Abstract:
In a simple model we investigate three competing orders: d-wave superconductivity, spin and orbital antiferromagnetism (AF) in the vicinity of a single vortex in a cuprate superconductor. We find that when the potential for the orbital AF $V_{d}$ is comparatively small, the spin-density-wave (SDW) order has an enhancement at the vortex core center, and the ''d-density-wave'' (DDW) order exhibits a rather weak behavior around the core and vanishing small away from the core. However, when $V_{d}$ becomes large, globally, the SDW order decreases and the DDW order increases; locally, not only the peak of the SDW order around the core still exists, though relatively suppressed, but also a local peak for the DDW order finally appears. Similar effects are also revealed for the features when varying doping. Comparisons with experiments are discussed.

Abstract:
The effect of a modulated magnetic field on the electronic structure of neutral graphene is examined in this paper. It is found that application of a small staggered modulated magnetic field does not destroy the Dirac-cone structure of graphene and so preserves its 4-fold zero-energy degeneracy. The original Dirac points (DPs) are just shifted to other positions in k space. By varying the staggered field gradually, new DPs with exactly the same electron-hole crossing energy as that of the original DPs, are generated, and both the new and original DPs are moving continuously. Once two DPs are shifted to the same position, they annihilate each other and vanish. The process of generation and evolution of these DPs with the staggered field is found to have a very interesting patten, which is examined carefully. Generally, there exists a corresponding branch of anisotropic massless fermions for each pair of DPs, resulting in that each Landau level (LL) is still 4-fold degenerate except the zeroth LL which has a robust $4n_t$-fold degeneracy with nt the number of pairs of DPs. As a result, the Hall conductivity $\sigma_{xy}$ shows a step of size $4n_te^2/h$ across zero energy.

Abstract:
Graphene properties can be manipulated by a periodic potential. Based on the tight-binding model, we study graphene under a one-dimensional (1D) modulated magnetic field which contains both a uniform and a staggered component. New chiral current-carrying edge states are generated at the interfaces where the staggered component changes direction. These edge states lead to an unusual integer quantum Hall effect (QHE) in graphene, which can be observed experimentally by a standard four-terminal Hall measurement. When Zeeman spin splitting is considered, a novel state is predicted where the electron edge currents with opposite polarization propagate in the opposite directions at one sample boundary, whereas propagate in the same directions at the other sample boundary.

Abstract:
The quantum Hall and longitudinal resistances in multi-terminal ferromagnetic graphene p-n junctions under a perpendicular magnetic field are investigated. In the Hall measurements, the transverse contacts are assumed to be located at p-n interface to avoid the mixing of edge states at the interface and the resulting quantized resistances are then topologically protected. According to the charge carrier type, the resistances in four-terminal p-n junction can be naturally divided into nine different regimes. The symmetric Hall and longitudinal resistances are observed, with lots of new robust quantum plateaus revealed due to the competition between spin splitting and local potentials.

Abstract:
Electron fully spin-polarized edge states in graphene emerged at the interfaces of a nonuniform magnetic field are studied numerically in a tight-binding model, with both the orbital and Zeeman-splitting effects of magnetic field considered. We show that the fully spin-polarized currents can be manipulated by a gate voltage. In order to make use of the fully spin-polarized currents in the spin related transport, a three-terminal experiment is designed and expected to export the fully spin-polarized currents. This may have important applications in spin based nanodevices.

Abstract:
We find that in a multi-orbital system with intraorbital and interorbital hopping integrals, the Hall conductance exhibits various topological quantum phase transitions (QPTs) induced by on-site orbital polarization: integer quantum Hall (IQH) plateau transitions, and topological Fermi liquid to IQH transitions. Such topological QPTs are demonstrated in two systems: a $p$-band spinless fermionic system realizable with ultracold atoms in optical lattice, and a $d$-band spinful fermionic system closely related to giant orbital Hall effects in transition metals and their compounds.

Abstract:
An intriguing magneto-transport property is demonstrated by tight-binding lattice electrons with Rashba spin-orbit coupling (SOC) in a magnetic field. With the flux strength $\phi={2\pi/N}$ ($N$ is an integer) and the Zeeman splitting fixed, when increasing the Rashba SOC $\lambda$, the spin-Hall and charge-Hall conductances (SHC and CHC) undergo four-step evolutions: the SHC shows size-dependent resonances and jumps at three critical $\lambda_{c}$'s, and changes its sign at $\lambda_{c1}$ and $\lambda_{c3}$; while the CHC exhibits three quantum jumps by $-Ne^2/h$, $+2Ne^2/h$ and $-Ne^2/h$. Such four-step evolutions are also reflected in topological characters and spin polarizations of edge states of a cylindrical system, and are robust against weak disorder.

Abstract:
The pattern transition induced by lattice anisotropy (LA) and magnetic impurities is computationally observed in near-optimally doped d-wave superconductors (DSCs). For the single impurity case, a transition from the checkerboard to stripe pattern can be induced even with a very weak LA. Moreover, the modulation period of eight lattice constants (8$a$) in the spin order coincides with neutron scattering data. For the two-impurity case, an orientation transition from the longitudinal impurity-pinned stripe into the transverse pattern is observed when the LA ratio reaches some critical value. At the critical point, it is found that the structures around magnetic impurities could restore checkerboard patterns. These results indicate that the formation of stripes in DSCs might induced by various effects, and could be tunable experimentally.

Abstract:
The transversal propagation of the edge states in a two-dimensional quantum spin Hall system are classified by decay characteristic quantity $\lambda$. Two different modes of the helical edge states exhibit distinct behaviors. The penetration depth is momentum dependent in normal edge states. The finite size gap decays monotonously with sample width, leading to the normal size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states with much shorter length. The finite size gap decays non-monotonously with width, leading to the anomalous finite size effect. Real materials are compared in the phase diagram, which explicitly demonstrates their differences. We also propose an intuitive way to search the special edge states in two-dimensional quantum spin Hall system.

Abstract:
The doping and frequency evolutions of the incommensurate spin response and the resonance mode are studied based on the scenario of the Fermi surface topology. We use the slave-boson mean-field approach to the $t-t^{\prime}-J$ model and including the antiferromagnetic fluctuation correction in the random-phase approximation. We find that the equality between the incommensurability and the hole concentration is reproduced at low frequencies in the underdoped regime. This equality observed in experiments was explained {\it only} based on the stripe model before. We also obtain the downward dispersion for the spin response and predict its doping dependence for further experimental testing, as well as a proportionality between the low-energy incommensurability and the resonance energy. Our results suggest a common origin for the incommensuration and the resonance peak based on the Fermi surface topology and the d-wave symmetry.