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We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes, thus confirming that these ephemeral points strengthen the standard VR barcode. 

The nitrogen-vacancy (NV) center in diamond is an increasingly popular quantum sensor for microscopy of electrical current, magnetization, and spins. However, efficient NV–sample integration with a robust, high-quality interface remains an outstanding challenge to realize scalable, high-throughput microscopy. In this work, we characterize a diamond micro-chip (DMC) containing a (111)-oriented NV ensemble and demonstrate its utility for high-resolution quantum microscopy. We perform strain imaging of the DMC and find minimal detrimental strain variation across a field of view of tens of micrometer. We find good ensemble NV spin coherence and optical properties in the DMC, suitable for sensitive magnetometry. We then use the DMC to demonstrate wide-field microscopy of electrical current and show that diffraction-limited quantum microscopy can be achieved. We also demonstrate the deterministic transfer of DMCs with multiple materials of interest for next-generation electronics and spintronics. Lastly, we develop a polymer-based technique for DMC placement. This work establishes the DMC's potential to expand the application of NV quantum microscopy in materials, device, geological, biomedical, and chemical sciences. 

Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant. 

Abstract One-dimensional chiral interface channels can be created at the boundary of two quantum anomalous Hall (QAH) insulators with different Chern numbers. Such a QAH junction may function as a chiral edge current distributer at zero magnetic field, but its realization remains challenging. Here, by employing an in-situ mechanical mask, we use molecular beam epitaxy to synthesize QAH insulator junctions, in which two QAH insulators with different Chern numbers are connected along a one-dimensional junction. For the junction between Chern numbers of 1 and −1, we observe quantized transport and demonstrate the appearance of the two parallel propagating chiral interface channels along the magnetic domain wall at zero magnetic field. For the junction between Chern numbers of 1 and 2, our quantized transport shows that a single chiral interface channel appears at the interface. Our work lays the foundation for the development of QAH insulator-based electronic and spintronic devices and topological chiral networks. 

A quantum anomalous Hall (QAH) insulator is a topological phase in which the interior is insulating but electrical current flows along the edges of the sample in either a clockwise or counterclockwise direction, as dictated by the spontaneous magnetization orientation. Such a chiral edge current eliminates any backscattering, giving rise to quantized Hall resistance and zero longitudinal resistance. Here we fabricate mesoscopic QAH sandwich Hall bar devices and succeed in switching the edge current chirality through thermally assisted spin–orbit torque (SOT). The well-quantized QAH states before and after SOT switching with opposite edge current chiralities are demonstrated through four- and three-terminal measurements. We show that the SOT responsible for magnetization switching can be generated by both surface and bulk carriers. Our results further our understanding of the interplay between magnetism and topological states and usher in an easy and instantaneous method to manipulate the QAH state. 

The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a 1-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra-Lenstra-Lovász algorithm from computational number theory. 

The interface between two different materials can show unexpected quantum phenomena. In this study, we used molecular beam epitaxy to synthesize heterostructures formed by stacking together two magnetic materials, a ferromagnetic topological insulator (TI) and an antiferromagnetic iron chalcogenide (FeTe). We observed emergent interface-induced superconductivity in these heterostructures and demonstrated the co-occurrence of superconductivity, ferromagnetism, and topological band structure in the magnetic TI layer—the three essential ingredients of chiral topological superconductivity (TSC). The unusual coexistence of ferromagnetism and superconductivity is accompanied by a high upper critical magnetic field that exceeds the Pauli paramagnetic limit for conventional superconductors at low temperatures. These magnetic TI/FeTe heterostructures with robust superconductivity and atomically sharp interfaces provide an ideal wafer-scale platform for the exploration of chiral TSC and Majorana physics. 

Abstract An axion insulator is a three-dimensional (3D) topological insulator (TI), in which the bulk maintains the time-reversal symmetry or inversion symmetry but the surface states are gapped by surface magnetization. The axion insulator state has been observed in molecular beam epitaxy (MBE)-grown magnetically doped TI sandwiches and exfoliated intrinsic magnetic TI MnBi₂Te₄flakes with an even number layer. All these samples have a thickness of ~ 10 nm, near the 2D-to-3D boundary. The coupling between the top and bottom surface states in thin samples may hinder the observation of quantized topological magnetoelectric response. Here, we employ MBE to synthesize magnetic TI sandwich heterostructures and find that the axion insulator state persists in a 3D sample with a thickness of ~ 106 nm. Our transport results show that the axion insulator state starts to emerge when the thickness of the middle undoped TI layer is greater than ~ 3 nm. The 3D hundred-nanometer-thick axion insulator provides a promising platform for the exploration of the topological magnetoelectric effect and other emergent magnetic topological states, such as the high-order TI phase.