Anomalies and fermion zero modes on strings and domain walls

We show that the mathematical relation between non-abelian anomalies in 2n dimensions, the parity anomaly in 2n+1 dimensions, and the Dirac index density in 2n+2 dimensions can be understood in terms of the physics of fermion zero modes on strings and domain walls. We show that the Dirac equation possesses chiral zero modes in the presence of strings in 2n+2 dimensions (such as occur in axion theories) or domain walls in 2n+1 dimensions. We show that the anomalies due to the chiral zero modes are exactly cancelled by anomalies due to the coupling of axion-like fields to the Dirac index density or by anomalies due to the induced topological mass term.     

On the Hebb–Wagner polarisation of SrTiO3 doped with redox-active ions

Hebb–Wagner polarisation is analysed for the case that internal redox-reactions complicate the situation. In the general case current–voltage characteristics deviate significantly from the Hebb–Wagner equation. Analytical results are derived and relevant approximations given. Theoretical and experimental current–voltage relations obtained for a 2.15×10¹⁸ cm⁻³ Fe-doped SrTiO₃ single crystal at P_O2=10⁵ Pa, agree reasonably well, and the results of the partial conductivities match findings obtained from other experiments. In addition, the fact is emphasised that a Hebb–Wagner-type current–voltage relation only requires the fulfilment of a power law for the non-blocked carrier concentrations with respect to the component partial pressure and not a field-free situation. In this case the Faraday constant appearing in the Hebb–Wagner equation has to be replaced by F, where is the product of the number of electrons necessary to ionise the gaseous component (n) times the absolute characteristic exponent (|N|). The condition of zero field, i.e. negligible chemical potential gradient with respect to the non-blocked species is identical to |N|=1/n. If |N|=const≠1/n, the internal field is non-zero but is still proportional to the gradient of the component potential.     

镓原子的Stark能级结构

里德堡原子的Stark效应在偶极偶极相互作用、量子信息和量子调控等方面具有潜在的应用前景. 本文首先根据零场时镓原子的能级数据, 通过非线性拟合方法获得了镓原子各态的量子亏损, 仔细分析了量子亏损随主量子数的变化特征; 然后利用Numerov算法计算了镓原子的径向波函数; 最后采用矩阵对角化方法, 数值计算了镓原子高里德堡态在场强范围F=0-3000 V·cm^{- 1}时n=7和n=18附近的Stark能级结构. 结果显示在主量子数n=7多重态以上的能级结构中, (n+1)P态的能级接近并大于nD态的能级, 在n=7多重态以下的能级结构中, (n+1)P态的能级接近并小于nD态的能级. 这一现象不同于通常的碱金属原子的Stark结构, 论文对该现象及其他Stark能级结构特征进行了详细分析, 为相关研究工作提供了重要参考价值.     

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we revisit the connection between the Riemann–Roch theorem and the zero-energy solutions of the two-dimensional Dirac equation in the presence of a delta-function-like magnetic field. Our main result is the resolution of a paradox—the fact that the Riemann–Roch theorem correctly predicts the number of zero-energy solutions of the Dirac equation despite counting what seem to be functions of the wrong type.     

Fermion zero modes in the presence of non-abelian vortices

The Dirac equation in the presence of non-abelian vortices is investigated . For isospinor fermions coupled to the vortex in the SU(2) Nielsen-Olesen model, there are normalisable well-behaved zero-energy solutions. When the fermions are in the adjoint representation there are no normalisable zero modes. The Z₂ vortex appearing in a SO(10) theory is explicitly constructed and it is shown that the heavy neutrino can be bound to it in a zero-energy state.     

基于介质环形柱结构的二维光子晶体中Dirac点的实现

利用偶然简并方法在二维正方格子介质环形柱结构光子晶体中成功实现了Dirac点, 并利用平面波展开法对实现Dirac点的过程进行了研究. 研究结果表明, 对于二维正方格子介质环形柱结构光子晶体, 在一定的外径R_O范围内(0.37a<R_O<0.5a), 当Dirac点存在时(n>1.4), 介质环内径R_I与外径R_O满足一个不随介质环折射率n变化的恒定关系式. 同时, Dirac点对应的光子约化频率f随折射率n及外径R_O的增大而减小. 利用所得的关系式对特定介质环折射率n条件下能实现Dirac点的环形光子晶体进行了预判设计.     

Band structures of bilayer graphene superlattices

We formulate a low energy effective Hamiltonian to study superlattices in bilayer graphene (BLG) using a minimal model which supports quadratic band touching points. We show that a one dimensional (1D) periodic modulation of the chemical potential or the electric field perpendicular to the layers leads to the generation of zero-energy anisotropic massless Dirac fermions and finite energy Dirac points with tunable velocities. The electric field superlattice maps onto a coupled chain model comprised of "topological" edge modes. 2D superlattice modulations are shown to lead to gaps on the mini-Brillouin zone boundary but do not, for certain symmetries, gap out the quadratic band touching point. Such potential variations, induced by impurities and rippling in biased BLG, could lead to subgap modes which are argued to be relevant to understanding transport measurements.     

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Pschl-Teller(tPT) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ(κ± 1)r-2.In view of spin and pseudo-spin(p-spin) symmetries,the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method(AIM).We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ.The non-relativistic limit is also obtained.     

Zero energy solutions of the Dirac equation in an N-pseudoparticle field

Zero-energy solutions of the Dirac equation are found for a fermion in the field of a Euclidean Yang-Mills pseudoparticle configuration with Pontryagin number N. In this field we find N non-singular normalizable zero-energy solutions.     

Anisotropic and valley-resolved beam-splitter based on a tilted Dirac system

We investigate theoretically valley-resolved lateral shift of electrons traversing an n–p–n junction bulit on a typical tilted Dirac system (8-Pmmn borophene). A gauge-invariant formula on Goos–Hänchen (GH) shift of transmitted beams is derived, which holds for any anisotropic isoenergy surface. The tilt term brings valley dependence of relative position between the isoenergy surface in n region and that in the p region. Consequently, valley double refraction can occur at the n–p interface. The exiting positions of two valley-polarized beams depend on the incident angle and energy of incident beam and barrier parameters. Their spatial distance D can be enhanced to be ten to a hundred times larger than the barrier width. Due to tilting-induced high anisotropy of the isoenergy surface, D depends strongly on the barrier orientation. It is always zero when the junction is along the tilt direction of Dirac cones. Thus GH effect of transmitted beams in tilted Dirac systems can be utilized to design anisotropic and valley-resolved beam-splitter.     

Dirac Quasinormal Modes of Static f(R) de Sitter Black Holes

Quasinormal modes(QNMs) for Dirac perturbations of f(R) black holes(BHs) are described in this paper,involving two types of f(R) solution: f(R)(Schwarzschild) BHs and f(R)(Maxwell) BHs. With the finite difference method, the stability of the f(R) black holes(BHs) is analysed and the threshold range of f(R)(Schwarzschild) BHs and f(R)(Maxwell) BHs is defined respectively. The results show that due to the presence of the correction factor R0, the damping rate of Dirac field decreases. Meanwhile, the influence of angular quantum number values |k| on the f(R) BHs is investigated. The results indicate that the QNMs oscillation becomes tenser and damping speed slowly decreases with|k| increasing. Furthermore, under the Dirac perturbation, the stability of f(R) solutions can be reflected in the manner of Dirac QNMs. The relationships between the QNMs and the parameters(|k|, charge Q and mass m) are discussed in massless, and massive cases, by contrast to the classical BHs.     

Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Gravitational quasi-normal modes of static R~2 Anti-de Sitter black holes

In this paper, we study the gravitational quasi-normal modes(QNMs) for a static R~2 black hole(BH) in Anti-de Sitter(AdS) spacetime. The corresponding master equation of odd parity is derived and the QNMs are evaluated by the Horowitz and Hubeny method. Meanwhile the stability of such BH is also discussed through the temporal evolution of the perturbation field. Here we mainly consider the coefficient λ, which is related to the radius of AdS black hole, on the QNMs of the R~2 AdS BH. The results show that the Re(ω) and |Im(ω)| of the QNMs increase together as |λ| increases for a given angular momentum number l. That indicates with a larger value of |λ| the corresponding R~2 AdS BH returns to stable much more quickly. The dynamic evolution of the perturbation field is consistent with the results derived by the Horowitz and Hubeny method. Since in the conformal field theory the QNMs can reflect its approach to equilibrium, so our related results could be referential to studies of the AdS/CFT conjecture. The relationship between λ and the properties of the static R~2 BH might be helpful for the development of R~2 gravitational theory.     

Zero-modes and thermodynamics of disordered spin-1/2 ladders

The influence of non-magnetic doping on the thermodynamic properties of two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a weak interchain coupling, the problem can be mapped onto a model of random mass Dirac (Majorana) fermions. We investigate in detail the structure of the fermionic states localized at an individual mass kink (zero-modes) in the framework of a generalized Dirac model. The low-temperature thermodynamic properties are dominated by these zero-modes. We use the single-fermion density of states, known to exhibit the Dyson singularity in the zero-energy limit, to construct the thermodynamics of the spin ladder. In particular, we find that the magnetic susceptibility χ diverges at T → 0 as 1/T ln²(1/T), and the specific heat behaves as C 1/ln³(1/T). The predictions on magnetic susceptibility are consistent with the most recent results of quantum Monte Carlo simulations on doped ladders with randomly distributed impurities. We also calculate the average staggered magnetic susceptibility induced in the system by such defects.     

Charge fractionalization in biased bilayer graphene

Fractional charge may arise when fermionic zero modes exist in a topological background field. In biased bilayer graphene (BBLG), the bias plays the role of the nontrivial background field. When semi-infinite BBLG with a zigzag edge is used, the dynamics induces an odd number of zero-energy modes, which, together with the conjugation symmetry between positive-?and negative-energy states, are the requisite conditions for fractionalization. Exploiting the trigonal interaction to isolate a given zero-energy mode on the zigzag edge, we consider extended and localized modes (the latter being obtained from a localized wavepacket generated by prior irradiation of the sample with an electromagnetic vortex). The valley degeneracy is lifted by a layer asymmetry, while an edge-induced spin polarization breaks the spin degeneracy. We describe scenarios for the detection of charge-[Formula: see text] edge states.     

Zero-energy states bound to a magnetic π-flux vortex in a two-dimensional topological insulator

We show that the existence of a pair of zero-energy modes bound to a vortex carrying a π-flux is a generic feature of the topologically non-trivial phase of the M–B model, which was introduced to describe the topological band insulator in HgTe quantum wells. We explicitly find the form of the zero-energy states of the corresponding Dirac equation, which contains a novel momentum-dependent mass term and describes a generic topological transition in a band insulator. The obtained modes are exponentially localized in the vortex-core, with the dependence of characteristic length on the parameters of the model matching the dependence extracted from a lattice version of the model. We consider in full generality the short-distance regularization of the vector potential of the vortex, and show that a particular choice yields the modes localized and simultaneously regular at the origin. Finally, we also discuss a realization of two-dimensional spin-charge separation through the vortex zero-modes.     

Zero modes for Dirac fermions on a sphere with fractional vortex

Normalized zero-energy states are shown to emerge for massless Dirac fermions in an external gauge field that gives rise to nonquantized vortices on a sphere. A field-theory model is used to describe electronic states of a fullerene-like molecule. In particular, we predict the existence of exactly one zero-energy mode due to a disclination. For 60° disclination, the normalized electron density at zero energy is found to behave as R ^?5/3, with R being the fullerene radius.     

Reversal of Klein reflection by magnetic barriers in bilayer graphene

Massless Dirac fermions in monolayer graphene exhibit total transmission when normally incident on a scalar potential barrier, a consequence of the Klein paradox originally predicted by O Klein for relativistic electrons obeying the 3 + 1 dimensional Dirac equation. For bilayer graphene, charge carriers are massive Dirac fermions and, due to different chiralities, electron and hole states are not coupled to each other. Therefore, the wavefunction of an incident particle decays inside a barrier as for the non-relativistic Schr?dinger equation. This leads to exponentially small transmission upon normal incidence. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission even at normal incidence. The general consequences of this behavior for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.     

(2+1)维光学孤子在饱和非线性介质中的模式特征

从非线性Schrdinger方程出发,应用数学解析的方法,详细讨论了在饱和非线性介质中(2+1)维空间光学孤子存在满足物理意义的自洽解的条件,给出数值计算所需要的边界条件。通过数值计算,给出了基模和一阶模在某一组参数下的部分模式的光场慢变包络、光强度的二维和三维分布的直观图形,以及相应能量(无量纲)。结果表明,孤子的存在不是任意的,而是依赖于一定的能量。当光脉冲能量不足以支持孤子的存在时,其解呈振荡形式,说明不存在孤子解。同时还给出介质的饱和参数、孤子在传播方向上的波矢k对孤子模式的影响等有意义的结论。     

The two-point correlation function of the one-dimensional matrix model

Following Kostov and Ben-Menahem, we calculate the two-puncture correlation function for the one-dimensional matrix model. We find that it depends on the details of discretization for all momenta p. Its only universal features are its vanishing as p → 0 and the appearance of double poles at |p| = n/√′, N = 1,2,…. We show how to derive these double poles in the conformal gauge treatment of Liouville gravity.