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In Uranium-based heavy fermion system, the 5f electrons display an intermediate character between partial localization and partial itinerancy, which makes the Kondo problem more complicated. Here we use scanning tunneling microscopy/spectroscopy to investigate the (001) surface of the Kondo lattice antiferromagnet USb2. Temperature-dependent dI/dV spectra from 4.7 K up to 140 K reveal several peak structures around the Fermi level. Two pronounced peaks are originated from the hybridization between the conduction and 5f electrons. We did not observe the mysteriously abrupt change of the electronic state at 45 K, which is previously reported by another STM group and attributed to a novel first-order like transition. Instead, we only observe continuous evolution of the dI/dV spectra with temperature. Furthermore, in some scanning regions, we find significant Sb atoms missing from the top layer, which gives us the opportunity to investigate the electronic structure of the U-terminated surface. For the U-terminated surface, an additional sharp peak emerges, which is closely related to the magnetic order., Comment: 12 pages, 3 figures

The Kondo effect typically arises from the spin-flip scattering between the localized magnetic moment of the impurity and the delocalized electrons in the metallic host, which leads to a variety of intriguing phenomena. Here, by using scanning tunnelling microscopy/spectroscopy (STM/STS), we present the Kondo effect and subatomic features of single U adatom on graphene/6H-SiC(0001). A dip spectral feature can be observed around the Fermi energy, which is termed as the "fingerprint" of the Kondo resonance in STS; in addition, two subatomic features with different symmetries: a three-lobe structure and a donghnut-like structure can be observed from the dI/dV maps. The Kondo resonance is only detectable within 5~\AA~of the lateral distance from the U atom center, which is much smaller than the distances observed in Co atoms on different surfaces, indicating the more localized 5$f$ states than 3$d$ orbitals. By comparing with density functional theory calculations, we find that the two subatomic features displaying different symmetries originate from the selective hybridization between U 6$d$, 5$f$ orbitals and the $p_z$ orbitals from two inequivalent C atoms of the multilayer graphene., Comment: 5 pages, 4 figures

In heavy-fermion compounds, the dual character of $f$ electrons underlies their rich and often exotic properties like fragile heavy quasipartilces, variety of magnetic orders and unconventional superconductivity. 5$f$-electron actinide materials provide a rich setting to elucidate the larger and outstanding issue of the competition between magnetic order and Kondo entanglement and, more generally, the interplay among different channels of interactions in correlated electron systems. Here, by using angle-resolved photoemission spectroscopy, we present detailed electronic structure of USb$_2$ and observed two different kinds of nearly flat bands in the antiferromagnetic state of USb$_2$. Polarization-dependent measurements show that these electronic states are derived from 5$f$ orbitals with different characters; in addition, further temperature-dependent measurements reveal that one of them is driven by the Kondo correlations between the 5$f$ electrons and conduction electrons, while the other reflects the dominant role of the magnetic order. Our results on the low-energy electronic excitations of USb$_2$ implicate orbital selectivity as an important new ingredient for the competition between Kondo correlations and magnetic order and, by extension, in the rich landscape of quantum phases for strongly correlated $f$ electron systems., Comment: 5 pages, 4 figures; submitted for publication on May 27, 2019

Hidden order in URu$_2$Si$_2$ has remained a mystery now entering its 4th decade. The importance of resolving the nature of the hidden order has stimulated extensive research. Here we present a detailed characterization of different surface terminations in URu$_2$Si$_2$ by angle-resolved photoemission spectroscopy, in conjunction with scanning tunneling spectroscopy and DMFT calculations that may unveil a new piece of this puzzle. The U-terminated surface is characterized by an electron-like band around the $\bar{X}$ point, while a hole-like band for the Si-terminated surface. We also investigate temperature evolution of the electronic structure around the $\bar{X}$ point from 11 K up to 70 K, and did not observe any abrupt change of the electronic structure around the coherence temperature (55 K). The $f$ spectral weight gradually weakens upon increasing temperature, still some $f$ spectral weight can be found above this temperature. Our results suggest that surface terminations in URu$_2$Si$_2$ are an important issue to be taken into account in future work., Comment: 7 pages, 5 figures

The electronic structure of the Kondo lattice CeIn3 has been studied by on-resonant angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy. A weakly dispersive quasiparticle band has been observed directly with an energy dispersion of 4 meV by photoemission, implying the existence of weak hybridization between the f electrons and conduction electrons. The hybridization is further confirmed by the formation of the hybridization gap revealed by temperature-dependent scanning tunneling spectroscopy. Moreover, we find the hybridization strength in CeIn3 is much weaker than that in the more two-dimensional compounds CeCoIn5 and CeIrIn5. Our results may be essential for the complete microscopic understanding of this important compound and the related heavy-fermion systems., Comment: 17pages, 5 figures

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L\'{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points., Comment: 25 pages, 25 figures, 7 tables

This article aims to develop a direct numerical approach to solve the space-fractional partial differential equations (PDEs) based on a new differential quadrature (DQ) technique. The fractional derivatives are approximated by the weighted linear combinations of the function values at discrete grid points on problem domain with the weights calculated via using three types of radial basis functions (RBFs) as test functions. The method in presence is robust, straight forward to apply, and highly accurate under the condition that the shape parameters of RBFs are well chosen. Numerical tests are provided to illustrate its validity and capability., Comment: 11 pages, 3 tables

In heavy-fermion compounds, f electrons show both itinerant and localized behaviour depending on the external conditions, and the hybridization between localized f electrons and itinerant conduction bands gives rise to their exotic properties like heavy-fermions, magnetic orders and unconventional superconductivity. Duo to the risk of handling radioactive actinide materials, the direct experimental evidence of the band structure evolution across the localized-itinerant and magnetic transitions for 5f electrons is lacking. Here, by using angle-resolved photoelectron spectroscopy, we revealed the dual nature (localized vs itinerant) and the development of two different kinds of heavy quasi-particle bands of 5f electrons in antiferromagnetic (AFM) USb2. Partially opened energy gaps were observed on one quasi-particle 5f band cross the AFM transition around 203 K, indicating that the magnetic orders in USb2 are of spin density wave (SDW) type similar to Cr. The localized 5f electrons and itinerant conduction bands hybridize to form another heavy quasi-particle band at about 120 K, and then open hybridization gaps at even lower temperature. Our results provide direct spectral demonstration of the localized-itinerant transition, hybridization and SDW transition of 5f electrons for uranium-based materials., Comment: 22 pages, 5 figures

This article studies a direct numerical approach for fractional advection-diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order $(0,1]$. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge-Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank-Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the concrete simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the existing algorithms in the open literature, numerical results finally illustrate the validity and accuracy., Comment: 18 pages, 7 tables, 14 figures

In this article, we propose an exponential B-spline collocation method to approximate the solution of the fractional sub-diffusion equation of Caputo type. The present method is generated by use of the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme in time and an efficient exponential B-spline based method in space. The unique solvability is rigorously discussed. Its stability is well illustrated via a procedure closely resembling the classic von Neumann approach. The resulting algebraic system is tri-diagonal that can rapidly be solved by the known algebraic solver with low cost and storage. A series of numerical examples are finally carried out and by contrast to the other algorithms available in the literature, numerical results confirm the validity and superiority of our method., Comment: 18 pages, 4 tables, 8 figures

The band structure, intra- and interband scattering processes of the electrons at the surface of a bismuth-bilayer on Bi$_2$Se$_3$ have been experimentally investigated by low-temperature Fourier-transform scanning tunneling spectroscopy. The observed complex quasiparticle interference patterns are compared to a simulation based on the spin-dependent joint density of states approach using the surface-localized spectral function calculated from first principles as the only input. Thereby, the origin of the quasiparticle interferences can be traced back to intraband scattering in the bismuth bilayer valence band and Bi$_2$Se$_3$ conduction band, and to interband scattering between the two-dimensional topological state and the bismuth-bilayer valence band. The investigation reveals that the bilayer band gap, which is predicted to host one-dimensional topological states at the edges of the bilayer, is pushed several hundred milli-electronvolts above the Fermi level. This result is rationalized by an electron transfer from the bilayer to Bi$_2$Se$_3$ which also leads to a two-dimensional electron state in the Bi$_2$Se$_3$ conduction band with a strong Rashba spin-splitting, coexisting with the topological state and bilayer valence band., Comment: 11 pages, 5 figures