Toroidal Topology Research Articles

Quantum states and spectra of small cylindrical and toroidal lattices

Abstract We examine the spectrum and quantum states of small lattices with cylindrical and toroidal topology subject to a scalar gauge potential that introduces a position dependent phase in the inter-site coupling. Equivalency of gauges assumed in infinite lattices is generally lost due to the periodic boundary conditions, and conditions that restore it are identified. We trace the impact of various system parameters including gauge choice, boundary conditions and inter-site coupling strengths, and an additional axial field. We find gauge dependent appearance of avoided crossings and persistent degeneracies, and we show their impact on the associated eigenstates. Smaller lattices develop prominent gaps in spectral lines associated with edge states, which are suppressed in the thermodynamic limit. Toroidal lattices have counterparts of most of the features observed in cylindrical lattices, but notably they display a transition from localization to delocalization determined by the relation between the field parameter and the number of lattice sites.

High-yield implosion modeling using the Frustraum: Assessing and controlling the formation of polar jets and enhancing implosion performance with applied magnetization

Frustraums have a higher laser-to-capsule x-ray radiation coupling efficiency and can accommodate a large capsule, thus potentially generating a higher yield with less laser energy than cylindrical Hohlraums for a given Hohlraum volume [Amendt et al., Phys. Plasmas 26, 082707 (2019]. Frustraums are expected to have less m = 4 azimuthal asymmetries arising from the intrinsic inner-laser-beam geometry on the National Ignition Facility. An experimental campaign at Lawrence Livermore National Laboratory to demonstrate the high-coupling efficiency and radiation symmetry tuning of the Frustraum has been under way since 2021. Simulations benchmarked against experimental data show that implosions using Frustraums can achieve more yield with higher ignition margins than cylindrical Hohlraums using the same laser energy. Hydrodynamic jets in capsules along the Hohlraum axis, driven by radiation-flux asymmetries in a Hohlraum with a gold liner on a depleted uranium (DU) wall, are present around stagnation, and these “polar” jets can cause severe yield degradation. The early-time Legendre mode P4<0 radiation-flux asymmetry is a leading cause of these jets, which can be reduced by using an unlined DU Hohlraum because the shape of the shell is predicted to be more prolate. Magnetization can increase the implosion robustness and reduce the required hotspot ρR for ignition; therefore, magnetizing the Frustraum can maintain the same yield while reducing the required laser energy or increase the yield using the same laser energy—all under the constraint that the ignition margin is preserved. Reducing polar jets is particularly important for magnetized implosions because of the intrinsic toroidal hotspot ion temperature topology.

Toroidal phase topologies within paraxial laser beams

Control of topologies in structured light fields with multi-degrees of freedom integrates fundamental optical physics and topological invariance. Beyond the simple phase vortex, three-dimensional (3D) topological singularities and related nonsingular textures have recently gained significant interest. Here, we experimentally demonstrate the creation of a family of toroidal phase topologies within paraxial laser beams. By employing single two-dimensional (2D) phase control, we generate propagating 3D topological textures, effectively embodying the topological configuration of a four-dimensional (4D) parameter space. The resulting light fields exhibit amplitude isosurfaces of toroidal vortices and hopfionic phase textures, both controlled by topological charges. The ability to prepare scalar phase textures of light offers new insights into the high-dimensional control of complex structured textures and may find significant applications in light-matter interactions, optical manipulation, and optical information encoding.

Arithmetic modeling of k-ary n-cubes and toroidal k-ary grids

The use of toroidal topologies offers many advantages in the computing field when it comes to both pattern spotting and path finding strategies. The former is covered by de Bruijn shapes, which permit to uniquely locate a single pattern throughout the shape. However, the latter is mainly carried out by k-ary n-cubes, which label node identifiers in a sequential order according to rows, columns, layers, and so on. This scheme facilitates the movement among nodes by just applying arithmetic operations, such as integer divisions and arithmetic modulo n. On the othe hand, toroidal k-ary grids are an alternative available in some specific cases, where determined patterns appear in all dimensions of each node, thus allowing to use those patterns to dictate paths to move among nodes. In this paper, the arithmetic bases of both path finding strategies have been presented and some pseudocode algorithms have been designed.

Engineering phase and density of Bose–Einstein condensates in curved waveguides with toroidal topology

We investigate the effects of ellipticity-induced curvature on atomic Bose–Einstein condensates confined in quasi-one-dimensional closed-loop waveguides. Our theoretical study reveals intriguing phenomena arising from the interplay between curvature and interactions. Density modulations are observed in regions of high curvature, but these modulations are suppressed by strong repulsive interactions. Additionally, we observe phase accumulation in regions with the lowest curvature when the waveguide with superflow has high eccentricity. Furthermore, waveguides hosting vortices exhibit dynamic transformations between states with different angular momenta. These findings provide insights into the behavior of atomic condensates in curved waveguides, with implications for fundamental physics and quantum technologies. The interplay between curvature and interactions offers opportunities for exploring novel quantum phenomena and engineering quantum states in confined geometries.

Strain‐Induced Toroidal Polar States in Wrinkled Ferroelectric Polymer by Phase‐Field Simulations

AbstractWrinkled ferroelectric polymer films with out‐of‐plane deformation exhibit exotic strain fields that differ from those of conventional epitaxial film systems, offering a new path toward manipulating electrical and mechanical behaviors in ferroelectrics. However, the direct observation of the domain structure evolution in ferroelectric polymer films during continuous deformation is challenging with current experimental approaches. In this study, the strain‐induced evolution of toroidal domain structures in organic poly(vinylidene fluoride‐ran‐trifluoroethylene) (P(VDF‐TrFE)) films with concentric toroidal wrinkle patterns is comprehensively investigated using phase‐field simulations. The results reveal morphology‐ and thickness‐dependent toroidal polar topology in wrinkled P(VDF‐TrFE) films. External strain loading can modulate this ferroelectric domain, which exhibits robust coupling behavior with a mixed strain field of periodic tensile and compressive strains. It is demonstrated that toroidal wrinkled ferroelectric films provide an effective and unique flexible platform for continuously controllable strain engineering in ferroelectrics and a theoretical framework with potential applications in flexible electronic devices.

MukBEF-dependent chromosomal organization in widened Escherichia coli.

The bacterial chromosome is spatially organized through protein-mediated compaction, supercoiling, and cell-boundary confinement. Structural Maintenance of Chromosomes (SMC) complexes are a major class of chromosome-organizing proteins present throughout all domains of life. Here, we study the role of the Escherichia coli SMC complex MukBEF in chromosome architecture and segregation. Using quantitative live-cell imaging of shape-manipulated cells, we show that MukBEF is crucial to preserve the toroidal topology of the Escherichia coli chromosome and that it is non-uniformly distributed along the chromosome: it prefers locations toward the origin and away from the terminus of replication, and it is unevenly distributed over the origin of replication along the two chromosome arms. Using an ATP hydrolysis-deficient MukB mutant, we confirm that MukBEF translocation along the chromosome is ATP-dependent, in contrast to its loading onto DNA. MukBEF and MatP are furthermore found to be essential for sister chromosome decatenation. We propose a model that explains how MukBEF, MatP, and their interacting partners organize the chromosome and contribute to sister segregation. The combination of bacterial cell-shape modification and quantitative fluorescence microscopy paves way to investigating chromosome-organization factors in vivo.

Meta-Atoms with Toroidal Topology for Strongly Resonant Responses

A conductive meta-atom of toroidal topology is studied both theoretically and experimentally, demonstrating a sharp and highly controllable resonant response. Simulations are performed both for a free-space periodic metasurface and a pair of meta-atoms inserted within a rectangular metallic waveguide. A quasi-dark state with controllable radiative coupling is supported, allowing to tune the linewidth (quality factor) and lineshape of the supported resonance via the appropriate geometric parameters. By conducting a rigorous multipole analysis, we find that despite the strong toroidal dipole moment, it is the residual electric dipole moment that dictates the electromagnetic response. Subsequently, the structure is fabricated with 3D printing and coated with silver paste. Importantly, the structure is planar, consists of a single metallization layer and does not require a substrate when neighboring meta-atoms are touching, resulting in a practical, thin and potentially low-loss system. Measurements are performed in the 5 GHz regime with a vector network analyzer and a good agreement with simulations is demonstrated.

Towards optical toroidal wavepackets through tight focusing of the cylindrical vector two dimensional spatiotemporal optical vortex.

Spatiotemporal optical vortices (STOVs) carrying transverse orbital angular momentum (OAM) are of rapidly growing interest for the field of optics due to the new degree of freedom that can be exploited. In this paper, we propose cylindrical vector two dimensional STOVs (2D-STOVs) containing two orthogonal transverse OAMs in both x-t and y-t planes for the first time, and investigate the tightly focusing of such fields using the Richards-Wolf vectorial diffraction theory. Highly confined spatiotemporal wavepackets with polarization structure akin to toroidal topology is generated, whose spatiotemporal intensity distributions resemble the shape of Yo-Yo balls. Tightly focused radially polarized 2D-STOVs will produce wavepackets towards transverse magnetic toroidal topology, while the focused azimuthally polarized 2D-STOVs will give rise to wavepackets towards transverse electric toroidal topology. The presented method may pave a way to experimentally generate the optical toroidal wavepackets in a controllable way, with potential applications in electron acceleration, nanophotonics, energy, transient light-matter interaction, spectroscopy, quantum information processing, etc.

Polarimetric signatures of hot spots in black hole accretion flows

Context. The flaring events observed in the Sagittarius A* supermassive black hole system can be attributed to the nonhomogeneous nature of the near-horizon accretion flow. Bright regions in this flow may be associated with density or temperature anisotropies, corresponding to so-called “bright spots” or “hot spots.” Such orbiting features may explain observations at infrared wavelengths, as well as recent findings at millimeter wavelengths. Aims. In this work, we study the emission from an orbiting equatorial bright spot, imposed on a radiatively inefficient accretion flow background, to find polarimetric features indicative of the underlying magnetic field structure and other system variables, including inclination angle, spot size, black hole spin, and more. Specifically, we investigate the impact of these parameters on the Stokes 𝒬 − 𝒰 signatures that commonly exhibit a typical double loop (pretzel-like) structure. Methods. Our semi-analytical model, describing the underlying plasma conditions and the orbiting spot, is built within the framework of the numerical radiative transfer code ipole, which calculates synchroton emission at 230 GHz. Results. We showcase the wide variety of 𝒬 − 𝒰 loop signatures and the relation between inner and outer loops. For the vertical magnetic field topology, the inner 𝒬 − 𝒰 loop is explained by the suppression of the synchrotron emission as seen by the distant observer. For the radial and toroidal magnetic field topologies, the inner 𝒬 − 𝒰 loop corresponds to the part of the orbit where the spot is receding with respect to the observer. Conclusions. Based on our models, we conclude that it is possible to constrain the underlying magnetic field topology with an analysis of the 𝒬 − 𝒰 loop geometry, particularly in combination with circular polarization measurements.

Deep networks on toroids: removing symmetries reveals the structure of flat regions in the landscape geometry* *This article is an updated version of: Pittorino F, Ferraro A, Perugini G, Feinauer C, Baldassi C and Zecchina R 2022 Deep networks on toroids: removing symmetries reveals the structure of flat regions in the landscape geometry Proc. 39th Int. Conf. Machine Learning vol

We systematize the approach to the investigation of deep neural network landscapes by basing it on the geometry of the space of implemented functions rather than the space of parameters. Grouping classifiers into equivalence classes, we develop a standardized parameterization in which all symmetries are removed, resulting in a toroidal topology. On this space, we explore the error landscape rather than the loss. This lets us derive a meaningful notion of the flatness of minimizers and of the geodesic paths connecting them. Using different optimization algorithms that sample minimizers with different flatness we study the mode connectivity and relative distances. Testing a variety of state-of-the-art architectures and benchmark datasets, we confirm the correlation between flatness and generalization performance; we further show that in function space flatter minima are closer to each other and that the barriers along the geodesics connecting them are small. We also find that minimizers found by variants of gradient descent can be connected by zero-error paths composed of two straight lines in parameter space, i.e. polygonal chains with a single bend. We observe similar qualitative results in neural networks with binary weights and activations, providing one of the first results concerning the connectivity in this setting. Our results hinge on symmetry removal, and are in remarkable agreement with the rich phenomenology described by some recent analytical studies performed on simple shallow models.

A pyrotoroidic transition in ferroelectric polymer

<h2>Summary</h2> Emergent topological textures coupled with ferroic orders have become the promising basis for exotic phenomena and application advances. While the number of materials observed with polar topological textures is growing, manipulation of these emergent polar topologies with the aid of external stimuli (e.g. electric field, strain, thermal perturbation) remains rare. Here, we report a pyrotoroidic transition accompanying the Curie transition in face-on lamellae of poly(vinylidene fluoride-<i>ran</i>-trifluoroethylene) (P(VDF-TrFE)), with a chirality reversal revealed in the polar topology. The existence of toroidal order in paraelectric polymers has been validated by further confirming the toroidal order in face-on lamellae of relaxor ferroelectric polymer induced by physical defects. This work demonstrates the manipulation of polar topological texture with the controls of the thermal condition. It should also expand the reach of toroidal polar topologies to a range of flexible materials and offer an insight into their formation.

Properties of neutral mesons in a hot and magnetized quark matter: Size-dependent effects

In this work we investigate the combined finite-size and thermomagnetic effects on the properties of neutral mesons in a hot medium, in the context of the Nambu--Jona-Lasinio model. In particular, by using the mean-field approximation and the Schwinger proper time method in a toroidal topology with periodic and antiperiodic conditions, we evaluate the chiral phase transition, the critical coupling, the constituent quark mass and meson observables like the ${\ensuremath{\pi}}^{0}$ and $\ensuremath{\sigma}$ meson masses and pion decay constant under the change of the size, temperature and strength of external magnetic field. The results indicate that the observables are strongly affected by the conjoint effects of relevant variables and also by the periodicity of the boundary conditions chosen, and the net result will depend on the balance of these competing phenomena.

Toroidal topology of population activity in grid cells

The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

How anti-de Sitter black holes reach thermal equilibrium

It is commonly known in the literature that large black holes in anti-de Sitter spacetimes (with reflective boundary condition) are in thermal equilibrium with their Hawking radiation. Focusing on black holes with event horizon of toroidal topology, we study a simple model to understand explicitly how this thermal equilibrium is reached under Hawking evaporation. It is shown that it is possible for a large toroidal black hole to evolve into a small (but stable) one.

Joule–Thomson expansion of the torus-like black hole

In this paper, we study the Joule–Thomson expansion of the torus-like black hole in extended phase space, where the cosmological constant is determined by the pressure. It is noteworthy that the torus-like black hole has a horizon of toroidal topology, rather than a horizon of spherical topology. We study both the inversion and isenthalpic curves of the torus-like black holes in the $$T-P$$ plane and find that their intersection coincides with the inversion point that distinguishes the heat in the cooling process. Compared to the Van der Waals fluid, the inversion curves of torus-like black holes do not end at any point and only lower inversion curves are available. Most importantly, we find that the $$P - V$$ diagram of the torus-like black hole has no inflection point, i.e., the torus-like black hole is always thermally stable. Therefore, the torus-like black hole has no critical point and critical temperature, and there is no ratio between the minimum inversion temperature and the critical temperature. This is completely different from the black hole with the horizon of spherical topology.

Supertoroidal light pulses as electromagnetic skyrmions propagating in free space

Topological complex transient electromagnetic fields give access to nontrivial light-matter interactions and provide additional degrees of freedom for information transfer. An important example of such electromagnetic excitations are space-time non-separable single-cycle pulses of toroidal topology, the exact solutions of Maxwell’s equations described by Hellwarth and Nouchi in 1996 and recently observed experimentally. Here we introduce an extended family of electromagnetic excitation, the supertoroidal electromagnetic pulses, in which the Hellwarth-Nouchi pulse is just the simplest member. The supertoroidal pulses exhibit skyrmionic structure of the electromagnetic fields, multiple singularities in the Poynting vector maps and fractal-like distributions of energy backflow. They are of interest for transient light-matter interactions, ultrafast optics, spectroscopy, and toroidal electrodynamics.

Scalar fields in causal dynamical triangulations

A typical geometry extracted from the path integral of a quantum theory of gravity may be quite complicated in the UV region. Even if a single configuration is not physical, its properties may be of interest to understand the details of its nature, since some universal features can be important for the physics of the model. If the formalism describing the geometry is coordinate independent, which is the case in the model studied here, such understanding may be facilitated by the use of suitable coordinate systems. In this article we use scalar fields that solve Laplace’s equation to introduce coordinates on geometries with a toroidal topology. Using these coordinates we observe what we identify as the cosmic voids and filaments structure, even if matter is only a tool to visualize the geometry. We also show that if the scalar fields we used as coordinates are dynamically coupled to geometry, they can change it in a dramatic way, leading to a modification of the spatial topology.

An Alternative Approach for Quantum Forces in Toroidal Topology

We investigate the Casimir effect in the context of a nontrivial topology by means of a generalized Matsubara formalism. This is performed in the context of a scalar field in D Euclidean spatial dimensions with d compactified dimensions. The procedure gives us the advantage of considering simultaneously spatial constraints and thermal effects. In this sense, the Casimir pressure in a heated system between two infinite planes is obtained and the results are compared with those found in the literature.

Size-effect at finite temperature in a quark–antiquark effective model in phase space

A quark–antiquark effective model is studied in a toroidal topology at finite temperature. The model is described by a Schrödinger equation with linear potential which is embedded in a torus. The following aspects are analyzed: (i) the nonclassicality structure using the Wigner function formalism; (ii) finite temperature and size-effects are studied by a generalization of Thermofield Dynamics written in phase space; (iii) in order to include the spin of the quark, Pauli-like Schrödinger equation is used; (iv) analysis of the size-effect is considered to observe the fluctuation in the ground state. The size effect goes to zero at zero, finite and high temperatures. The results emphasize that the spin is a central aspect for this quark–antiquark effective model.