learning of continuous-variable quantum states

The characterization of continuous-variable quantum states is crucial for applications in quantum communication, sensing, simulation, and computing. However, a full characterization of multimode quantum states requires a number of experiments that grows exponentially with the number of modes. Here we propose an alternative approach where the goal is not to reconstruct the full quantum state, but rather to estimate its characteristic function at a given set of points. For multimode states with reflection symmetry, we show that the characteristic function at M points can be estimated using only O(logM) copies of the state, independently of the number of modes. When the characteristic function is known to be positive, as in the case of squeezed vacuum states, the estimation is achieved by an experimentally friendly setup using only beamsplitters and homodyne measurements. Published by the American Physical Society 2024

Observational features of reflection asymmetric black holes

The Kerr spacetime is symmetric with respect to a well-defined equatorial plane. When testing the equatorial reflection symmetry of an isolated black hole, one is at the same time testing the Kerr hypothesis in General Relativity. In this work, we investigate the possible observational features when a Keplerian disk is surrounding a rotating black hole without reflection symmetry. When such symmetry is broken, generically, the photon trajectories around the black hole and the Keplerian orbits on the accretion disk are distorted vertically away from the equatorial plane by an amount that depends on their distance to the black hole. In the reflection asymmetric spacetime we are considering, these two kinds of orbits are distorted in opposite directions. Interestingly, while the size and shape of black hole shadows closely resemble those of Kerr black holes, distinct observational characteristics can emerge in the disk image and emission line profiles. When observing the disk edge-on, a pronounced concave shape may appear along its innermost edge on the incoming side. Furthermore, distinctive horn-like features might be observed on the spectral line profile at the blue-shifted side. These special features can serve as compelling indicators of the reflection asymmetry present in rotating black holes.

Nonideal Magnetohydrodynamic Instabilities in Protoplanetary Disks: Vertical Modes and Reflection Asymmetry

Magnetized disk winds and wind-driven accretion are an essential and intensively studied dispersion mechanism of protoplanetary disks (PPDs). However, the stability of these mechanisms has yet to be adequately examined. This paper employs semi-analytic linear perturbation theories under nonideal magnetohydrodynamics, focusing on disk models whose magnetic diffusivities vary by a few orders of magnitude from the disk midplane to its surface. Linear modes are distinguished by their symmetry with respect to the midplane. These modes have qualitatively different growth rates: symmetric modes almost always decay, while at least one antisymmetric mode always has a positive growth rate. This growth rate decreases faster than the Keplerian angular velocity with cylindrical radius R in the disk and scales steeper than R −5/2 in the fiducial disk model. The growth of antisymmetric modes breaks the reflection symmetry across the disk equatorial plane, and may occur even in the absence of the Hall effect. In the disk regions where fully developed antisymmetric modes occur, accretion flows appear only on one side of the disk, while disk winds occur only on the other. This may explain the asymmetry of some observed PPD outflows.

Boundary condition and reflection anomaly in $2+1$ dimensions

It is known that the 2+12+1d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The problem is the absence of a corresponding outgoing wave to a specific incoming wave into the boundary, which leads to the non-conservation of the energy. For 16 Majorana fermions, it is possible to connect every incoming wave to an outgoing wave without breaking the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in 3+13+1 dimensions.

Harbor Detection in Polarimetric SAR Images Based on Context Features and Reflection Symmetry

Harbor Detection in Polarimetric SAR Images Based on Context Features and Reflection Symmetry

Synthetic Weyl points in plasmonic chain with simultaneous inversion and reflection symmetry breaking

Synthetic Weyl points in plasmonic chain with simultaneous inversion and reflection symmetry breaking

An Optimal Family of Eighth-Order Methods for Multiple-Roots and Their Complex Dynamics

We present a new family of optimal eighth-order numerical methods for finding the multiple zeros of nonlinear functions. The methodology used for constructing the iterative scheme is based on the approach called the ‘weight factor approach’. This approach ingeniously combines weight functions to enhance convergence properties and stability. An extensive convergence analysis is conducted to prove that the proposed scheme achieves optimal eighth-order convergence, providing a significant improvement in efficiency over lower-order methods. Furthermore, the applicability of these novel methods to some real-world problems is demonstrated, showcasing their superior performance in terms of speed and accuracy. This is illustrated through a series of three examples involving basins of attraction with reflection symmetry, confirming the dominance of the new methods over existing counterparts. The examples highlight not only the robustness and precision of the proposed methods but also their practical utility in solving the complex nonlinear equations encountered in various scientific and engineering domains. Consequently, these eighth-order methods hold great promise for advancing computational techniques in fields that require the resolution of multiple roots with high precision.

The role of task on the human brain's responses to, and representation of, visual regularity defined by reflection and rotation

Identifying and segmenting objects in an image is generally achieved effortlessly and is facilitated by the presence of symmetry: a principle of perceptual organisation used to interpret sensory inputs from the retina into meaningful representations. However, while imaging studies show evidence of symmetry selective responses across extrastriate visual areas in the human brain, whether symmetry is processed automatically is still under debate. We used functional Magnetic Resonance Imaging (fMRI) to study the response to and representation of two types of symmetry: reflection and rotation. Dot pattern stimuli were presented to 15 human participants (10 female) under stimulus-relevant (symmetry) and stimulus-irrelevant (luminance) task conditions. Our results show that symmetry-selective responses emerge from area V3 and extend throughout extrastriate visual areas. This response is largely maintained when participants engage in the stimulus irrelevant task, suggesting an automaticity to processing visual symmetry. Our multi-voxel pattern analysis (MVPA) results extend these findings by suggesting that not only spatial organisation of responses to symmetrical patterns can be distinguished from that of non-symmetrical (random) patterns, but also that representation of reflection and rotation symmetry can be differentiated in extrastriate and object-selective visual areas. Moreover, task demands did not affect the neural representation of the symmetry information. Intriguingly, our MVPA results show an interesting dissociation: representation of luminance (stimulus irrelevant feature) is maintained in visual cortex only when task relevant, while information of the spatial configuration of the stimuli is available across task conditions. This speaks in favour of the automaticity for processing perceptual organisation: extrastriate visual areas compute and represent global, spatial properties irrespective of the task at hand.

Numerical study of a generic ship's airwake for understanding bi‐stability mechanism

A distinguishing feature of the bi-stable wake is that the wake persists in either of two preferred states for a sufficiently long time. Aiming to understand the persistence mechanism, this paper numerically investigates the airwake characteristics of the Chalmers ship model (CSM) using large eddy simulation with a wall-adapting local-eddy viscosity model and is complemented by experimental testings for validations. There are two cases of interest: (i) the baseline CSM with a sharp-edged superstructure front that induces massive boundary layer separation; (ii) the front-rounded (FR) CSM with suppressed flow separation. During a characteristic time ( $t^*$ ) of 1142 (26.5 s), the baseline case has a frequently switching wake, whereas the FR wake maintains a stable asymmetric structure with only one switch attempt. To understand the different wake behaviours, the study starts by analysing wake flow structures, vortex cores and the wake dynamics, followed by investigating the instantaneous flow physics. Results suggest that the baseline wake has a weak bi-stable pattern, whereas the FR wake behaves similarly to a reflectional symmetry breaking state of a potential bi-stable wake. The wake switching is found to be driven by the tilting of (vertical-oriented) $z$ -vorticity sheets from either side of the base toward the centre. This tilting behaviour is subjected to the high-magnitude vorticity that sheds from the upstream flow separation at the front sharp edges. With the sharp edges rounded in the FR case, the upstream vorticity is mitigated and the tilting effect is significantly reduced, leading to a more stable wake structure. The reasoning provided in the paper potentially explains the persistence mechanism of the bi-stable wake.

Discrete Space Deep Reinforcement Learning Algorithm Based on Support Vector Machine Recursive Feature Elimination

Algorithms for training agents with experience replay have advanced in several domains, primarily because prioritized experience replay (PER) developed from the double deep Q-network (DDQN) in deep reinforcement learning (DRL) has become a standard. PER-based algorithms have achieved significant success in the image and video domains. However, the exceptional results observed in images and videos are not as effective in many domains with simple action spaces and relatively small states, particularly in discrete action spaces with sparse rewards. Moreover, most advanced techniques may improve sampling efficiency using deep learning algorithms rather than reinforcement learning. However, there is growing evidence that deep learning algorithms cannot generalize during training. Therefore, this study proposes an algorithm suitable for discrete action space environments that uses the sample efficiency of PER based on DDQN but incorporates support vector machine recursive feature elimination (SVM-RFE) without enhancing the sampling efficiency through deep learning algorithms. The proposed algorithm exhibited considerable performance improvements in classical OpenAI Gym environments that did not use images or videos as inputs. In particular, simple discrete space environments with reflection symmetry, such as Cart–Pole, exhibited a faster and more stable learning process. These results suggest that the application of SVM-RFE, which leverages the orthogonality of support vector machines (SVMs) across learning patterns, can be appropriate when the data in the reinforcement learning environment demonstrate symmetry.

Inflationary Cosmology from Anti-de Sitter Wormholes.

We propose a new type of wave function for the Universe computed from the Euclidean path integral, with asymptotically anti-de Sitter (AdS) boundary conditions. In the semiclassical limit, it describes a Euclidean (half-)wormhole geometry, exhibiting a local maximum of the scale factor at the surface of reflection symmetry, giving rise to an expanding Universe upon analytic continuation to Lorentzian signature. We find that these Euclidean wormholes set natural initial conditions for inflation and that the semiclassical Wheeler-DeWitt wave function can favor a long-lasting inflationary epoch, resolving a well-known issue of the no-boundary proposal. Because of the asymptotic AdS conditions in the Euclidean past, they raise the possibility of describing the physics of inflating cosmologies and their perturbations within the context of holography.