Topological Properties Research Articles

Some Topological Properties on The Cartesian Product of Fuzzy b-Metric Space

The aim of this paper is to introduce the concept of the Cartesian product of two fuzzy b-metric spaces with related properties. We have introduced the property that the Cartesian product of two complete fuzzy b-metric spaces is a complete fuzzy b-metric space in terms of the product t-norm (.), as well as the minimum t-norm (Λ). The properties of convergent and Cauchy sequence are introduced by using this concept.

Topological Metasurface for Spin‐Decoupled Wavefront Manipulation of Terahertz Surface Plasmons

AbstractOn‐chip wavefront manipulation of terahertz surface plasmons is crucial for the miniaturization and practical application of terahertz technology. Recently, topological metasurfaces have emerged as a promising avenue for spin‐decoupled wavefront manipulation, leveraging the unique topological properties of non‐Hermitian matrices near their singular points. Despite their innovative phase control mechanisms, topological metasurfaces have not yet been explored for surface plasmon devices. Here, spin‐decoupled wavefront manipulation of terahertz surface plasmons is experimentally achieved using freestanding topological metasurfaces. This approach promises to expand the control methods of surface plasmons and broaden the application scenarios of topological metasurfaces, providing a new paradigm for the design of integrated on‐chip terahertz devices.

Distance functions and filter topological residuated lattices1

In this paper, we firstly extend C. C. chang’s distance functions from MV-algebras into residuated lattices. But in general, the functions may not be a distance function on residuated lattices. We introduce weak involutory residuated lattices, in which Chang’s function is a pseduo distance function. Moreover we prove that the functions become distance functions on involutory residuated lattices. Secondly by use of the function and a lattice filter, we define F-ball on residuated lattices, and we prove that the set of all F-balls forms a base of a topology τLF on an involutory residuated lattice. Moreover we show that the topology τLF on an involutory residuated lattice L makes L to be a topological residuated lattice. At last, we use filters instead of lattice filters to set up filter topologies on an involutory residuated lattice, and study the properties of the filter topologies.

Nonlinear Optical Properties of the Topological Material Bi2Se3 Family for the Application of an Ultrafast Pulse Laser Based on the Occupied and Unoccupied Band Structures.

The Bi2Se3 family can exhibit many intriguing topological insulator properties, including a narrow bandgap and strong surface states, which show excellent nonlinear optical properties. Thinning bulk Bi2Se3 family materials to create a low-cost photoelectric modulation device and explaining the mechanisms of nonlinear optical differences in different types of materials remain challenges. Based on liquid-phase exfoliation technology and tapered fiber, this work prepared optoelectronic modulation devices for various samples within the Bi2Se3 family, quantitatively compared their nonlinear optical properties, and analyzed the sources of differentiation using the occupied and unoccupied multiband structure theory. The results correspond well to the phenomena observed in the ultrafast laser, which will provide strong support for the design of higher performance photoelectric devices based on topological insulators.

Anomalous Floquet Phases. A resonance phenomena

Floquet topological phases emerge when systems are periodically driven out-of-equilibrium. They gained attention due to their external control, which allows to simulate a wide variety of static systems by just tuning the external field in the high frequency regime. However, it was soon clear that their relevance goes beyond that, as for lower frequencies, anomalous phases without a static counterpart are present and the bulk-to-boundary correspondence can fail. In this work we discuss the important role of resonances in Floquet phases. For that, we present a method to find analytical solutions when the frequency of the drive matches the band gap, extending the well-known high frequency analysis of Floquet systems. With this formalism, we show that the topology of Floquet phases with resonances can be accurately captured in analytical terms. We also find a bulk-to-boundary correspondence between the number of edge states in finite systems and a set of topological invariants in different frames of reference, which crucially do not explicitly involve the micromotion. To illustrate our results, we periodically drive a SSH chain and a π-flux lattice, showing that our findings remain valid in various two-band systems and in different dimensions. In addition, we notice that the competition between rotating and counter-rotating terms must be carefully treated when the undriven system is a semi-metal. To conclude, we discuss the implications to experimental setups, including the direct detection of anomalous topological phases and the measurement of their invariants.

Single-Sample Networks Reveal Intra-Cytoband Co-Expression Hotspots in Breast Cancer Subtypes

Breast cancer is a heterogeneous disease comprising various subtypes with distinct molecular characteristics, clinical outcomes, and therapeutic responses. This heterogeneity evidences significant challenges for diagnosis, prognosis, and treatment. Traditional genomic co-expression network analyses often overlook individual-specific interactions critical for personalized medicine. In this study, we employed single-sample gene co-expression network analysis to investigate the structural and functional genomic alterations across breast cancer subtypes (Luminal A, Luminal B, Her2-enriched, and Basal-like) and compared them with normal breast tissue. We utilized RNA-Seq gene expression data to infer gene co-expression networks. The LIONESS algorithm allowed us to construct individual networks for each patient, capturing unique co-expression patterns. We focused on the top 10,000 gene interactions to ensure consistency and robustness in our analysis. Network metrics were calculated to characterize the topological properties of both aggregated and single-sample networks. Our findings reveal significant fragmentation in the co-expression networks of breast cancer subtypes, marked by a change from interchromosomal (TRANS) to intrachromosomal (CIS) interactions. This transition indicates disrupted long-range genomic communication, leading to localized genomic regulation and increased genomic instability. Single-sample analyses confirmed that these patterns are consistent at the individual level, highlighting the molecular heterogeneity of breast cancer. Despite these pronounced alterations, the proportion of CIS interactions did not significantly correlate with patient survival outcomes across subtypes, suggesting limited prognostic value. Furthermore, we identified high-degree genes and critical cytobands specific to each subtype, providing insights into subtype-specific regulatory networks and potential therapeutic targets. These genes play pivotal roles in oncogenic processes and may represent important keys for targeted interventions. The application of single-sample co-expression network analysis proves to be a powerful tool for uncovering individual-specific genomic interactions.

Spectral Signatures of Nontrivial Topology in a Superconducting Circuit

Topology, like symmetry, is a fundamental concept in understanding general properties of physical systems. In condensed matter, nontrivial topology may manifest itself as singular features in the energy spectrum or the quantization of electrical properties such as conductance and magnetic flux. Using microwave spectroscopy, we determine that a superconducting circuit with three Josephson tunnel junctions in parallel can possess degeneracies indicative of nontrivial topology. We identify three topological invariants, one of which is related to a hidden quantum mechanical supersymmetry. Measurements show that devices fabricated in different topological regimes fall on a simple phase diagram, which should be robust to junction imperfections and geometric inductance. Josephson tunnel junction circuits, which are readily fabricated with conventional microlithography techniques, allow access to a wide range of topological systems that may have no condensed matter analog. Notable spectral features of these circuits, such as degeneracies and flat bands, may find use in quantum information, sensing, and metrology. Published by the American Physical Society 2024

Topological properties of single-particle states decaying into a continuum due to interaction

We investigate how topological Chern numbers can be defined when single-particle states hybridize with continua. We do so exemplarily in a bosonic Haldane model at zero temperature with an additional on-site decay of one boson into two and the conjugate fusion of two bosons into one. Restricting the Hilbert space to two bosons at maximum, the exact self-energy is accessible. We use the bilinear Hamiltonian H0 corrected by the self-energy Σ to compute Chern numbers by two different approaches. The results are gauged against a full many-body calculation in the Hilbert space where possible. We establish numerically and analytically that the effective Hamiltonian Heff=H0(k⃗)+Σ(ω,k⃗) reproduces the correct many-body topology if the considered band does not overlap with the continuum. In case of overlaps, one can extend the definition of the Chern number to the non-Hermitian Heff and there is evidence that the Chern number changes at exceptional points. But the bulk-boundary correspondence appears to be no longer valid and edge modes delocalize. Published by the American Physical Society 2024

Topology in a one-dimensional plasmonic crystal: the optical approach

Abstract In this paper we study the topology of the bands of a plasmonic crystal composed of graphene and of a metallic grating. Firstly, we derive a Kronig–Penney type of equation for the plasmonic bands as function of the Bloch wavevector and discuss the propagation of the surface plasmon polaritons on the polaritonic crystal using a transfer-matrix approach considering a finite relaxation time. Second, we reformulate the problem as a tight-binding model that resembles the Su–Schrieffer–Heeger (SSH) Hamiltonian, one difference being that the hopping amplitudes are, in this case, energy dependent. In possession of the tight-binding equations it is a simple task to determine the topology (value of the winding number) of the bands. This allows to determine the existense or absence of topological end modes in the system. Similarly to the SSH model, we show that there is a tunable parameter that induces topological phase transitions from trivial to non-trivial. In our case, it is the distance d between the graphene sheet and the metallic grating. We note that d is a parameter that can be easily tuned experimentally simply by controlling the thickness of the spacer between the grating and the graphene sheet. It is then experimentally feasible to engineer devices with the required topological properties. Finally, we suggest a scattering experiment allowing the observation of the topological states.

Photonic hook shaping achieved by the micro-nano fiber array based Janus cylindrical metalens

Abstract Near-field focusing of an electromagnetic wave on the assembly of hexagonally asymmetric arranged close-contact nanofibers into a cylindrical Janus metalens is considered for different fiber sizes and optical properties. We propose combining the meta-material concept with a photonic hook based on the finite-difference time-domain technique. Simulation results show that the cylindrical meta-photonic Janus structure produces the focal region of enhanced optical intensity, which exists in the spatial form of a localized structured light known as a photonic hook. A detailed study of all key parameters of a photonic hook depending on the Janus metalens topology and optical properties of the fiber filling is carried out. The structure conditions have been determined for the beam waist of a photonic hook that is less than the diffraction limit. This Janus cylindrical metalens may contribute to the versatile control of photonic hook generation in the applications of bio-photonics and optical microscopes.

NIMG-11. DISCOVERING POSTOPERATIVE APHASIA RECOVERY IN GLIOMA PATIENTS: THE ROLE OF THE RIGHT BROCA’S HOMOLOG AND VERIFICATION

Abstract OBJECTIVE This study aimed to elucidate the neurological mechanisms that support the recovery of postoperative aphasia by examining alterations in language and extended networks in patients with glioma. Additionally, we sought to determine the causal relationship between the right Broca’s homolog and language recovery using neuro-navigated repetitive transcranial magnetic stimulation (nrTMS). METHODS Sixty-two patients with left hemisphere gliomas were retrospectively analyzed and categorized into aphasia and non-aphasia groups based on Aphasia Quotient (AQ) scores. Resting-state functional MRI (rs-fMRI) data were collected and analyzed to investigate differences in functional connectivity (FC) and topological properties of language-related networks. Additionally, eight patients with surgery-induced aphasia were prospectively treated with nrTMS, targeting the node identified from the retrospective analysis. RESULTS Significant increases in FC between the left auditory cortex and the right inferior frontal gyrus in the Executive Control Network were observed in the non-aphasia group. Nodal properties, such as the nodal efficiency of the right Broca’s homolog, were independently associated with postoperative language function. The nrTMS treatment group demonstrated significant improvements in AQ scores, with correlations between increased nodal efficiency and positive language outcomes. CONCLUSION The right hemisphere, specifically the right inferior frontal gyrus, plays a compensatory role in maintaining language function after glioma resection. These findings suggest the therapeutic potential of nrTMS for enhancing postoperative language recovery, offering insights into individualized patient care and rehabilitation strategies for patients with glioma.

Observation of planar Hall effect in the topological insulator NaCd4As3

The observation of the planar Hall effect (PHE) illuminates the spin textures and topological properties of materials, indicating potential applications in quantum computing and electronic devices. Here, we present a study on the planar Hall transport of topological insulator NaCd4As3 single crystals. When the magnetic field is rotated within the sample plane relative to the current direction, we observe remarkable planar Hall resistivity and giant planar anisotropic magnetoresistance (AMR), both consistent with the theoretical expression of the PHE. Further analysis reveals that the orbital magnetoresistance effect, unrelated to surface electrons from topological surface states or bulk electrons from nontrivial Berry phases, lays a dominant role in the PHE in NaCd4As3. Additionally, the AMR ratio reaches −43% at 3 K under 14 T and remains −9% at room temperature, markedly exceeding that of traditional ferromagnetic metals. These findings provide a platform for understanding the PHE mechanism in topological insulators and highlight the potential of NaCd4As3 for angle and magnetic field detection applications.

Some Insights into the Sierpiński Triangle Paradox

We realize that a Sierpiński arrowhead curve (SAC) fills a Sierpiński gasket (SG) in the same manner as a Peano curve fills a square. Namely, in the limit of an infinite number of iterations, the fractal SAC remains self-avoiding, such that SAC⊂SG. Therefore, SAC differs from SG in the same sense as the self-avoiding Peano curve PC⊂0,12 differs from the square. In particular, the SG has three-line segments constituting a regular triangle as its border, whereas the border of SAC has the structure of a totally disconnected fat Cantor set. Thus, in contrast to the SG, which has loops at all scales, the SAC is loopless. Consequently, although both patterns have the same similarity dimension D=ln⁡3/ln⁡2, their connectivity dimensions are different. Specifically, the connectivity dimension of the self-avoiding SAC is equal to its topological dimension dlSAC=d=1, whereas the connectivity dimension of the SG is equal to its similarity dimension, that is, dlSG=D. Therefore, the dynamic properties of SG and SAC are also different. Some other noteworthy features of the Sierpiński triangle are also highlighted.

Inner rates of finite morphisms

Let (X, 0) be a complex analytic surface germ embedded in (Cn,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\mathbb {C}}^n,0)$$\\end{document} with an isolated singularity and Φ=(g,f):(X,0)⟶(C2,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi =(g,f):(X,0) \\longrightarrow ({\\mathbb {C}}^2,0)$$\\end{document} be a finite morphism. We define a family of analytic invariants of the morphism Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document}, called inner rates of Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document}. By means of the inner rates we study the polar curve associated with the morphism Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} when fixing the topological data of the curve (gf)-1(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(gf)^{-1}(0)$$\\end{document} and the surface germ (X, 0), allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function f:(X,0)⟶(C,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:(X,0) \\longrightarrow ({\\mathbb {C}},0)$$\\end{document}. The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ (X, 0) and of the curve (gf)-1(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(gf)^{-1}(0)$$\\end{document}.

Topological material in the III–V family: Heteroepitaxial InBi on InAs

InBi(001) is formed epitaxially on InAs(111)-A by depositing Bi onto an In-rich surface. Angle-resolved photoemission measurements reveal topological electronic surface states, close to the M¯ high symmetry point. This demonstrates a heteroepitaxial system entirely in the III–V family with topological electronic properties. InBi shows coexistence of Bi and In surface terminations, in contradiction with other III–V materials. For the Bi termination, the study gives a consistent physical picture of the topological surface electronic structure of InBi(001) terminated by a Bi bilayer rather than a surface formed by splitting to a Bi monolayer termination. Theoretical calculations based on relativistic density functional theory and the one-step model of photoemission clarify the relationship between the InBi(001) surface termination and the topological surface states, supporting a predominant role of the Bi bilayer termination. Furthermore, a tight-binding model based on this Bi bilayer termination with only Bi–Bi hopping terms, and no Bi–In interaction, gives a deeper insight into the spin texture. Published by the American Physical Society 2024

Mother-Child Closeness and Adolescent Structural Neural Networks: A Prospective Longitudinal Study of Low-Income Families.

Mother-child closeness, a mutually trusting and affectionate bond, is an important factor in shaping positive youth development. However, little is known about the neural pathways through which mother-child closeness are related to brain organization. Utilizing a longitudinal sample primarily from low-income families (N=181; 76% African American youth and 54% female), this study investigated the associations between mother-child closeness at ages 9 and 15 and structural connectivity organization (network integration, robustness, and segregation) at age 15. The assessment of mother-child closeness included perspectives from both mother and child. The results revealed that greater mother-child closeness is linked with increased global efficiency and transitivity, but not modularity. Specifically, both the mother's and child's report of closeness at age 15 predicted network metrics but report at age 9 did not. Our findings suggest that mother-child closeness is associated with neural white matter organization, as adolescents who experienced greater mother-child closeness displayed topological properties indicative of more integrated and robust structural networks.

Topological Hall Effect Instigated in Kagome Mn$_{3-x}$Sn due to Mn-deficit Induced Noncoplanar Spin Structure.

Magnetic topological semimetals are manifestations of the interplay between electronic and magnetic phases of matter, leading to peculiar characteristics such as the anomalous Hall effect (AHE) and the topological Hall effect (THE). Mn$_{3}$Sn is a time-reversal symmetry-broken (TRS) magnetic Weyl semimetal showing topological characteristics within the Kagome lattice network. This study reveals a large topological Hall effect in Mn$_{2.8}$Sn (6\% Mn deficit Mn$_3$Sn) at room temperature in the $xy$-plane, despite being an antiferromagnet. We argue that the magnetocrystalline anisotropy induced noncoplanar spin structure is responsible for the observed topological Hall effect in these systems. Further, the topological properties of these systems are highly anisotropic, as we observe a large anomalous Hall effect in the $zx$-plane. We find that Fe doping at the Mn site, Mn$_{3-x}$Fe$_x$Sn ($x$=0.2, 0.25, and 0.35), tunes the topological properties of these systems. These findings promise the realization of potential topotronic applications at room temperature.

Topological magnon in exchange frustration driven incommensurate spin spiral of kagome-lattice YMn6Sn6

YMn6Sn6 consists of two types of Mn-based kagome planes stacked along the c-axis having a complex magnetic interaction. We report a spin reconstruction in YMn6Sn6 from ferromagnetic (FM) into a combination of two incommensurate spin spirals (SSs) originating from two different types of Mn kagome planes driven by frustrated magnetic exchanges along the c-axis with the inclusion of the Hubbard U. The pitch angle and wave vector of the incommensurate SSs are ∼89.3∘ and ∼ (0 0 0.248), respectively, which are in excellent agreement with the experiment. We employ an effective model Hamiltonian constructed out of exchange interactions to capture the experimentally observed nonequivalent nature of the two incommensurate SSs which also explain the FM-SS crossover due to antiferromagntic spin exchange with correlation. We further report the existence of a topological magnon with spin-orbit coupling in the incommensurate SS phase of YMn6Sn6 by calculating the topological invariants and Berry curvature profile. The location of the Dirac magnon in the energy landscape at 73 meV matches with another experimental report. We demonstrate the accuracy of our results by highlighting the experimental features in YMn6Sn6. Published by the American Physical Society 2024

Particle-attached bacterial communities are more susceptible to seasonal environmental fluctuations in mesotrophic than eutrophic tropical reservoirs.

Particle-attached bacterial (PAB) communities play pivotal roles in water organic matter decomposition, nutrient cycling, and the natural self-purification processes. However, we know little about their responses to seasonal environmental fluctuations, under eutrophication in reservoir ecosystems. In this study, we studied the shifts of PAB communities to seasonal environmental fluctuations in tropical China. Trophic state index (TSI) indicated that the studied reservoirs ranged from mesotrophic to eutrophic state with a gradual increase in TSI from 31 to 58. In eutrophic reservoirs, Cyanobacteria, especially Raphidiopsis raciborskii, significantly increased in its relative abundance from wet to dry season, but Synechococcales and Microcystaceae decreased. In contrast, the relative abundance of Clostridia, Bacilli, Coriobacteriia, Enterobacteriales, and Vibrionales were more susceptible to seasonal environmental fluctuations in mesotrophic than eutrophic reservoirs. PAB co-occurrence relationships in mesotrophic reservoirs varied more greatly in response to seasonal environmental fluctuations, compared with eutrophic reservoirs, in terms of topological properties of connectedness, average degree, robustness and vulnerability. Our results further demonstrated that the seasonal stability of PAB co-occurrence relationships was strongly correlative with TSI through mediating key bacterial taxa and community biodiversity. We proposed that eutrophication dramatically reduced the seasonal variation of PAB community compositions and co-occurring relationships in reservoir ecosystems.

Anisotropic quantum transport in a programmable photonic topological insulator

Quantum transport in materials describes the behavior of particles at the quantum level. Topological materials exhibit nontrivial transport properties with topological invariants, leading to the emergence of protected states that are immune against disorders at the material boundaries. In many real-world materials, especially those with anisotropic crystal structures, the transport properties can vary significantly along different directions within the material bulk. Here, we experimentally observe counterintuitive quantum transport phenomena in anisotropic topological insulators with controllable anisotropy and disorder, implemented on a programmable topological photonic chip. We examine phase transition from the topological phase to the Anderson phase, between which a new quasi-diffusive phase emerges. Anisotropic topological transport demonstrates unconventional superior robustness in the bulk mode compared to the edge mode, in the presence of disorder and loss in realistic systems. Peculiar topological transport with sophisticated gradient anisotropy, emulating stretched topological materials, occurs at the gradient domain wall that can be reconfigured. Our findings provide fresh insights into the intricate interplay between anisotropy within the bulk and robustness at the boundary of topological materials, which could lead to advancements in the field of topological material science and the development of topological devices with tailored functionalities.