In crystalline systems, charge polarization is related to Zak's phase determined by bulk band topology. Nontrivial charge polarization induces robust edge states accompanied with fractional charge. In Su-Schrieffer-Heeger (SSH) model, it is known that the strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. Here, we consider a bi-atomic honeycomb lattice to introduce such strong modulation, i.e. A$_3$B sheet. By tuning hopping ratio and onsite potential difference between A and B atoms, we show that topological phase transition characterized by Zak's phase occurs. Furthermore, we propose that C$_3$N and BC$_3$ are the possible realistic materials on the basis of first-principles calculations. Both of them display topological edge states induced by Zak's phase without spin-orbital couplings and external fields unlike conventional topological insulators.
We consider the end states of a half-filled rectangular armchair graphene ribbon (RAGR) in a staggered potential. Taking electron-electron interactions into account we find that, as the strength of the staggered potential varies, three types of couplings between the end states can occur: antiferromagnetic without or with spin splitting, and paramagnetic without spin-splitting. We find that a spin-splitting is present only in the staggered potential region $0<Δ<Δ_c$. The transition from the antiferromagnetic state at $Δ=0$ to the paramagnetic state goes through an intermediate spin-split antife…
We show that monolayer graphene intrinsically hosts higher-order topological corner states, in which electrons are localized topologically at atomic sizes. The emergence of the topological corner states in graphene is due to a nontrivial product of the Zak phases for two independent directions, which can be handily calculated graphically by using the bulk wavefunctions. We give an explicit expression that indicates the existence of topological corner states for various geometric edges and corner angles. We also demonstrate the nontrivial localization nature of the topological corner states in …
We show the emergence of topological Bogoliubov bosonic excitations in the relatively strong coupling limit of an LC (inductance-capacitance) one-dimensional quantum circuit. This dimerized chain model reveals a ${\cal Z}_2$ local symmetry as a result of the counter-rotating wave (pairing) terms. The topology is protected by the sub-lattice symmetry, represented by an anti-unitary transformation. We present a methos to measure the winding of the topological Zak phase across the Brillouin zone by a reflection measurement of (microwave) light. Our method probes bulk quantities and can be impleme…
Zak phase, i.e. the Berry phase acquired during an adiabatic motion of a Bloch particle across the Brillouin zone, provides a measure of the topological invariant of Bloch bands in one-dimensional crystalline potentials. Here a photonic structure, based on engineered lattices of evanescently-coupled optical waveguides, is proposed to detect Zak phase difference of photons undergoing Bloch oscillations in topologically distinct Bloch bands of dimerized superlattices.
In this paper, we discuss the characteristic features of one-dimensional topological insulators with inversion symmetry but noncentered inversion axis in the unit cell, for any choice of the unit cell. In these systems, the global inversion operation generates a k-dependent inversion operator within the unit cell and this implies a nonquantized Zak's phase both for the trivial and nontrivial topological phases. By relating the Zak's phase with the eigenvalues of modified parity operators at the inversion invariant momenta, a corrected quantized form of the Zak's phase is derived. We show that …
Geometric phases that characterize the topological properties of Bloch bands play a fundamental role in the modern band theory of solids. Here we report on the direct measurement of the geometric phase acquired by cold atoms moving in one-dimensional optical lattices. Using a combination of Bloch oscillations and Ramsey interferometry, we extract the Zak phase - the Berry phase acquired during an adiabatic motion of a particle across the Brillouin zone - which can be viewed as an invariant characterizing the topological properties of the band. For a dimerized optical lattice, which models poly…
A mechanical diamond, a classical mechanics of a spring-mass model arrayed on a diamond lattice, is discussed topologically. Its frequency dispersion possesses an intrinsic nodal structure in the three-dimensional Brillouin zone (BZ) protected by the chiral symmetry. Topological changes of the line nodes are demonstrated associated with modification of the tension. The line nodes projected into two-dimensional BZ form loops which are characterized by the quantized Zak phases by 0 and $π$. With boundaries, edge states are discussed in relation to the Zak phases and winding numbers. It establish…
Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here, we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent time…
One-dimensional lattices with chiral symmetry are known to possess quantized Zak phase and nontrivial topological phases. Here it is shown that quantized Zak phase and nontrivial edge states, partially protected by inversion symmetry rather than chiral symmetry, can be observed and probed in the bulk exploiting continuous-time photonic quantum walk in zig-zag waveguide arrays. Averaged beam displacement measurements can detect quantized Zak phase and non-trivial topological phases in the extended Su-Schrieffer-Heeger model with broken chiral symmetry.
Zak phase, which refers to the Berry's phase picked up by a particle moving across the Brillouin zone, characterizes the topological properties of Bloch bands in one-dimensional periodic system. Here the Zak phase in dimerized one-dimensional locally resonant metamaterials is investigated. It is found that there are some singular points in the bulk band across which the Bloch states contribute π to the Zak phase, whereas while in the rest of the band the contribution is nearly zero. These singular points associated with zero reflection are caused by two different mechanisms: the dimerization-i…
Recent experimental studies have found that magnetic impurities deposited on superconducting monolayer NbSe$_2$ generate coupled Yu-Shiba-Rusinov bound states. Here we consider ferromagnetic chains of impurities which induce a Yu-Shiba-Rusinov band and harbor Majorana bound states at the chain edges. We show that these topological phases are stabilized by strong Ising spin-orbit coupling in the monolayer and examine the conditions under which Majorana phases appear as a function of distance between impurities, impurity spin projection, orientation of chains on the surface of the monolayer, and…
Different from the two-dimensional (2D) topological insulator, the 2D topological crystalline insulator (TCI) phase disappears when the mirror symmetry is broken, e.g., upon placing on a substrate. Here, based on a new family of 2D TCIs - SnTe and PbTe monolayers - we theoretically predict the realization of the quantum anomalous Hall effect with Chern number C = 2 even when the mirror symmetry is broken. Remarkably, we also demonstrate that the considered materials retain their large-gap topological properties in quantum well structures obtained by sandwiching the monolayers between NaCl laye…
Employing ab initio electronic calculations, we propose a new type of two-dimensional (2D) topological insulator (TI), monolayer (ML) low buckled (LB) mercury telluride (HgTe) and mercury selenide (HgSe), with tunable band gaps. Monolayer LB HgTe undergoes a transition to a topological nontrivial phase under the appropriate in-plane tensile strain (ε > 2.6%) due to the combination effects of strain and spin orbital coupling (SOC). Under the 2.6%< ε <4.2% tensile strain, the band inversion and topological nontrivial gap are induced by the SOC. For ε >4.2%, the band inversion is already realized…
Nonlinearity induced topological properties in nonlinear lattice systems are studied in both momentum space and real space. Experimentally realizable through the Kerr effect on photonic waveguide systems, our working model depicts on-site nonlinearity added to the Su-Schrieffer-Heeger (SSH) model plus a chiral-symmetry breaking term. Under the periodic-boundary condition, two of the nonlinear energy bands approach the energy bands of the chiral-symmetric SSH model as nonlinearity strength increases. Further, we account for a correction to the Zak phase and obtain a general expression for nonli…
We systematically examine the topological, valleytronic, and optical properties of experimentally accessible PbS (001) few-layers, with a focus on the monolayer. With even-odd layer-dependent oscillations and without band inversions, the band gaps cover a wide spectrum from infrared to visible, making the few-layers promising for optoelectronics. Intriguingly, the uniaxial (biaxial) compressive strain can tune the trivial monolayer into a $\mathcal{Z}_2$-topological (topological crystalline) insulator, which may be utilized for controllable low-power electronic devices. Although elliptical dic…
Nonlinear Hall quantum oscillations to probe topological Brown-Zak fermions in graphene moiré systems
Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology（北京理工大学物理学院先进光电子量子结构与测量教育部重点实验室）；School of Physics and Electronics, Hunan University（湖南大学物理与电子信息学院）；School of Physical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院大学物理科学学院及中国科学院拓扑量子计算卓越创新中心）；International Center for Quantum Materials, Beijing Institute of Technology, Zhuhai（北京理工大学珠海国际量子材料中心）
We theoretically investigate the chiral topological excitons emerging in the monolayer transition metal dichalcogenides, where a bulk energy gap of valley excitons is opened up by a position dependent external magnetic field. We find two emerging chiral topological nontrivial excitons states, which exactly connects to the bulk topological properties, i.e., Chern number =2. The dependences of the spectrum of the chiral topological excitons on the width of the magnetic field domain wall as well as the magnetic filed strength are numerically revealed. The chiral topological valley excitons are no…
Monolayer molybdenum disulfide (MoS2) has a honeycomb crystal structure. We consider the triangular sublattice of molybdenum atoms and introduce a simple tight-binding Hamiltonian for studying the phase transition and topological superconductivity in MoS2 under uniaxial strain. It is shown that spin-singlet p+ip wave phase is a topological superconducting phase which possesses nonzero Chern numbers. When the chemical potential is bigger (smaller) than the spin-orbit coupling (SOC) strength the Chern number is equal to four (two) and otherwise it is equal to zero. Also, it is shown that, when t…
The topology in different dimensions has attracted enormous interests, e.g. the Zak phase in 1D systems, the Chern number in 2D systems and the Weyl points or nodal lines in the systems with higher dimensions. It would be fantastic to find the connection of different topology in different dimensions from one simple model and reveal the deep physical picture behind them. In this work, we propose a new model which starts from a binary-layered 1D photonic crystal, and by introducing synthetic dimensions the topology of higher dimension systems could appear. From this model, we find that the topol…
We study the electronic structure and topological properties of monolayer and ABC-stacked multilayer of graphyne-$N$, which are a family of planar carbon sheets consisting of $sp$ and $sp_2$-bonding. By using the density-functional theory and the effective continuum model, we find a striking even-odd effect in the dependence of the band structure on $N$ (the number of carbon-carbon triple bonds between neighboring benzene rings). Specifically, even-$N$ graphyne monolayer has doubly-degenerate conduction and valence bands near the Fermi energy, and in its ABC multilayer, the band inversion of t…
We present a two-dimensional (2D) lattice model that exhibits a nontrivial topological phase in the absence of the Berry curvature. Instead, the Berry connection provides the topological nontrivial phase in the model, whose integration over the momentum space, the so-called 2D Zak phase, yields a fractional wave polarization in each direction. These fractional wave polarizations manifest themselves as degenerated edge states with opposite parities in the model.
We study the unconventional topological phases of polaritons inside a cavity waveguide, demonstrating how strong light-matter coupling leads to a breakdown of the bulk-edge correspondence. Namely, we observe an ostensibly topologically nontrivial phase, which unexpectedly does not exhibit edge states. Our findings are in direct contrast to topological tight-binding models with electrons, such as the celebrated Su-Schrieffer-Heeger (SSH) model. We present a theory of collective polaritonic excitations in a dimerized chain of oscillating dipoles embedded inside a photonic cavity. The added degre…
We propose a physical model based on disordered (a hole punched inside a material) monolayer transition metal dichalcogenides (TMDs) to demonstrate a large-gap quantum valley Hall insulator. We find an emergence of bound states lying inside the bulk gap of the TMDs. They are strongly affected by spin-valley coupling, rest- and kinetic- mass terms and the hole size. In addition, in the whole range of the hole size, at least two in-gap bound states with opposite angular momentum, circulating around the edge of the hole, exist. Their topological insulator (TI) feature is analyzed by the Chern num…
Magnons dominate the magnetic response of the recently discovered insulating ferromagnetic two dimensional crystals such as CrI$_3$. Because of the arrangement of the Cr spins in a honeycomb lattice, magnons in CrI$_3$ bear a strong resemblance with electronic quasiparticles in graphene. Neutron scattering experiments carried out in bulk CrI$_3$ show the existence of a gap at the Dirac points, that has been conjectured to have a topological nature. Here we propose a theory for magnons in ferromagnetic CrI$_3$ monolayers based on an itinerant fermion picture, with a Hamiltonian derived from fir…
Monolayer transition metal dichalcogenides in the T' phase promise to realize the quantum spin Hall (QSH) effect at room temperature, because they exhibit a prominent spin-orbit gap between inverted bands in the bulk. Here we show that the binding energy of electron-hole pairs excited through this gap is larger than the gap itself in MoS2, a paradigmatic material that we investigate from first principles by many-body perturbation theory (MBPT). This paradoxical result hints at the instability of the T' phase against the spontaneous generation of excitons, and indeed we find that it gives rise …
To magnetize surfaces of topological insulators without damaging their topological feature is a crucial step for the realization of the quantum anomalous Hall effect (QAHE), and still remains as a challenging task. Through density functional calculations, we found that adsorption of a two-dimensional van der Waals (2D-vdW) ferromagnetic insulator CrI3 monolayer can create a sizable spin splitting at the Dirac point of the topological surface states of Bi2Se3 films. Furthermore, general rules that connect different quantum and topological parameters are established through model analyses. This …
The weak interlayer Van Der Waals material \{beta}-Bi2Pd has recently been established as a strong topological superconductor candidate with unconventional spin-triplet pairing and Majorana zero modes at vortices. In this letter, we study the topological characters and the superconducting pairing, which are still obscure in monolayer \{beta}-Bi2Pd, in light of our effective theoretical model. We find that the non-Rashba spin-orbital coupling plays a critical role in realizing and tuning various novel topological natures. In particular, the spin-triplet p-wave superconducting pairing with Major…
We investigate topological phases of monolayer jacutingaite (Pt$_2$HgSe$_3$) that arise when considering the competing effects of spin-orbit coupling (SOC), magnetic exchange interactions, and staggered sublattice potential V. The interplay between the staggered potential and exchange field offers the possibility of attaining different topological phases. By analyzing the Berry curvatures and computing the Chern numbers and Hall conductivities, we demonstrate that the system is time-reversal-symmetry-broken quantum spin Hall insulator when $m_b<λ_{so}$, where $m_b$ is the exchange field operat…
We explore the electronic structure, orbital character and topological aspect of a monolayer MoS$_2$ nanoribbon using tight-binding (TB) and low-energy (${\bm k}\cdot{\bm p} $) models. We obtain a mid-gap edge mode in the zigzag ribbon of monolayer MoS$_2$, which can be traced back to the topological properties of the bulk band structure. Monolayer MoS$_2$ can be considered as a valley Hall insulator. The boundary conditions at armchair edges mix the valleys on the edges, and a gap is induced in the edge modes. The spin-orbit coupling in the valence band reduces the hybridization of the bulk s…
Recent experimental evidence for the quantum spin Hall (QSH) state in monolayer WTe$_2$ has bridged two of the most active fields of condensed matter physics, 2D materials and topological physics. This 2D topological crystal also displays unconventional spin-torque and gate-tunable superconductivity. While the realization of QSH has demonstrated the nontrivial topology of the electron wavefunctions of monolayer WTe$_2$, the geometrical properties of the wavefunction, such as the Berry curvature, remain unstudied. On the other hand, it has been increasingly recognized that the Berry curvature p…
Generally, the topological corner state in two-dimensional second-order topological insulator (2D SOTI) is equivalent to the well-known domain wall state, originated from the mass-inversion between two adjacent edges with phase shift of pi. In this work, go beyond this conventional physical picture, we report a fractional mass-kink induced 2D SOTI in monolayer FeSe with canted checkerboard antiferromagnetic (AFM) order by analytic model and first-principles calculations. The canted spin associated in-plane Zeeman field can gap out the quantum spin Hall edge state of FeSe, forming a fractional …
We propose a realizable device design for an all-electrical robust valley filter that utilizes spin protected topological interface states hosted on monolayer 2D-Xene materials with large intrinsic spin-orbit coupling. In contrast with conventional quantum spin-Hall edge states localized around the $X$-points, the interface states appearing at the domain wall between topologically distinct phases are either from the $K$ or $K^{'}$ points, making them suitable prospects for serving as valley-polarized channels. We show that the presence of a large band-gap quantum spin Hall effect enables the s…
In various topological phases, nontrivial states appear at the boundaries of the system. In this paper, we investigate anomalous dielectric response caused by such states caused by the pi Zak phase. First, by using the one-dimensional Su-Schrieffer-Heeger model, we show that, when the system is insulating and the Zak phase is pi, the polarization suddenly rises to a large value close to e/2, by application of an external electric field. The pi Zak phase indicates existence of half-filled edge states, and we attribute this phenomenon to charge transfer between the edge states at the two ends of…
The monolayer FeSe with a thickness of one unit cell grown on a single-crystal SrTiO$_{3}$ substrate (FeSe/STO) exhibits striking high-temperature superconductivity with transition temperature $T_{c}$ over 65K reported by recent experimental measurements. In this work, through analyzing the distinctive electronic structure, and providing systematic classification of the pairing symmetry , we find that both $s$-and $p$-wave pairing with odd parity give rise to topological superconducting states in monolayer FeSe, and the exotic properties of $s$-wave topological superconducting states have clos…
The layered semimetal WTe_2 has recently been found to be a two-dimensional topological insulator (2D TI) when thinned down to a single monolayer, with conducting helical edge channels. We report here that intrinsic superconductivity can be induced in this monolayer 2D TI by mild electrostatic doping, at temperatures below 1 K. The 2D TI-superconductor transition can be easily driven by applying a just a small gate voltage. This discovery offers new possibilities for gate-controlled devices combining superconductivity and topology, and could provide a basis for quantum information schemes base…
Three decades ago, in a celebrated work, Zak found an expression for the geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. Unfortunately Zak's expression suffers from two flaws: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Here we explain that these flaws in Zak's expression arise from the assumption that the electron's adiabatic motion is cyclic in the sense of re…
We evidence the possibility for coherent electrical manipulation of the spin orientation of topologically protected edge states in a low-symmetry quantum spin Hall insulator. By using a combination of ab-initio simulations, symmetry-based modeling, and large-scale calculations of the spin Hall conductivity, it is shown that small electric fields can efficiently vary the spin textures of edge currents in monolayer 1T'-WTe2 by up to a 90-degree spin rotation, without jeopardizing their topological character. These findings suggest a new kind of gate-controllable spin-based device, topologically …
The topological states of matters arising from the nontrivial magnetic configuration provide a better understanding of physical properties and functionalities of solid materials. Such studies benefit from the active control of spin orientation in any solid, which is yet known to rarely take place in the two-dimensional (2D) limit. Here we demonstrate by the first-principles calculations that spin-orientation dependent topological states can appear in the geometrically frustrated monolayer antiferromagnet. Different topological states including quantum anomalous Hall (QAH) effect and time-rever…
We report the recent progress on the theoretical aspects of monolayer topological insulators including silicene, germanene and stanene, which are monolayer honeycomb structures of silicon, germanium and tin, respectively. They show quantum spin Hall effects in nature due to the spin orbit interaction. The band gap can be tuned by applying perpendicular electric field, which induces a topological phase transition. We also analyze the topological properties of generic honeycomb systems together with the classification of topological insulators. Phase diagram of topological insulators and superco…
We propose a platform to realize nodal topological superconductors in a superconducting monolayer of MoX$_2$ (X$=$S, Se, Te) using an in-plane magnetic field. The bulk nodal points appear where the spin splitting due to spin-orbit coupling vanishes near the $\pm \boldsymbol{K}$ valleys of the Brillouin zone, and are six or twelve per valley in total. In the nodal topological superconducting phase, the nodal points are connected by flat bands of zero-energy Andreev edge states. These flat bands, which are protected by chiral symmetry, are present for all lattice-termination boundaries except zi…
Motivated by the experiments on double monolayer graphene that observe a variety of fractional quantum Hall states [Liu et al., Nat. Phys. 15, 893 (2019); Li et al., Nat. Phys. 15, 898 (2019)], we study the special setting in which two monolayers have different areas. It has not been considered before and allows us to construct a class of exotic topological states. The elementary excitations of these states do not carry fractional charges but obey fractional statistics. This is in sharp contrast to all previously studied cases, where the two properties are intimately connected and serve as hal…
We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z(k), which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum k at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k for which edge states exist in a graphene ribbon is then…
We study topological features of interacting spin- 1 2 particles in one-dimensional state-dependent optical lattices. Due to the co-translational symmetry, we introduce the center-of-mass Zak phase with the help of center-of-mass momentum. There appear topological bound states composed by two particles in different spin states via tuning hopping and interaction strengths. Under symmetric open boundary conditions, topological edge bound-states appear as a result of the non-trivial center-ofmass Zak phase of bound-state band, which is protected by the center-of-mass inversion symmetry. The inter…
Topological materials are fertile ground for investigating topological phases of matter and topological phase transitions. In particular, the quest for novel topological phases in 2D materials is attracting fast growing attention. Here, using Floquet-Bloch theory, we propose to realize chiral topological phases in 2D hexagonal FeX2 (X=Cl, Br, I) monolayers under irradiation of circularly polarized light. Such 2D FeX2 monolayers are predicted to be dynamical stable, and exhibit both ferromagnetic and semiconducting properties. To capture the full topological physics of the magnetic semiconducto…
By means of the first-principles calculations and magnetic topological quantum chemistry, we demonstrate that the low energy physics in the checkerboard antiferromagnetic (AFM) monolayer FeSe, very close to an AFM topological insulator that hosts robust edge states, can be well captured by a double-degenerate fragile topologically flat band just below the Fermi level. The Wilson loop calculations identify that such fragile topology is protected by the $S_{4z}$ symmetry, which gives rise to an AFM higher-order topological insulator that support the bound state with fractional charge $e/2$ at th…
