One leading question for the application of graphene in nanoelectronics is how electronic properties depend on the size at the nanoscale. Direct observation of the quantized electronic states is central to conveying the relationship between electronic structures and local geometry. Scanning tunneling spectroscopy was used to measure differential conductance dI/dV patterns of nanometer-size graphene islands on an Ir(111) surface. Energy-resolved dI/dV maps clearly show a spatial modulation, indicating a modulated local density of states due to quantum confinement, which is unaffected by the edge configuration. We establish the energy dispersion relation with the quantized electron wave vector obtained from a Fourier analysis of dI/dV maps. The nanoislands preserve the Dirac Fermion properties with a reduced Fermi velocity.

Experimental investigations of spin-polarized electron confinement in nanostructures by scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) are reviewed. To appreciate the experimental results on the electronic level, the physical basis of STM is elucidated with special emphasis on the correlation between differential conductance, as measured by STS, and the electron density of states, which is accessible in ab initio theory. Experimental procedures which allow one to extract the electron dispersion relation from energy-dependent and spatially resolved STM and STS studies of electron confinement are reviewed. The role of spin polarization in electron confinement is highlighted by both experimental and theoretical insights, which indicate variation of the spin polarization in sign and magnitude on the nanometer scale. This review provides compelling evidence for the necessity to include spatial-dependent spin-resolved electronic properties for an in-depth understanding and quantitative assessment of electron confinement in magnetic nanostructures and interaction between magnetic adatoms.

Low-dimensionality in magnetic materials often leads to noncollinear magnetic order, such as a helical spin order and skyrmions, which have received much attention because of envisioned applications in spin transport and in future data storage. Up to now, however, the real-space observation of the noncollinear magnetic order has been limited mostly to systems involving a strong spin-orbit interaction. Here we report a noncollinear magnetic order in individual nanostructures of a prototypical magnetic material, bilayer iron islands on Cu (111). Spin-polarized scanning tunnelling microscopy reveals a magnetic stripe phase with a period of 1.28 nm, which is identified as a one-dimensional helical spin order. Ab initio calculations identify reduced-dimensionality-enhanced long-range antiferromagnetic interactions as the driving force of this spin order. Our findings point at the potential of nanostructured magnets as a new experimental arena of noncollinear magnetic order stabilized in a nanostructure, magnetically decoupled from the substrate.

We performed scanning tunneling microscopy/spectroscopy (STM/S) on the edges of monolayer graphene islands grown on Ir(111). Constant current STM images reveal that the atomic corrugation at the graphene edges correlates with the moiré pattern of graphene on Ir(111). The graphene islands terminate with a zigzag edge and moiré-periodic kinks in the regions of the on-top stacked carbon rings. Graphene edges near an Ir(111) monatomic step also show the formation of periodic kinks. STS indicates a significant spatial dependence of the differential conductance dI/dV near graphene edges. We observe a considerably reduced differential conductance at the graphene island edges. We tentatively ascribe these observations to the electronic interaction of graphene with the Ir(111) substrate, which affects the differential conductance differently near graphene edges as compared to the inner part of a graphene island.

In non-abelian gauge theories without matter fields, expectation values of large Wilson loops and loop correlation functions are difficult to compute through numerical simulation, because the signal-to-noise ratio is very rapidly decaying for increasing loop sizes. Using a multilevel scheme that exploits the locality of the theory, we show that the statistical errors in such calculations can be exponentially reduced. We explicitly demonstrate this in the SU(3) theory, for the case of the Polyakov loop correlation function, where the efficiency of the simulation is improved by many orders of magnitude when the area bounded by the loops exceeds 1 fm 2 .