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We show in this paper that the technologically relevant field-like spin–orbit torque (SOT) shows resilience against the geometrical effect of electron backscattering. As a device grows smaller in size, the effect of geometry on physical properties like spin torque, and hence switching current could place a physical limit on the continued shrinkage of such a device — a necessary trend of all memory devices (MRAM). The geometrical effect of curves has been shown to impact quantum transport and topological transition of Dirac and topological systems. In our work, we have ruled out the potential threat of line curves degrading the effectiveness of SOT switching. In other words, SOT switching will be resilient against the influence of curves that line the circumferences of defects in the events of electron backscattering, which commonly happens in the channel of modern electronic devices. [ABSTRACT FROM AUTHOR]

Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics, chaotic science, and non-Euclidean geometry. We report the sharp difference between the regular and chaotic motions of cavity photons subjected to the varying space curvature. While the island modes of regular motion rise in the phase diagram in the curved space, the chaotic modes show special mechanisms to adapt to the space curvature, including the fast diffusion of ray dynamics, and the localization and hybridization of the Husimi wavepackets among different periodic orbits. These observations are unique effects enabled by the combination of the chaotic trajectory, the wave nature of light, and the non-Euclidean orbital motion, and therefore make the system a versatile optical simulator for chaotic science under quantum mechanics in curved space-time. [ABSTRACT FROM AUTHOR]

Self-assembly of nanoscale objects is of essential importance in materials science, condensed matter physics, and biophysics. Curvature modifies the principles and sequence of self-assembly in Euclidean space, resulting in unique and more complex structures. Understanding self-assembly behavior in curved space is not only instrumental for designing structural building blocks and assembly processes from a bottom-up perspective but is also critically important for delineating various biological systems. In this review, we summarize efforts made to unveil the physical nature of self-assembly in curved space through experiments and simulations. First, we outline the differences in the physical nature of self-assembly between curved space and Euclidean space by presenting relevant results of experiments and simulations. Second, we explore the principles of self-assembly in curved space at multiple scales and interactions, elucidating important factors that govern the self-assembly process from the perspectives of confinement and structural building blocks. Finally, we enumerate practical applications and control strategies for self-assembly in curved space and outline the challenges and prospects in this field. We hope that this review will encourage further efforts toward fundamental research and broaden the potential applications of designed assemblies in curved space. [ABSTRACT FROM AUTHOR]

ABSTRACTTransformation optics have been an essential paradigm to manipulate electromagnetic waves on the subwavelength scale and have brought various functional photonic architectures into integrated photonic chips. On the other hand, in the spirit of analogical thinking, classical and quantum simulations of general relativity have been extensively studied in diverse physical systems. In this review, we summarize recent advances in analogical gravitation based on integrated photonic chips with the aid of transformation optics. Meanwhile, different types of transformation optical structures, such as gradient waveguides, metasurface waveguides, waveguides on curved space and gradient waveguide arrays, emulating a variety of phenomena in curved space are reviewed, including the gravitational lensing of black holes, Einstein rings, cosmic strings, the particle pair evolution near the event horizon and so on. Furthermore, perspectives for the study of analogical gravitation based on integrated photonic chips are discussed.

Chaos, namely exponential sensitivity to initial conditions, is generally considered a nuisance, inasmuch as it prevents long‐term predictions in physical systems. Here, an easily accessible approach to undo deterministic chaos and tailor ray trajectories in arbitrary 2D optical billiards by introducing spatially varying refractive index therein is presented. A new refractive index landscape is obtained by a conformal mapping, which makes the trajectories of the chaotic billiard fully predictable and the billiard fully integrable. Moreover, trajectory rectification can be pushed a step further by relating chaotic billiards with non‐Euclidean geometries. Two examples are illustrated by projecting billiards built on a sphere as well as the deformed spacetime outside a Schwarzschild black hole, which respectively lead to all periodic orbits and spiraling trajectories remaining away from the boundaries of the transformed 2D billiards/cavities. An implementation of this method is proposed, which enables real‐time control of chaos and can further contribute to a wealth of potential applications in the domain of optical microcavities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce matrix operator algebra involving a universal curvature constant and using the Dunkl derivative. Consequently, the Dirac equation can be written without spin connections. Iterating the Dirac equation gives the Klein–Gordon equation in its canonical form without first-order Dunkl derivatives. This leads to a new form for the static metric based on the Dunkl operator in curved space. [ABSTRACT FROM AUTHOR]

The elaboration of a general formalism on quantum spin entanglement in curved space–time is presented by a system of two particles described by wave packets moving in a gravitational field (GF). This formulation allows us to study different models in curved space–time. In this work, the non-commutative Reissner–Nordström model is considered. The spin entanglement of a system of two spin 1/2 particles is discussed. With particularity that contains multiple and various physical parameters, allowing for a detailed study of this purely quantum phenomenon in different frames of space and geometry or both at the same time. [ABSTRACT FROM AUTHOR]

This article is the beginning of a reexpresson, and partial revision, of my book Black Light. The questions that I discuss in this article are listed in the below list of sections. In Sections 7, 8, 27, and 28, I discuss my discovery of black light. The theoretical discovery of black light (that is, the thought experimental discovery of black light): The black spatial field in, for example, a “dark” room is actually, I argue, black light; and it is emitted from everything in the spatial field of the room (that is, the relatively empty space, and all objects). If, hypothetically, the black light in the above room was removed, we, when we would look into the spatial field of the room, would be blind, despite that we have the capacity to see. The observational discovery of black light: There is no such thing as a “colorless” visual field for observers: A “colorless” visual field would be a visual field of blindness for observers, despite that they have vision, and that their eyes would be open. The black visual field is not, as is commonly stated, “the absence of photons,” “the absence of visible light,” and the, as such, absence of color: If it were, then it would be “colorless” (that is, not black), and, as such, a visual field of blindness for observers. The experimental confirmation of black light via neurophysics: In Section 28, I demonstrate that particular EEG experimentation that was done on test-subjects in various conditions provides evidence or proof that the black of the black visual field that strikes our retinas is black light.

Light propagation on a two-dimensional curved surface embedded in a three-dimensional space has attracted increasing attention as an analog model of four-dimensional curved spacetime in the laboratory. Despite recent developments in modern cosmology on the dynamics and evolution of the universe, investigation of nonlinear dynamics of light on non-Euclidean geometry is still scarce, with fundamental questions, such as the effect of curvature on deterministic chaos, challenging to address. Here, we study classical and wave chaotic dynamics on a family of surfaces of revolution by considering its equivalent conformally transformed flat billiard, with nonuniform distribution of the refractive index. We prove rigorously that these two systems share the same dynamics. By exploring the Poincare surface of section, the Lyapunov exponent, ´ and the statistics of eigenmodes and eigenfrequency spectrum in the transformed inhomogeneous table billiard, we find that the degree of chaos is fully controlled by a single, curvature-related geometric parameter of the curved surface. A simple interpretation of our findings in transformed billiards, the “fictitious force,” allows us to extend our prediction to other classes of curved surfaces. This powerful analogy between two a priori unrelated systems not only brings forward an approach to control the degree of chaos, but also provides potentialities for further studies and applications in various fields, such as billiards design, optical fibers, or laser microcavities. [ABSTRACT FROM AUTHOR]

Topological insulators (TIs) with robust boundary states against perturbations and disorders have boosted intense research in classical systems. In general, two-dimensional (2D) TIs are designed on a flat surface with special boundary to manipulate the wave propagation. In this work, we design a 2D curved acoustic TI by perforation on a curved rigid plate to localize the edge state by means of the geometric potential effect, which provide a unique approach for manipulating waves. We experimentally demonstrate that the topological edge state in the bulk gap is modulated by the curvature of space into a localized mode, and the corresponding pressure distributions are confined at the position with the maximal curvature. Moreover, we experimentally verify the localized edge state is still topologically protected by introducing defects near the localized position. To understand the underlying mechanism for the localization of the topological edge state, a tight-binding model considering the geometric potential effect is proposed. The interaction between the geometrical curvature and topology in the system provides a novel scheme for manipulating and trapping wave propagation along the boundary of curved TIs, thereby offering potential applications in flexible devices.

In this paper we have investigated the general condition of self-adjointness of the generalised momentum operators and we have shown that it highly depends on the metric of the space. We have also discussed the domain of the generalised momentum operators at boundary conditions. [ABSTRACT FROM AUTHOR]

This article is the beginning of a reexpresson, and partial revision, of my book Black Light. The questions that I discuss in this article are listed in the below list of sections. In Sections 7, 8, 27, and 28, I discuss my discovery of black light. The theoretical discovery of black light (that is, the thought experimental discovery of black light): The black spatial field in, for example, a “dark” room is actually, I argue, black light; and it is emitted from everything in the spatial field of the room (that is, the relatively empty space, and all objects). If, hypothetically, the black light in the above room was removed, we, when we would look into the spatial field of the room, would be blind, despite that we have the capacity to see. The observational discovery of black light: There is no such thing as a “colorless” visual field for observers: A “colorless” visual field would be a visual field of blindness for observers, despite that they have vision, and that their eyes would be open. The black visual field is not, as is commonly stated, “the absence of photons,” “the absence of visible light,” and the, as such, absence of color: If it were, then it would be “colorless” (that is, not black), and, as such, a visual field of blindness for observers. The experimental confirmation of black light via neurophysics: In Section 28, I demonstrate that particular EEG experimentation that was done on test-subjects in various conditions provides evidence or proof that the black of the black visual field that strikes our retinas is black light. [ABSTRACT FROM AUTHOR]

Even in free space, which is commonly considered of as a flat space–time in most settings, the degree of polarization of a partially spatially coherent light beam changes as it travels. Similarly, the polarization degree would change when a partially spatially coherent light beam propagates in a curved space–time. The difference of the polarization degree between the curved space and flat space can reveal the essential structure of the curved space. In this work, we consider a simplest case of curved space known as Schwarzschild spacetime. We can simulate the Schwarzschild space–time as an optical material with an effective refractive index. The difference of the polarization degree of a light beam propagating in curved space and flat space can be achieved up to 5%, which is detectable in practical measurement. In addition, we have found that the partially spatially coherent light source is necessary for obtaining significant changes in polarization degree. Our results provide an alternative method to estimate the Schwarzschild radius of a massive object with the optical polarization degree measurement. • We consider the partially spatially coherent light in a three-dimensional curved space with the Schwarzschild space–time structure. • The polarization degree change is about 5 % away from the case of light propagation in a flat space, and this change is detectable in practical measurements. [ABSTRACT FROM AUTHOR]

We show how fabricated semiconductor heterostructures enable quantum particles to exhibit resilience to curved space, achieved through a suitable position-dependent mass distribution within the semiconductor. Following the proposed approach, we show that the first-order approximation of curved space requires a position-dependent mass distribution that is common for semiconductor heterostructures. For more complex structures of curved space, one has to consider a more exotic fabricated semiconductor. • Semiconductor heterostructure can resist space curvature for quantum particles. • Resisting curved space by heterostructures restores standard commutation relation. • Quantum potential quantifies heterostructure-curved space interplay. [ABSTRACT FROM AUTHOR]

• Identifies issue of "body coherence" in curved-space VR games. • Presents solution to body coherence problem in curved-space billiards. • Discusses "native-inhabitant view" vs. "tourist view" in curved-space VR. • Simplifies understanding of projection transformations in curved-space VR. [Display omitted] Virtual-reality simulations of curved space are most effective and most fun when presented as a game (for example, curved-space billiards), so the user not only has something to see in the curved space, but also has something fun to do there. However, such simulations encounter a geometrical problem: they must track the player's hands as well as her head, and in curved space the effects of holonomy would quickly lead to violations of body coherence. That is, what the player sees with her eyes would disagree with what she feels with her hands. This article presents a solution to the body coherence problem, as well as several other questions that arise in interactive VR simulations in curved space (radians vs. meters, visualization of the projection transformation, native-inhabitant view vs. tourist view, and mental models of curved space). [ABSTRACT FROM AUTHOR]

The effect of the geometry (deviation from the flat space) on the quantum evolution of the momentum and position of a free particle is discussed. It is shown that beginning with a wave-packet of minimum uncertainty (a Gaussian wave), there is a usual increase in the product of the volume uncertainties in the momentum and position space, as seen in the quantum mechanics on a flat spaces. But there is also a contribution from geometry. The leading order of this contribution is calculated. [ABSTRACT FROM AUTHOR]

The extrinsic and intrinsic curvature of a two-dimensional waveguide influences wave propagation therein. While this can already be apprehended from a geometric point of view in terms of geodesics generalizing straight lines as the shortest distance between any two points, in wave optics interference phenomena strongly govern the field evolution, too. Radii of curvature in the order of the wavelength of light modify the local effective refractive index by altering the mode profile. Macroscopic radii only influence light propagation for non-vanishing intrinsic (or Gaussian) curvature. A positive Gaussian curvature leads to refocusing and thus an imaging behavior, whereas negative Gaussian curvature forces the field profile to diverge exponentially. These effects can be explained by an effective transverse potential acting on the electromagnetic field distribution’s envelope. This can also be extended to nonlinear beam propagation. In this review paper we give a thorough introduction to differential geometry in two-dimensional manifolds and its incorporation with Maxwell’s equations. We report on first fundamental experiments in this newly emerging field, which may lead to applications in integrated optical circuits. The close conceptual analogy to phenomena in four-dimensional spacetime with constant curvature as well as toy-models of the Schwarzschild metric and a wormhole topology are also discussed.

The governing equations of Maxwell electrodynamics in multi-dimensional spaces are derived from the variational principle of least action, which is applied to the action function of the electromagnetic field. The Hamiltonian approach for the electromagnetic field in multi-dimensional pseudo-Euclidean (flat) spaces has also been developed and investigated. Based on the two arising first-class constraints, we have generalized to multi-dimensional spaces a number of different gauges known for the three-dimensional electromagnetic field. For multi-dimensional spaces of non-zero curvature the governing equations for the multi-dimensional electromagnetic field are written in a manifestly covariant form. Multi-dimensional Einstein’s equations of metric gravity in the presence of an electromagnetic field have been re-written in the true tensor form. Methods of scalar electrodynamics are applied to analyze Maxwell equations in the two and one-dimensional spaces.

In this work, we review two methods used to approach singular Hamiltonians in (2 + 1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modeled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square-integrable and then can be associated with a physical state. However, a study based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.

In this paper, we extend the theory of the q -deformed quantum mechanics in one dimension 2 1 into three-dimensional case. We relate the q -deformed quantum theory to the quantum theory in a curved space. We discuss the diagonal metric based on q -addition in the Cartesian coordinate system and core radius of neutron star. We also discuss the diagonal metric based on q -addition in the spherical coordinate system and q -deformed Heisenberg atom model. [ABSTRACT FROM AUTHOR]

We study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. Furthermore, the resistivity of wrinkled graphene sheets is calculated for different average space curvatures and shown to obey a linear relation. The study is carried out with a quantum lattice Boltzmann method, solving the Dirac equation on curved manifolds. [ABSTRACT FROM AUTHOR]

The history of feminist geography in Hungary coincides with the 25 year-long history of Gender, Place and Culture. Authorities denied the existence of gender inequality in the era of state socialism, which was the primary obstacle to the spread of gender studies. The political changes that had occurred after 1989 had removed most obstacles, but feminist geography emerged with a delay relative to other disciplines. Its first two decades was characterised by struggles and compromises within and against the geographical discipline in order for it to win recognition. The 25 year-long history of feminist studies has, however, been completely broken by legislation proposed by the current government suggesting a ban on masters programs in gender studies. In this article, I trace the situation of feminist geography in Hungary by applying the concept ʻcurved spaceʼ. This concept adapted from modern physics claims that mass creates a gravitational field, i.e. it bends 4-dimensional ʻspacetimeʼ. My argument is that the situation of feminist geography in Hungary can be interpreted as an embodiment of ʻcurved spaceʼ. Using this analogy, I argue that the current Hungarian government has amassed such a huge amount of power that has enabled it to curve the space of feminist geographical knowledge production. It has established a quasi-dictatorship that resembles the one that impeded the evolution of gender/feminist geography in the state socialist era. Therefore, only broad-based solidarity can help create opposition to the current government's attacks against gender studies. [ABSTRACT FROM AUTHOR]

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially, the field equations of the four quantum-fields in the quantum mechanics. Apparently, the paper and General Relativity both confirm and succeed to the Cartesian academic thought of 'the space is the extension of substance'. [ABSTRACT FROM AUTHOR]

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

We report a new type of spin–orbit coupling (SOC) called geometric SOC. Starting from the relativistic theory in curved space, we derive an effective nonrelativistic Hamiltonian in a generic curve embedded into flat three dimensions. The geometric SOC is O ( m ^−1 ), in which m is the electron mass, and hence much larger than the conventional SOC of O ( m ^−2 ). The energy scale is estimated to be a hundred meV for a nanoscale helix. We calculate the current-induced spin polarization in a coupled-helix model as a representative of the chirality-induced spin selectivity. We find that it depends on the chirality of the helix and is of the order of 0.01 ℏ per nm when a charge current of 1 μ A is applied.

Based on the numerical solution of the Quantum Lattice Boltzmann Method in curved space, we predicted the onset of a quantized alternating current on curved graphene sheets. This numerical prediction was verified analytically via a set of semi-classical equations that related the Berry curvature to real space curvature. The proposed quantized oscillating current on curved graphene could form the basis for the implementation of quantum information-processing algorithms.

Dynamic density functional theory calculations of fluid–fluid demixing on spherical geometries are characterized via their angular power spectrum as well as via the Minkowski functionals (MFs) of their binarized fluid density fields. MFs form a complete set of additive, motion invariant and continuous morphological measures sensitive to nonlinear (spatial) correlations. The temporal evolution of the fluid density fields is analyzed for different sphere sizes and mixing compositions. The demixing process in the stages of early spinodal decomposition and consecutive domain growth can be characterized by both methods and a power-law domain growth $L(t)\propto {t}^{\alpha }$ is evidenced for the MF measures. The average domain size obtained by the structure factor only responds to the late stage domain growth of the demixing process. MFs provide refined insights into the demixing process: they allow the detection of distinct stages in the early spinodal decomposition, provide a precise measure of the relative species composition of the mixture and, most importantly: after a proper rescaling, they allow the detection of a universal demixing behavior for a wide range of mixture fractions and for different sphere sizes.

We present a mathematical model for physical systems. A large class of functions is built through the functional quantization method and applied to the geometric study of the model. Quantized equations of motion along the Hamiltonian vector field are built up. It is seen that the procedure in higher dimension carries more physical information. The metric tensor appears to induce an electromagnetic field into the system and the dynamical nature of the electromagnetic field in curved space arises naturally. In the end, an explicit formula for the curvature tensor in the quantized space is given. [ABSTRACT FROM AUTHOR]