Topological Classification Research Articles

Topological insulators exhibit boundary states protected by bulk band topology, a principle first established in quantum systems and later extended to classical waves, including phononics. Conventionally, an $n$-dimensional bulk with nontrivial topology hosts $(n-1)$-dimensional topologically protected boundary states, which may be further gapped out by breaking the symmetry that protects them, potentially leading to the emergence of $(n-2)$-dimensional, or even lower-dimensional topological states, as in higher-order topological insulators. In this work, we introduce an alternative mechanism for \textcolor{blue}{gapping out topological states and forming new topological modes within the resulting gap without further unit-cell symmetry breaking or dimension reduction.} Using one- and two-dimensional Su-Schrieffer-Heeger (SSH) models, we show that controlled repositioning of topological domain walls enables the construction of hierarchical unit cells that gap out the original domain-wall states while preserving the underlying symmetry. This process produces \textcolor{blue}{higher-hierarchical-level topological states}, characterized by a generalized winding number, and can be iterated to realize multiple - potentially infinite - hierarchical levels of topological states. Our approach expands the conventional topological classification and offers a versatile route for engineering complex networks of protected modes in higher dimensions.

The deformation mechanisms of classic lattice topologies (e.g., Cubic, Diamond, Octet, and Double Pyramid lattices) and their specific density-dependent mechanical properties have already been thoroughly explored by the scientific community. This study details a novel approach to designing lattices by generating the topologies that correspond to the voids of these classic lattice designs. This is achieved by using a Boolean operation performed to create a solid topology from the original voided fraction. The resultant topologies are proposed to be named Inverse lattices. Static structural numerical analysis shows that this process may generate significant changes in the lattice deformation mechanism and stiffness. For this effect, elastic properties such as the Specific modulus and Apparent Poisson’s ratio were determined as a function of Specific density. Specifically, for Octet and Double Pyramid inverse lattice topologies, results show a reduction in stiffness by promoting a change to a bending deformation mechanism. However, the inverse Diamond inverse lattice topologies present a higher stiffness (i.e., specific modulus) relative to the original classic design. This new lattice model may be a promising design for future lattice applications.

This paper introduces ασ-sets as a new class of generalized topological structures that extend α-open sets to countable unions with controlled boundary behavior. We prove these structures form a σ-algebra under union operations and exhibit strong hereditary properties in subspaces. Our investigation establishes fundamental preservation properties under continuous mappings and homeomorphisms, with characteristic behavior in product spaces. The framework bridges classical topological concepts with refined local-to-global properties while preserving critical topological invariants. We demonstrate applications in digital topology and image processing, particularly for texture analysis and pattern recognition, where ασ-sets effectively capture complex boundary behaviors. Through counterexamples and characterization theorems, we precisely position these structures within the broader topological landscape, providing new tools for topological classification problems in image analysis.

Topological classification of insulators: I. Non-interacting spectrally-gapped one-dimensional systems

We present experiments and simulations on cyclically sheared colloidal gels, and probe their behaviour on several different length scales. For experimental gels formed by colloid-polymer mixtures, the shearing induces structural changes, which are quantified by the topological cluster classification, bond-order parameters, and the pore size distribution. These results are mirrored in computer simulations of a model gel-former: for cyclic shear with amplitudes up to 4%, local structural analysis shows that the material evolves down the energy landscape under shearing, and the average pore size increases. We also analyze mechanical responses including the stress and the dissipation rate, revealing a crossover between elastic and plastic responses as the strain amplitude is increased. Depending on the parameters, we observe both increased compliance after shearing (thixotropy), and reduced compliance (strain hardening). We simulate creeping flow under constant shear stress, for gels that were previously subject to cyclic shear, showing that strain-hardening also increases gel stability. This response depends on the orientation of the applied shear stress, revealing that the cyclic shear imprints anisotropic structural features into the gel.

We obtain a topological classification of smooth structurally stable flows on four-dimensional closed manifolds whose wandering set contains isolated trajectories connecting saddle equilibria (heteroclinic curves). For dimensional reasons, heteroclinic curves of such flows belong to the intersection of invariant manifolds of saddles of neighboring Morse indices. We assume that the non-wandering set of the flows under consideration consists of exactly one source, one sink, and an arbitrary number of saddles, the dimension of whose unstable manifolds is equal to 1 and 2. We obtain necessary and sufficient conditions for the topological equivalence of such flows and present an algorithm for realizing a representative in each class of topological equivalence. In particular, we show that in the considered class of flows on the sphere S 4 there exists exactly one class of topological equivalence of flows with a single heteroclinic curve and a countable set of topologically nonequivalent flows with three heteroclinic curves. The latter result contrasts with the three-dimensional situation, where for a similar class of flows there are only finitely many equivalence classes for each number of heteroclinic curves.

Thermodynamic scalar curvature and topological classification in accelerating charged AdS black holes under rainbow gravity

Topological classification of global dynamics of planar polynomial Hamiltonian systems with separable variables

Abstract In this study, we explore a spherically symmetric charged black hole (BH) with a negative cosmological constant under the influence of a Kalb–Ramond field background. We compute the photon sphere and shadow radii, validating our findings using observational data from the Event Horizon Telescope, with a particular emphasis on the shadow images of Sagittarius A*. Furthermore, we investigate the greybody factors, emission rate, and partial absorption cross section. It is shown that the Lorentz-violating parameter l ¯ has an important effect on the absorption cross section. Our analysis also includes an examination of the topological charge, temperature-dependent topology, and generalized free energy. In particular, we regard the AdS charged BH with an antisymmetric tensor background as a topological defect in the thermodynamic space, then the system has the same topological classification to the charged Reissner–Nordström–AdS BH.

Laughlin charge pumping has provided critical insights into the topological classification of individual materials, but remains largely unexplored in topological junctions. We explore Laughlin charge pumping in junctions composed of a chiral topological superconductor sandwiched between two quantum anomalous Hall insulators, driven by an adiabatically varying magnetic flux. Here, charge pumping can be mediated merely by chiral Dirac modes or by the interplay of chiral Dirac and chiral Majorana modes (CMMs). In the former case, a variation of one magnetic flux quantum induces the pumping of a unit charge, as the chiral Dirac mode accumulates the full flux-induced phase. In contrast, in the latter case, pumping a unit charge requires a variation of fractional magnetic flux quanta, determined by the device geometry and the parity of the number of enclosed superconducting vortices. This unique feature results from the charge-neutral and zero-momentum nature of zero-energy CMMs. Our work offers an experimentally viable pathway toward detecting CMMs and could also inspire further research into Laughlin charge or spin pumping in diverse topological junctions, which are now within experimental reach.

The diffraction spectra of the Hat and Spectre monotile tilings, which are known to be pure point, are derived and computed explicitly. This is done via model set representatives of self-similar members in the topological conjugacy classes of the Hat and the Spectre tiling, which are the CAP and the CASPr tiling, respectively. This is followed by suitable reprojections of the model sets to represent the original Hat and Spectre tilings, which also allows to calculate their Fourier–Bohr coefficients explicitly. Since the windows of the underlying model sets have fractal boundaries, these coefficients need to be computed via an exact renormalization cocycle in internal space.

In topology, averaging over local geometrical details reveals robust global features. These are crucial in physics for understanding quantized bulk transport and exotic boundary effects of linear wave propagation in (meta-)materials. Beyond linear Hamiltonian systems, topological physics strives to characterize open (non-Hermitian) and interacting systems. Here, we establish a framework for the topological classification of driven-dissipative nonlinear systems by defining a graph index for their Floquet semiclassical equations of motion. Our index builds upon the topology of vector flows and encodes the particle-hole nature of excitations around all out-of-equilibrium stationary states. Thus, we uncover the topology of nonlinear resonator's dynamics under external and parametric forcing. Our framework sheds light on the topology of driven-dissipative phases, including under- to overdamped responses and symmetry-broken phases linked to population inversion. We therefore expose the pervasive link between topology and nonlinear dynamics, with broad implications for interacting topological insulators, topological solitons, neuromorphic networks, and bosonic codes.

We study optical manifestations of multigap band topology in multiband superconductors with a nontrivial topological Euler class. We introduce a set of lattice models for non-Abelian superconductors with the Euler invariant signified by a nontrivial quantum geometry. We then demonstrate that the topological Bogoliubov excitations realized in these models provide for a characteristic first-order optical response distinct from those of the other known topological superconductors. We find that the spectral distribution of the optical conductivity universally admits a topological jump originating from the Euler class in the presence of d-wave superconducting pairings and naturally differs from the features induced by the quantum geometry in the noninteracting bands without pairing terms. Further to uncovering observable signatures in first-order optical conductivities, we showcase that the higher-order optical responses of the non-Abelian Euler superconductor can result in enhanced nonlinear currents that fingerprint the exotic topological invariant. Finally, by employing a diagrammatic approach, we generalize our findings beyond the specific models of Euler superconductors.

Abstract Let S be a fine and saturated (fs) log scheme, and let F be a group scheme over the underlying scheme of S which is étale locally representable by (1) a finite dimensional $\mathbb{Q}$ -vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group. We give a full description of all the higher direct images of F from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of F vanish; and in case (2) the first higher direct image of F vanishes and the nth ( $n\gt 1$ ) higher direct image of F is isomorphic to the $(n-1)$ -th higher direct image of $F\otimes_{{\mathbb Z}}{\mathbb Q}/{\mathbb Z}$ . In the end, we make some computations when the base is a standard henselian log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.

Magnetic gears (MGs) are attracting increasing attention in power transmission systems due to their contactless operation principles, low frictional losses, and high efficiency. However, the broad application potential of these technologies requires a comprehensive evaluation of engineering parameters, such as material selection, energy efficiency, and structural design. This review focuses solely on solid-core magnetic gear systems designed using laminated electrical steels, soft magnetic composites (SMCs), and high-saturation alloys. This review systematically examines the topological diversity, torque transmission principles, and the impact of various core materials, such as electrical steels, soft magnetic composites (SMCs), and cobalt-based alloys, on the performance of magnetic gear systems. Literature-based comparative analyses are structured around topological classifications, evaluation of material properties, and performance analyses based on losses. Additionally, the study highlights that aligning material properties with appropriate manufacturing methods, such as powder metallurgy, wire electrical discharge machining (EDM), and precision casting, is essential for the practical scalability of magnetic gear systems. The findings reveal that coaxial magnetic gears (CMGs) offer a favorable balance between high torque density and compactness, while soft magnetic composites provide significant advantages in loss reduction, particularly at high frequencies. Additionally, application trends in fields such as renewable energy, electric vehicles (EVs), aerospace, and robotics are highlighted.

We investigate the transonic accretion flow in the spacetime of a supermassive black hole (BH) coupled to an anisotropic dark matter fluid, as proposed by Cardoso et al. We essentially compare the accretion properties of the Cardoso BH with those of an isolated Schwarzschild BH. The Cardoso BH is described by the halo mass (M H) and its characteristic length scale (a 0). Various classes of accretion solution topologies (e.g., A and W-types) are obtained by solving the dynamical equations of the flow in a fully general relativistic framework. We find that the global accretion solutions in the identified solution topologies are substantially influenced by the halo parameters (M H, a 0) when the halo mass is high or the dark matter distribution is concentrated near the black hole. In this high compactness regime, different observational signatures of the accretion disc, like the spectral energy distribution (SED) and bolometric disc luminosity, are found to exhibit considerable deviations from the known results in the Schwarzschild BH model. Furthermore, we obtain shock-induced accretion solutions, where different shock properties, such as the shock radius (r sh), flow mass density (ρ) compression, and electron temperature (Te ) compression across the shock front, are potentially altered from those in the Schwarzschild BH model when the halo compactness is high. Interestingly, the existing shock parameter space, defined by the flow specific angular momentum (λ) and energy (E), is largely reduced for higher halo compactness compared to that of the Schwarzschild BH. These unique features offer a possible valuable tool for characterizing the presence or absence of a dark matter halo around a galactic black hole.

Abstract In this study, we develop universal thermodynamic topological classes for the static black holes in the context of the Conformal Killing Gravity. Our findings indicate that the Conformal Killing Gravity significantly reconstructs the thermodynamic properties of both the smallest inner and the largest outer black hole states. Additionally, it considerably alters the thermodynamic stability of black holes across both high-temperature and low-temperature regimes. This analysis shows that different CKG parameter settings will lead to $$W^{0+}$$ W 0 + $$(\lambda >0)$$ ( λ > 0 ) and $$W^{1+}$$ W 1 + $$(\lambda <0)$$ ( λ < 0 ) categories for the charged AdS black hole, the Reissner–Nordstr $$\ddot{o}$$ o ¨ m black hole in Conformal Killing Gravity is classified into the $$W^{0+}$$ W 0 + and $$W^{1+}$$ W 1 + categories. Furthermore, we examine the specific scenario where charge is neglected. The study reveals that within the framework of Conformal Killing Gravity, the Schwarzschild black hole similar to the Schwarzschild-AdS black hole, can be classified into the $$W^{1-}$$ W 1 - and $$W^{0-}$$ W 0 - categories. This work provides key insights into the fundamental nature of quantum gravity theory.

We investigated the thermodynamic topology of quantum-corrected AdS-Reissner-Nordström black holes in Kiselev spacetime using non-extensive entropy formulation derived from Loop Quantum Gravity (LQG). Through systematic analysis, we examined how the Tsallis parameter λ influences topological charge classification with respect to various equation of state parameters. Our findings revealed a consistent pattern of topological transitions: for λ=0.1, the system exhibited a single topological charge (ω=−1) with total charge W=−1, as λ increased to 0.8, the system transitioned to a configuration with two topological charges (ω=+1,−1) and total charge W=0. When λ=1, corresponding to the Bekenstein–Hawking entropy limit, the system displayed a single topological charge (ω=+1) with W=+1, signifying thermodynamic stability. The persistence of this pattern across different fluid compositions—from exotic negative pressure environments to radiation—demonstrates the universal nature of quantum gravitational effects on black hole topology.

Open-forest lawns – comprising sparse tree canopies and herbaceous ground layers – play vital ecological, esthetic, and social roles in urban green spaces, yet lack a systematic spatial typology to inform design and application. This study employs a case study approach to develop a spatial typology for open-forest lawn landscapes grounded in topological principles. It identifies three spatial archetypes – node-based, pathway-driven, and island-based – and selects 12 representative sites from urban parks in Hangzhou, Zhejiang, and Shanghai, China, for analysis. The study examines the spatial characteristics of each archetype, along with their corresponding design logic and thematic intentions. The results show that node-based spaces create a sense of ceremony through enclosed interfaces; pathway-driven spaces encourage dynamic exploration through linear infiltration; and island-based spaces rely on discrete clusters to foster diverse forms of interaction. Moreover, these three spatial forms can be layered to establish a coordinated relationship spanning design themes, spatial sequences, and planting configurations. From both theoretical and practical perspectives, the proposed typological framework offers a spatially oriented reference and guidance for open-forest lawn design.

Elucidating the topology of host-guest complexes is essential for the rational design of supramolecular assemblies. Building on the recent success of data-driven approaches, we evaluate the combination of ion mobility-mass spectrometry (IMS-MS), density functional theory (DFT) featurization, and machine learning to predict and classify the binding modes of 1:1 complexes formed between cucurbit[6]uril (CB6) and diamine guests. Training a regression model with DFT-derived molecular descriptors and experimentally determined collisional cross sections (CCS) enables predicting the CCS of host-guest complexes with a diverse set of diamine guests. The predicted values naturally separate in two distinct groups corresponding respectively to inclusion and exclusion complexes, thereby enabling topology classification. This approach demonstrates that DFT-featurization and IMS-MS data capture well host-guest topology and provide a framework for the data-driven design of supramolecular assemblies.