It is well known that inversion symmetry in one-dimensional (1D) systems leads to the quantization of the geometric Zak phase to values of either 0 or π. When the system has particle-hole symmetry, this topological property ensures the existence of zero-energy interface states at the interface of two bulk systems carrying different Zak phases. In the absence of inversion symmetry, the Zak phase can take any value and the existence of interface states is not ensured. We show here that the situation is different when the unit cell contains multiple degrees of freedom and a hidden inversion symmetry exists in a subspace of the system. As an example, we consider a system of two Su-Schrieffer-Heeger (SSH) chains coupled by a coupler chain. Although the introduction of coupler chain breaks the inversion symmetry of the system, a certain hidden inversion symmetry ensures the existence of a decoupled $2\times2$ SSH Hamiltonian in the subspace of the entire system and the two bands associated with this subspace have quantized Zak phases. These "quantized" bands in turn can provide topological boundary or interface states in such systems. Since the entire system has no inversion symmetry, the bulk-boundary correspondence may not hold exactly. The above is also true when next-nearest-neighbor hoppings are included. Our systems can be realized straightforwardly in systems such as coupled single-mode optical waveguides or coupled acoustic cavities.
As reflection symmetry or space-time inversion symmetry is preserved, with a non-contractible integral loop respecting the symmetry in the Brilliouin zone, Berry phase is quantized in proper basis. Topological nodal lines can be enclosed in the integral loop and $π$-Berry phase topologically protects the nodal lines. In this work, we show that to have quantized Berry phase restricted by the symmetry in any crystal structure, we choose to use the cell-periodic convention and define the origin point in the real space at one of the reflection (inversion) centers. In addition, $π$-Berry phase is n…
We show how the presence of inversion symmetry in a one-dimensional (1D) lattice model is not a sufficient condition for a quantized Zak's phase. This is only the case when the inversion axis is at the center of the unit cell. When the inversion axis is not at the center, the modified inversion operator within the unit cell gains a k-dependence in some of its matrix elements which adds a correction term to the usual Zak's phase expression1, making it in general deviate from its quantized value. A general expression that recovers a quantized Zak's phase in a lattice model with a unit cell of ar…
Landau level lasers have the advantage of tunability of the laser frequency by means of the external magnetic field. The crucial prerequisite of such a laser is a population inversion between optically coupled Landau levels. Efficient carrier-carrier and carrier-phonon scattering generally suppresses this effect in conventional materials. Based on microscopic calculations, we predict for the first time the occurrence of a long-lived population inversion in Landau-quantized graphene and reveal the underlying many-particle mechanisms. To guide the experimental demonstration, we present optimal c…
We propose that the noncentrosymmetric LiGaGe-type hexagonal $ABC$ crystal SrHgPb realizes a new type of topological semimetal that hosts both Dirac and Weyl points in momentum space. The symmetry-protected Dirac points arise due to a band inversion and are located on the sixfold rotation $z$-axis, whereas the six pairs of Weyl points related by sixfold symmetry are located on the perpendicular $k_z=0$ plane. By studying the electronic structure as a function of the buckling of the HgPb layer, which is the origin of inversion symmetry breaking, we establish that the coexistence of Dirac and We…
Two dimensional inversion symmetry ($180^{\circ}$ rotations in the ``Hall plane'' that hosts the incompressible electron fluid that exhibits the quantized Hall effect) is identified as its fundamental unbroken symmetry. A consequence is that the integers $p$ and $q$ which define both the Landau level filling factor $ν$ = $p/q$ and the elementary fractional charge $\pm e/q$ of topological excitations, cannot have a common divisor greater than 2.
Low-dimensional materials with broken inversion symmetry and strong spin-orbit coupling can give rise to fascinating quantum phases and phase transitions. Here we report coexistence of superconductivity and ferromagnetism below 2.5\,K in the quasi-one dimensional crystals of non-centrosymmetric (TaSe$_4$)$_3$I (space group: $P\bar{4}2_1c$). The unique phase is a direct consequence of inversion symmetry breaking as the same material also stabilizes in a centro-symmetric structure (space group: $P4/mnc$) where it behaves like a non-magnetic insulator. The coexistence here upfront contradicts the…
Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We a…
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 …
Similar to Weyl fermions, a recently discovered topological fermion "triple point" can be generated from the splitting of Dirac fermion while the system has inversion symmetry (IS) breaking or time reversal symmetry (TRS) breaking. Inducing triple points in IS breaking symmorphic systems have been well studied, whereas in TRS breaking symmorphic systems have not yet. In this work, we extend the theory of searching triple points to all symmorphic magnetic systems. We list all $k$ paths of all symmorphic systems which allow the existence of triple points. With this systematic study, we also find…
The coexistence of Rashba and Dresselhaus spin-orbit interactions (SOIs) in semiconductor quantum wells leads to an anisotropic effective field coupled to carriers' spins. We demonstrate a gate-controlled anisotropy in Aharonov-Casher (AC) spin interferometry experiments with InGaAs mesoscopic rings by using an in-plane magnetic field as a probe. Supported by a perturbation-theory approach, we find that the Rashba SOI strength controls the AC resistance anisotropy via spin dynamic and geometric phases and establish ways to manipulate them by employing electric and magnetic tunings. Moreover, a…
We establish the correspondence between the fractional charge bound to a vortex in a textured lattice and the relevant bulk band topology in two-dimensional (2D) topological crystalline insulators. As a representative example, we consider the Kekule textured graphene whose bulk band topology is characterized by a 2D $\mathbb{Z}_{2}$ topological invariant $ν_{\rm 2D}$ protected by inversion symmetry. The fractional charge localized at a vortex in the Kekule texture is shown to be related to the change in the bulk topological invariant $ν_{\rm 2D}$ around the vortex, as in the case of the Su-Sch…
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.
Realizing stable two-dimensional (2D) Dirac points against spin-orbit coupling (SOC) has attracted much attention because it provides a platform to study the unique transport properties. In previous work, Young and Kane [Phys. Rev. Lett. \textbf{115}, 126803 (2015)] proposed stable 2D Dirac points with SOC, in which the Berry curvature and edge states vanish due to the coexistence of inversion and time-reversal symmetries. Herein, using the tight-binding model and k$\cdot$p effective Hamiltonian, we present that 2D Dirac points can survive in the presence of SOC without inversion symmetry. Suc…
Here we report the investigation of the anomalous Hall effect in the magnetically doped topological insulator (V,Bi,Sb)2Te3. We find it contains two contributions of opposite sign. Both components are found to depend differently on carrier density, leading to a sign inversion of the total anomalous Hall effect as a function of applied gate voltage. The two contributions are found to have different magnetization reversal fields, which in combination with a temperature dependent study points towards the coexistence of two ferromagnetic orders in the system. Moreover, we find that the sign of tot…
We demonstrate the coexistence of pseudospin- and valley-Hall-like edge states in a photonic crystal with $C_{3v}$ symmetry, which is composed of three interlacing triangular sublattices with the same lattice constants. By tuning the geometry of the sublattices, three complete photonic band gaps with nontrivial topology can be created, one of which is due to the band inversion associated with the pseudospin degree of freedom at the $Γ$ point and the other two due to the gapping out of Dirac cones associated with the valley degree of freedom at the $K, K'$ points. The system can support tri-ban…
SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s-wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetry-protected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introduci…
Two-dimensional ($2$D) semi-Dirac systems, such as $2$D black phosphorus and arsenene, can exhibit a rich topological phase transition between insulating, semi-Dirac, and band inversion phases when subjected to an external modulation. How these phase transitions manifest within the quantum transport and shot noise signatures remain an open question thus far. Here, we show that the Fano factor converges to the universal $F\approx0.179$ at the semi-Dirac phase, and transits between the sub-Poissonian ($F\approx1/3$) and the Poissonian shot noise ($F\approx1$) limit at the band inversion and the …
The Weyl semimetal MoTe$_2$ offers a rare opportunity to study the interplay between Weyl physics and superconductivity. Recent studies have found that Se substitution can boost the superconductivity up to 1.5K, but suppress the Td structure phase that is essential for the emergence of Weyl state. A microscopic understanding of possible coexistence of enhanced superconductivity and the Td phase has not been established so far. Here, we use scanning tunneling microscopy (STM) to study a optimally doped new superconductor MoTe$_{1.85}$Se$_{0.15}$ with bulk Tc ~ 1.5K. By means of quasiparticle in…
Band topology of anomalous quantum Hall insulators can be precisely addressed by computing Chern numbers of constituent non-degenerate bands that describe quantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chern numbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands of materials preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, without detailed knowledge of underlying basis? We affirmatively answer this question by employing a non-Abelian generalization of Stokes' theorem and…
Coexistence of Chiral Majorana Edge States and Bogoliubov Fermi Surfaces in Two-Dimensional Nonsymmorphic Dirac Semimetal/Superconductor Heterostructures
Following a previous work [Shen, Ma, Xie and Zhang, Phys. Rev. Lett. 92, 256603 (2004)] on the resonant spin Hall effect, we present detailed calculations of the spin Hall conductance in two-dimensional quantum wells in a strong perpendicular magnetic field. The Rashba coupling, generated by spin-orbit interaction in wells lacking bulk inversion symmetry, introduces a degeneracy of Zeeman-split Landau levels at certain magnetic fields. This degeneracy, if occuring at the Fermi energy, will induce a resonance in the spin Hall conductance below a characteristic temperature of order of the Zeeman…
In general, the stability of a band crossing point indicates the presence of a quantized topological number associated with it. In particular, the recent discovery of three-dimensional Dirac semimetals in Na$_{3}$Bi and Cd$_{3}$As$_{2}$ demonstrates that a Dirac point with four-fold degeneracy can be stable as long as certain crystalline symmetries are supplemented in addition to the time-reversal and inversion symmetries. However, the topological charges associated with Na$_{3}$Bi and Cd$_{3}$As$_{2}$ are not clarified yet. In this work, we identify the topological charge of three-dimensional…
Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here we identify the analog of quadrupole order in Maxwell's equations for a photonic crystal (PhC) and identify quadrupole topological photonic crystals formed through a band inversion process. Unlike prior studies relying on threaded…
To date, almost all of the discussions on topological insulators (TIs) have focused on two- and three-dimensional systems. One-dimensional (1D) TIs manifested in real materials, in which localized spin states may exist at the end or near the junctions, have largely been unexplored. Previous studies have considered the system of gapped graphene nanoribbons (GNRs) possessing spatial symmetries (e.g. inversion) with only termination patterns commensurate with inversion- or mirror-symmetric unit cells. In this work, we prove that a symmetry-protected $\mathbb{Z}_{2}$ topological classification exi…
We identify the existence of various symmetry-protected topological states in one-dimensional superlattices with periodically modulated hopping amplitudes or on-site potentials, which can be characterized by the quantized Berry phase $π$ or the emergence of a pair of degenerate boundary states. It is shown that there may exist three types of topological phases, which are protected by the inversion symmetry, the chiral symmetry, and both of them, respectively, depending on the modulations, the odd or even modulation period. The connection between the hopping and potential modulations is also di…
We study corner states on a flat band in the square lattice. To this end, we introduce a two dimensional model including Su-Schrieffer-Heeger type bond alternation responsible for corner states as well as next-nearest neighbor hoppings yielding flat bands. The key symmetry of the model for corner states is space-time inversion ($\cal PT$) symmetry, which guarantees quantized Berry phases. This implies that edge states as well as corner states would show up if boundaries are introduced to the system. We also argue that an infinitesimal $\cal PT$ symmetry-breaking perturbation could drive flat b…
