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In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $\delta\rightarrow0$ or $\delta\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case., Comment: 20 pages, 39 figures

In this paper, we conduct rigorous error analysis of the Lie-Totter time-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equation with a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodic boundary conditions on a $d$-dimensional torus $\mathbb T^d$. Different from existing works based on regularisation of the nonlinear term $ f(u)\approx f^\varepsilon(u)=u\ln (|u| + \varepsilon )^2,$ we directly discretize the LogSE with the understanding $f(0)=0.$ Remarkably, in the time-splitting scheme, the solution flow map of the nonlinear part: $g(u)= u {\rm e}^{-{\rm} i t \ln|u|^{2}}$ has a higher regularity than $f(u)$ (which is not differentiable at $u=0$ but H\"older continuous), where $g(u)$ is Lipschitz continuous and possesses a certain fractional Sobolev regularity with index $0

Kagome lattice hosts a plethora of quantum states arising from the interplay of topology, spin-orbit coupling, and electron correlations. Here, we report symmetry-breaking electronic orders tunable by an applied magnetic field in a model Kagome magnet FeSn consisting of alternating stacks of two-dimensional Fe3Sn Kagome and Sn2 honeycomb layers. On the Fe3Sn layer terminated FeSn thin films epitaxially grown on SrTiO3(111) substrates, we observe trimerization of the Kagome lattice using scanning tunneling microscopy/spectroscopy, breaking its six-fold rotational symmetry while preserving the transitional symmetry. Such a trimerized Kagome lattice shows an energy-dependent contrast reversal in dI/dV maps, which is significantly enhanced by bound states induced by Sn vacancy defects. This trimerized Kagome lattice also exhibits stripe modulations that are energy-dependent and tunable by an applied in-plane magnetic field, indicating symmetry-breaking nematicity from the entangled magnetic and charge degrees of freedom in antiferromagnet FeSn.

Two-dimensional (2D) magnets exhibit unique physical properties for potential applications in spintronics. To date, most 2D ferromagnets are obtained by mechanical exfoliation of bulk materials with van der Waals interlayer interactions, and the synthesis of single or few-layer 2D ferromagnets with strong interlayer coupling remains experimentally challenging. Here, we report the epitaxial growth of 2D non-van der Waals ferromagnetic bilayer FeSb on SrTiO3(001) substrates stabilized by strong coupling to the substrate, which exhibits in-plane magnetic anisotropy and a Curie temperature above 300 K. In-situ low-temperature scanning tunneling microscopy/spectroscopy and density-functional theory calculations further reveal that a Fe Kagome layer terminates the bilayer FeSb. Our results open a new avenue for further exploring emergent quantum phenomena from the interplay of ferromagnetism and topology for application in spintronics.

A two dimensional topological insulator exhibits helical edge states topologically protected against single particle backscattering. Such protection breaks down, however, when electron electron interactions are significant or when edge reconstruction occurs, leading to suppressed density of states (DOS) at Fermi level that follows universal scaling with temperature and energy, characteristic of Tomonaga Luttinger liquid (TLL). Here, we grow multilayer FeSe on SrTiO3 by molecular beam epitaxy, and observe robust edge states at both {100}Se and {110}Se steps using scanning tunneling microscopy/spectroscopy. We determine the DOS follows a power law, resulting in Luttinger parameter K of 0.26 +/- 0.02 and 0.43 +/- 0.07 for {100}Se and {110}Se edges, respectively. The smaller K for the {100}Se edge also indicates strong correlations, attributed to ferromagnetic ordering likely present due to checkerboard antiferromagnetic fluctuations in FeSe. These results demonstrate TLL in FeSe helical edge channels, providing an exciting model system for novel topological excitations arising from superconductivity and interacting helical edge states., Comment: 28 pages

Quantum materials, particularly Dirac materials with linearly dispersing bands, can be effectively tuned by strain-induced lattice distortions leading to a pseudo-magnetic field that strongly modulates their electronic properties. Here, we grow kagome magnet FeSn films, consisting of alternatingly stacked Sn$_2$ honeycomb (stanene) and Fe$_3$Sn kagome layers, on SrTiO$_3$(111) substrates by molecular beam epitaxy. Using scanning tunneling microscopy/spectroscopy, we show that the Sn honeycomb layer can be periodically deformed by epitaxial strain for film thickness below 10 nm, resulting in differential conductance peaks consistent with Landau levels generated by a pseudo-magnetic field greater than 1000 T. Our findings demonstrate the feasibility of strain engineering the electronic properties of topological magnets at the nanoscale., Comment: 46 pages

In this paper, we introduce a new family of orthogonal systems, termed as the M\"{u}ntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $\|x\|^{2\theta+2\mu-2} (1-\|x\|^{2\theta})^{\alpha}$ with the parameters $\alpha>-1, \mu>- 1/2$ and $\theta>0$ in the $d$-dimensional unit ball $x\in {\mathbb B}^d=\big\{x\in\mathbb{R}^d: r=\|x\|\leq1\big\}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schr\"odinger's operators with fractional potentials. We demonstrate that the use of such non-standard basis functions can not only tailor to the singularity of the solutions but also lead to sparse linear systems which can be solved efficiently., Comment: 22 pages, 7 figures

Scanning tunneling microscopy (STM) is a powerful technique for imaging atomic structure and inferring information on local elemental composition, chemical bonding, and electronic excitations. However, traditional methods of visual inspection can be challenging for such determination in multi-component alloys, particularly beyond the dilute limit due to chemical disorder and electronic inhomogeneity. One viable solution is to use machine learning to analyze STM data and identify patterns and correlations that may not be immediately apparent through visual inspection alone. Here, we apply this approach to determine the Se/S concentration in superconducting single-layer FeSe1-xSx alloy epitaxially grown on SrTiO3(100) substrate by molecular beam epitaxy. First, defect-related dI/dV tunneling spectra are identified by the K-means clustering method, followed by singular value decomposition to distinguish between those from S and Se. Such analysis provides an efficient and reliable determination of local elemental composition, and further reveals correlations of nanoscale chemical inhomogeneity to superconductivity in single-layer iron chalcogenide films.

As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about $95\%$ range of $[-1,1]$ except for a small neighbourhood near the singular point $x=a.$ In this paper, we rigorously show that the Jacobi expansion for a more general class of $\Phi$-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired $\log n$-factor in the pointwise error estimate for the Legendre expansion recently stated in Babu\u{s}ka and Hakula [5] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates., Comment: 34 pages, 10 figures

By substituting S into single-layer FeSe/SrTiO3, chemical pressure is applied to tune its paramagnetic state that is modeled as an incoherent superposition of spin-spiral states. The resulting electronic bands resemble an ordered checkerboard antiferromagnetic structure, consistent with angle-resolved photoemission spectroscopy measurements. Scanning tunneling spectroscopy reveals a gap evolving from U-shaped for FeSe to V-shaped for FeS with decreasing size, attributed to a d-wave superconducting state for which nodes emerge once the gap size is smaller than the effective spin-orbit coupling., Comment: need revision

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)$ with constant $\sigma\not=0$ and usual initial conditions at $t=-1,$ are associated with the generalised Bessel polynomials (GBPs). The essential idea of the analysis is to properly construct the basis functions for the solution and its dual spaces so that the matrix of the $m$th derivative is an identity matrix, and the mass matrix is then identical or approximately equals to the Jacobi matrix of the three-term recurrence of GBPs with specific integer parameters. This allows us to characterise the eigenvalue distributions and identify the eigenvectors. As a by-product, we are able to answer some open questions related to the very limited known results on the collocation method at Legendre points (studied in 1980s) for the first-order IVP, by reformulating it into a Petrov-Galerkin formulation. Moreover, we present two stable algorithms for computing zeros of the GBPs, and develop a general space-time spectral method for evolutionary PDEs using either the matrix diagonalisation, which is restricted to a small number of unknowns in time due to the ill-conditioning but is fully parallel, or the QZ decomposition which is numerically stable for a large number of unknowns in time but involves sequential computations. We provide ample numerical results to demonstrate the high accuracy and robustness of the space-time spectral methods for some interesting examples of linear and nonlinear wave problems., Comment: 25 pages, 31 figures