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Periodical corrugation in dielectric slab transfers the two waveguide modes at zero Bloch wave number into a leaky resonant mode and a symmetry protected bound states in the continuum (BIC) with small frequency detune. The leaky resonant mode can be directly excited by weak linearly polarized normally incident optical field. In the presence of Kerr nonlinearity, the BIC can be indirectly excited by an optical bistable response. Two types of bistable operations are considered. For the first type, the intensity of the incident field gradually increases to exceed a critical value, and then decreases to zero. For the second type, the intensity is fixed, while the linear polarization angle of the incident field gradually increases to exceed a critical value, and then decreases to 0$^{o}$. Theoretically, the indirectly excited BIC can store the optical energy without loss, even though the intensity of the incident field decreases to zero. Incidence of an optical field with double frequency or orthogonal linear polarization can erase the stored optical field by destroying the BIC. The proposed optical system could function as optical storage and switching device., Comment: 6 figures

Wavy dielectric grating hosts bound states in the continuum (BICs) at nonzero Bloch wave number. For oblique incident optical field with parameters near to the BICs, the reflectance spectrum exhibits ultra-sharp Fano line shape, and the reflected beam has large negative Goos-H$\ddot{a}$nchen shift, due to excitation of the corresponding quasi-BIC with negative group velocity. Under incidence of Gaussian beam with sizable beam width, the excited quasi-BIC could travel a long distance along the direction of the Goos-H$\ddot{a}$nchen shift, designated as $L_{GH}$, before the energy is completely radiated. If the length between the termination of the wavy shape and the focus of the incident Gaussian beam is smaller than $L_{GH}$, sizable energy flux can be coupled into the waveguide mode of the flat dielectric slab that is connected to the wavy dielectric grating. Measurement of the energy flux of the waveguide mode can sense the variation of the refractive index of the background medium. The proposed sensing scheme can be integrated with waveguide in optical circuit., Comment: 7 figures

We theoretically studied the Floquet state of gated bilayer graphene, which is irradiated by normally incident focused Gaussian beam with orbital angular momentum (OAM). According to the Floquet theory, in-plane and out-of-plane electric field of the OAM beam periodically perturbs the intralayer and interlayer hopping, which are equivalent to the presence of effective staggered sublattice potential and next-nearest neighboring interlayer hopping within the light spot region, respectively. The combination of the effective terms and the gated voltage form an effective trapping potential, which hosts localized quantum states with energy level being within the energy gap of the non-irradiated gated bilayer graphene. The energy spectrum can be tuned by the amplitude of the optical beam and additional static magnetic field. By engineering the parameters, valley-polarized two-fold degenerated zero energy Floquet states can be induced., Comment: 4 figures

We have theoretically proposed a highly compact refractive-index sensor consisted of edge-cavity and line-defect waveguide in two-dimensional photonic crystal. The sensing object is completely outside of the single enclosed surface of the sensor. The edge-cavity is designed by engineering the spatial distribution of the cutoff frequency of edge modes. The coupling between the edge-cavity and the waveguide is maximized by optimizing the radius of the rods between them, so that the transmittance spectrum through the waveguide has a sharp anti-peak. As the refractive index of the sensing object changes, the resonant wavelength of the edge-cavity is changed, which in turn changes the wavelength of the anti-peak. The sensitivity of the sensor is up to 40 nm/RIU, and the footprint of the sensor is only 40 $\mu m^{2}$. Because the transmittance spectrum is determined by the overlap between the sensing object and the highly localized resonant mode, the sensor can also perceive spatial distribution of refractive index in the sensing object., Comment: 5 figures

A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic curve. The universal family of Collino cycles naturally gives rise to a normal function, whose induced monodromy relates to the hyperelliptic Johnson homomorphism. Colombo computed this monodromy explicitly and made this relation precise. We recast this in the perspective of relative completion. In particular, we use Colombo's result to construct Collino classes, which are cohomology classes of hyperelliptic mapping class groups with coefficients in a certain symplectic representation. We also determine the dimension of their span in the case of the level two hyperelliptic mapping class group.

In this study, we theoretically and numerically investigate the resonant modes and reflectance of an optical grating consisting of a wavy dielectric slab by applying the spectral element method. The presence of the wavy shape transforms the waveguide modes into leaky resonant modes. A few resonant modes with specific longitudinal wave number have infinitely large Q factor, while the other resonant modes have finite Q factor. For the leaky resonant mode with zero longitudinal wave number, the Q factor is inversely proportional to the amplitude of the wavy shape. An array of multiple low-Q wavy gratings has a high reflectance in a large bandwidth. A double-layer wavy grating forms a Fabry-Perot cavity, which hosts Fabry-Perot bound states in the continuum (BICs) at the resonant frequency. The Q-factor of the Fabry-Perot cavity can be tuned by adjusting the distance between the two wavy slabs. The wavy shape could be generated by a vibrational wave in a flat dielectric slab so that the BICs mode and wideband reflectance could be controlled on-demand., Comment: 7 figures

Twistronic has recently attracted tremendous attention because the twisting can engineer the bilayer graphene-like materials into varying types of strongly correlated phases. In this paper, we study the twisting of bilayer graphene (BLG) quantum dots (QDs) with hexagonal shape and zigzag edges. In the untwisted BLG-QDs, the zigzag edges of graphene host spontaneous magnetism with varying magnetic configurations. As a BLG-QD being adiabatically twisted, the quantum state evolves as a function of the twisting angle. If the twisting angle changes across certain critical value, the magnetic configuration of the quantum state sharply changes. For the twisting process with increasing or decreasing twisting angle, the number and value of the critical twisting angles are different. Thus, the twisting process with the twisting angle increasing and decreasing back and forth could enter a hysteresis loop. The twisting of BLG QDs with adatom is also investigated. The tuning features of the magnetic configuration of the twisted BLG-QDs could be applied for graphene-based quantum memory devices., Comment: 5 figures

Second order topological insulator can be engineered from two-dimensional materials with strong spin-orbit coupling and in-plane Zeeman field. In proximity to superconductor, topological superconducting phase could be induced in the two-dimensional materials, which host Majorana corner modes at the intersection between two zigzag edges. Two types of tight binding models in hexagonal lattice, which include $p_{z}$ or $p_{x,y}$ orbit(s) in each lattice site, are applied to engineer two-dimensional materials in topological superconducting phase. In both models, the condition that induces the second order topological superconductor requires nonuniform value of either in-plane Zeeman fields or superconductor pairing parameters in two sublattices. The finite size effect of the second model is weaker than that of the first model., Comment: 5 figures

Combination of proximity-induced superconductivity and ferromagnetic exchange field in a two-dimensional square-lattice antiferromagnet with spin-orbit coupling and nonsymmorphic symmetry can induce a topological superconductor phase with chiral Majorana edge states. The lattice model of the Bogoliubov-de Gennes (BdG) Hamiltonian was applied to study the phase diagram of bulks and chiral Majorana edge states in nanoribbons. By numerically studying the phase diagram, we found that the non-uniformity of either the superconducting pairing parameters or the exchange field at the two sublattices is necessary to induce a topological superconductor phase with chiral Majorana edge states. The BdG Chern number of certain topological superconductor phases is $\pm1$ or $\pm3$, such that the corresponding nanoribbons have one or three pairs of chiral Majorana edge states, respectively., Comment: 4 figures

In this paper, we theoretically study the Floquet engineering of two dimensional photonic waveguide arrays in three types of lattices: honeycomb lattice with Kekule distortion, breathing square lattice and breathing Kagome lattice. The Kekule distortion factor or the breathing factor in the corresponding lattice is periodically changed along the axial direction of the photonic waveguide with frequency $\omega$. Within certain ranges of $\omega$, the Floquet corner states in the Floquet band gap of quasi-energy spectrum are found, which are localized at the corner of the finite two-dimensional arrays. Due to particle-hole symmetric in the model of honeycomb and square lattice, the quasi-energy level of the Floquet $\pi$ corner states is $\pm\omega/2$. On the other hand, Kagome lattice does not have particle-hole symmetric, so that the quasi-energy level of the Floquet $\pm2\pi/3$ corner states is near to $\pm1\omega/3$. The corner states are either protected by crystalline symmetry or reflection symmetry. The finding of Floquet fractional-$\pi$ corner states could provide more options for engineering of on-chip photonic devices., Comment: 7 figures

Zigzag nanoribbons of monolayer graphene-like two-dimensional materials host spontaneous edge magnetism at the zigzag terminations, whose configuration controls the band gap. In this article, the edge magnetism of zigzag nanoribbons of bilayer hexagonal crystals are studied. The specific models of bilayer graphene and bilayer silicene are both studied. As the gated voltage increases, the total energy level, magnetic structure and band structures of the ground state and the first quasi-stable excited state are tuned. For certain region of the gated voltage, the band gaps of spin up and down are opened and closed, respectively, so that the systems are in the spin-polarized metallic phase. The spin-polarized conducting edge band of the bilayer nanoribbons could be applied for gate tunable spintronic devices., Comment: 7 figures

We theoretically investigated the Floquet states of bilayer graphene heterostructure under the irradiation by horizontally incident circular polarized light. The in-plane and out-of-plane electric field of the light periodically perturbs the intra-layer and inter-layer hopping, respectively. For circular polarized light, the two components of the electric field has $\pi/2$ phase difference, so that the two types of hopping are periodically perturbed with the $\pi/2$ phase difference, which modify the effective inter-layer hopping. We focus on the model of bilayer graphene in the heterostructure of antiferromagnetic van der Walls spin valve. The amplitude of the irradiation can tune the band gap and topological properties of the bulk state. The spin-polarized quantum anomalous Hall phase with Chern number being one is predicted. The incident angle of the irradiation can tune the band gap and dispersion of the edge states in zigzag nanoribbons., Comment: 6 figures

The topological phases of graphene with spin-orbit coupling, an exchange field, and a staggered-sublattice potential determine the properties of the edge states of the zigzag nanoribbon. In the presence of the Hubbard interaction, the spontaneous magnetization at the zigzag terminations induces sizable magnetic moments at the lattice sites in the bulk region. Thus, the exchange field and staggered-sublattice potential in the bulk region are effectively changed, which in turn change the topological phase. Within a certain parameter regime, quasi-stable excited states of the zigzag nanoribbon exist, which have a different magnetism configuration at the zigzag terminations from the ground state. The quasi-stable excited states could effectively suppress the finite size effect of the topological edge states. The investigation of the topological edge states in the presence of interaction helps the engineering of spintronic nanodevices based on realistic materials., Comment: 8 figures

The two dimensional square lattice antiferromagnet with spin-orbit coupling and nonsymmorphic symmetry is recently found to be topological insulator (TI). We theoretically studied the Floquet states of the antiferromagnetic crystal with optical irradiation, which could be applicable in opto-spintronic. An optical irradiation with circular polarization induces topological phase transition into quantum Anomalous Hall (QAH) phase with varying Chern number. At the phase boundaries, the Floquet systems could be semimetal with one, two or three band valleys. A linear polarized optical field induces effective antiferromagnetic exchange field, which change the phase regime of the TI. At the intersection of two phase boundaries, the bulk band structure is nearly flat along one of the high symmetry line in the first Brillouin zone, which result in large density of states near to the Fermi energy in bulk and nanoribbons., Comment: 7 figures

In the presence of the Hubbard interaction, graphene zigzag nanoribbons have spontaneous edge magnetism with anti-parallel configuration, whose amplitude can be tuned by a transversal electric field. As the electric field increases or decreases across a critical value, the edges are demagnetized or re-magnetized, respectively. A magnetic field at each edge determines the orientation of the re-magnetization. Thus, a combination of slowly varying transversal electric field and magnetic field in monolayer graphene zigzag nanoribbon could drive the quantum system into a bistability loop. The same phenomenon can be induced in a bilayer/monolayer zigzag nanoribbon without the magnetic field, because the non-symmetry superexchange interaction controls the orientation of the re-magnetization. By this way, the quantum system is switched between ground state and quasi-stable excited state with different magnetism, band structures and conductance. This feature could be used to develop graphene-based spintronic nano-devices without magnetic field., Comment: 4 figures

The non-Hermitian Su-Schrieffer-Heeger spinless Fermion model with interacting terms is studied by exact diagonalization. The model is derived from the spin chain with damping Dzyaloshinskii-Moriya interaction. The presence of the interaction induce anomalous spectral function. The skin effect is exhibited by the non-diagonal terms of the Green's function, i.e. the spatial correlation. The interaction modifies the degeneracy between the ground state and the first excited state of the half filled systems, as well as between the ground state of the half and half plus one filled systems. The energy bands of the higher excited states within the topological regime have complicated pattern, depending on the sign and amplitude of the interaction., Comment: 6 figures

Silicene have been proposed to host multiple topological phases depending on the environmental parameters such as gated voltage and proximity induced exchange field. Two typical topological phase are quantum spin Hall phase and quantum anomalous Hall phase. We found that in the presence of antiferromagnetic exchange field and (or) gated voltage, the bilayer silicene host a topological phase that the Chern numbers of the two spin are different in both sign and magnitude, .i.e the anomalous Hall effects of the two spins are imbalanced. The topological phase is generated by triple band inversion that changes the Chern number of the upper and lower bands by $\pm3$. The phase host large anomalous spin Hall respondence. Meanwhile, because of the large trigonal warping in bilayer silicene near to the triple band inversion regime, the Magnus Hall effect is large. This finding enriches the topological systems of bilayer silicene as candidate for spintronic devices.

The Kekule pattern of the (anti)ferromagnetic exchange field on monolayer MoS2 can be induced by proximity to the (111) surface of BiFeO3 on both sides. The three-band tight binding model of the MoS2 layer with Kekule patterned exchange field is applied to describe the heterostructures. The tight binding model is justified by the first principle calculations. The magnetization orientations of the substrates control the pattern of the exchange field, which then switches the band structures of the lowest zigzag edge states between being metallic and insulating. The lowest four zigzag edge bands provide conducting channels with a spin-polarized current. Optical excitation of carriers in these bands generates sizable spin and charge currents, which are theoretically modeled by the perturbation solution of the semiconductor Bloch equation. The injected spin currents have multiple resonant peaks at a few frequencies, which can be switched off by rotating the magnetization orientations of the substrates., Comment: 5 figures

Twisted bilayer graphenes with magical angle exhibit strongly correlated electronic properties because of the isolated flat band at the Fermi level. We studied the twisted bilayer graphene with substrates on both layers. The substrate induce Kekul$\acute{e}$ distortion for each graphene layer. The systems are investigated by both continuous Dirac Fermion model and tight binding model. The two investigations give similar conclusion that isolated flat band with ultra-narrow bandwidth could appear., Comment: 5 figures

Twist bilayer graphenes with magical angle have nearly flat band, which become strongly correlated electron systems. Herein, we propose another system based on strained bilayer graphene that have flat band at the intrinsic Fermi level. The top and bottom layers are uniaxially stretched along different directions. When the strength and directions of the strain satisfy certain condition, the periodical lattices of the two layers are commensurate to each other. The regions with AA, AB and BA stacking arrange in a triangular lattice. With magical strain, the bands around the intrinsic Fermi level are nearly flat and have large gap from the other bands. This system could provide more feasible platform for graphene-based integrated electronic system with superconductivity., Comment: 4 figures

Networks of graphene-based topological domain walls function as nano-scale interferometers of zero-line modes, with magnetic field and(or) scalar potential as the controlling parameters. In the absence of externally applied magnetic or electrical field, strain induces pseudomagnetic field and scalar potential in graphene, which could control the interferometers more efficiently. Two types of strains are considered: (i) Horizontally bending the graphene nanoribbon into circular arc induces nearly uniform pseudomagnetic field; (ii) Helicoidal graphene nanoribbon exhibit nonuniform pseudomagnetic field. Both types of strain induce small scalar potential due to dilatation. The interferometers are studied by transport calculation of the tight binding model. The transmission rates through the interferometer depend on the strain parameters. An interferometer with three loops is designed, which could completely switch the transmitting current from one export to the other., Comment: 5 figures

Spin valves consisting of heterostructures of single-layer hexagonal crystal on an antiferromagnetic substrate or of bilayer hexagonal crystal intercalated between two (anti)ferromagnetic insulators, with the current-in-plane geometry, are proposed. The two-dimensional hexagonal crystals such as graphene, silicene, germanene, and stanene are modeled by the tight binding model of honeycomb lattice. The magnetization orientation of the antiferromagnetic substrate(s) controls the band gap and topological properties of bulk, which in turn control the transport of three types of spin valve geometries: (i) the in-plane transport of bulk; (ii) the transport of topological edge states along nanoribbon with bulk gap; (iii) the transport of chiral edge state along domain wall. The heterostructures are investigated by a tight binding model with an (anti)ferromagnetic exchange field, Hubbard interaction and(or) spin-orbital coupling. For the first type of spin valve geometry, the Hubbard interaction could enlarged the effective band gap of bulk, which in turn improve the sensitivity of the spin valves to the antiferromagnetic exchange field. For the second and third types of spin valve geometries, the topological phase diagrams of varying types of heterostructures with spin-orbital coupling serve as guideline for designing the spin valve. The coexistence of the Hubbard interaction and the spin-orbital coupling could enlarge the topological gap in bulk and improve the quality of the chiral edge states at the domain walls between regions with different topological numbers., Comment: 13 figures

Two-dimensional Floquet systems consisting of irradiated valley-polarized metal are investigated. For the corresponding static systems, we consider two graphene models of valley-polarized metal with either a staggered sublattice or uniform intrinsic spin-orbital coupling, whose Dirac point energies are different from the intrinsic Fermi level. If the frequency of irradiation is appropriately designed, the largest dynamical gap (first-order dynamical gap) opens around the intrinsic Fermi level. In the presence of the irradiation, two types of edge state appear at the zigzag edge of semi-infinite sheet with energy within the first-order dynamical gap: the Floquet edge states and the strongly localized edge states. In narrow zigzag nanoribbons, the Floquet edge states are gapped out by the finite size effect, and the strongly localized edge states remain gapless. As a result, the conducting channels of the nanoribbons consist of the strongly localized edge states. Under the first and second model, the strongly localized edge states carry one-way spin polarized and one-way charge current around the intrinsic Fermi level, respectively. Thus, the narrow zigzag nanoribbons of the first and second model have asymmetric spin and charge transmission rates, respectively. Quantum-transport calculations predict sizable pumped currents of charge and spin, which could be controlled by the Fermi level., Comment: 7 figures

Topologically protected zero-line modes appear at the interface between two regions of the monolayer graphene in quantum anomalous Hall phase with different Chern number. In the presence of staggered sublattice potential, the band gaps of the two valleys become different, and the phase diagram defined by the Chern number has an additional regime of topologically trivial phase. The interface between the topologically trivial and non-trivial regions hosts zero-line mode in only one valley. By tuning the exchange field, three types of interface that host zero-line modes in selected valley(s) are formed. The nano-devices consisted of Y-shape junctions of the three types of interface exhibit the functions of valley splitting, merging or filtering for the incident currents., Comment: 6 figures

Domain wall in bilayer graphene with Rashba spin-orbital coupling and staggered sublattice potentials, at the interface between two domains with different gated voltages, is studied. Varying type of zero-line modes are identified, including zero-line mode with pure spin filtering effect. The Y-shape current partition at the junction among three different domains are proposed.

Gated heterostructures containing bilayer graphene with staggered sublattice potentials are investigated by tight binding model with Rashba spin-orbital coupling and Hubbard interaction. The topological phase diagrams depend on the combinations of substrates and the Hubbard interaction. The presence of the staggered sublattice potential favor the topological phase transition with small Rashba spin-orbital coupling strength. The presence of the Hubbard interaction modified the topological phase boundaries, increasing the minimal spin-orbital coupling strength for topological phase transition. A phase space of topological semi-metal with indirect band gap is identified in the non-interacting systems. For the bilayer graphene with different staggered sublattice potentials in the two layers, the conditions for the zigzag nanoribbons to host edge polarized chiral edge states are discussed. The conditions require moderate or vanishing Rashba spin-orbital coupling strength, as well as proper range of the gate voltage. The conditions for the systems with and without the Hubbard interaction are compared. The edge polarization can be controlled by the gate voltage., Comment: 6 figures

We develop an explicit algebriac de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$. This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of holomorphic modular forms that were previously studied by Brown and Manin. These newly constructed iterated integrals provide all 'multiple modular values' defined by Brown., Comment: Revised mostly the last 3 sections, updated introduction and references