Topological aspects of graphene

Topological aspects of the electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Presence of the pair of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is generic for a class of two-dimensional lattices that interpolate between square and π-flux lattices. Topological stability guarantees persistence of the peculiar QHE. (ii) While we have the massless Dirac dispersion only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behaviour accompanied by huge jumps in the QHE with a sign change. (iii) We establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is another topological effect. We have also explicitly shown that the E=0 edge states in honeycomb in zero magnetic field persist in magnetic field. (iv) We have also identified a topological origin of the fermion doubling in terms of the chiral symmetry.     

Geometric quantization of topological gauge theories

We show that Abelian gauge theories in 2+1 space-time dimensions with the introduction of a topological Chern-Simons term can be quantized with the use of the symplectic formalism. The consistency of our results are verified by the agreement with the ones from the Dirac case.     

Topological transport in Dirac electronic systems:A concise review

Various novel physical properties have emerged in Dirac electronic systems, especially the topological characters protected by symmetry. Current studies on these systems have been greatly promoted by the intuitive concepts of Berry phase and Berry curvature, which provide precise definitions of the topological phases. In this topical review, transport properties of topological insulator(Bi2Se3), topological Dirac semimetal(Cd3As2), and topological insulator-graphene heterojunction are presented and discussed. Perspectives about transport properties of two-dimensional topological nontrivial systems,including topological edge transport, topological valley transport, and topological Weyl semimetals, are provided.     

Fermionic Zero Modes in Self-dual Vortex Background on a Torus

We study fermionic zero modes in the self-dual vortex background on an extra two-dimensional Riemann surface in （5＋1） dimensions. Using the generalized Abelian-Higgs model, we obtain the inner topological structure of the self-dual vortex and establish the exact self-duality equation with topological term. Then we analyze the Dirac operator on an extra torus and the effective Lagrangian of four-dimensional fermions with the self-dual vortex background. Solving the Dirac equation, the fermionic zero modes on a torus with the self-dual vortex background in two simple cases are obtained.     

On a new type of massless Dirac fermions in crystalline topological insulators

A new type of massless Dirac fermions in crystalline three-dimensional topological insulators (three-dimensional → two-dimensional situation) has been predicted. The spectrum has fourfold degeneracy at the top of the two-dimensional Brillouin zone (M point) and twofold degeneracy near the M point. Crystal symmetry along with the time reversal invariance in three-dimensional topological insulators allows fourfold degenerate Dirac cones, which are absent in the classification of topological features in R.-J. Slager et al., Nat. Phys. 9, 98 (2013). The Hamiltonian in the cited work does not contain Dirac singularities with more than twofold degeneracy. For this reason, the corresponding topological classification is incomplete. The longitudinal magnetic field in the spinless case holds the massless dispersion law of fermions and does not lift fourfold degeneracy. In the spinor case, the magnetic field lifts fourfold degeneracy, holding only twofold degeneracy, and results in the appearance of a band gap in the spectrum of fermions.     

Topological surface states in lead-based ternary telluride Pb(Bi(1-x)Sb(x))2Te4

We have performed angle-resolved photoemission spectroscopy on Pb(Bi(1-x)Sb(x))2Te4, which is a member of lead-based ternary tellurides and has been theoretically proposed as a candidate for a new class of three-dimensional topological insulators. In PbBi2Te4, we found a topological surface state with a hexagonally deformed Dirac-cone band dispersion, indicating that this material is a strong topological insulator with a single topological surface state at the Brillouin-zone center. Partial replacement of Bi with Sb causes a marked change in the Dirac carrier concentration, leading to the sign change of Dirac carriers from n type to p type. The Pb(Bi(1-x)Sb(x))2Te4 system with tunable Dirac carriers thus provides a new platform for investigating exotic topological phenomena.     

拓扑绝缘体的普适电导涨落

拓扑绝缘体因其无能量耗散的拓扑表面输运而备受关注, 揭示拓扑表面态因其 的贝利相位而产生的拓扑输运现象, 将有助于拓扑绝缘体相关器件的应用开发. 本文回顾了普适电导涨落(UCF) 揭示拓扑绝缘体奇异输运性质的研究进展. 通过调控温度、角度、门电压、垂直磁场和平行磁场等外部参量, 实现了对拓扑绝缘体的UCF 效应的系统研究, 证实了拓扑绝缘体中二维UCF 的输运现象, 并通过尺寸标度规律获得了UCF 的拓扑起源的实验证据, 讨论了拓扑表面态的UCF 的统计对称规律. 从而实现了对拓扑绝缘体UCF 效应的较为完整的理解.     

Electron fractionalization in two-dimensional graphenelike structures

Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry. In this Letter, we show that fractionally charged topological excitations exist on graphenelike structures, where quasiparticles are described by two flavors of Dirac fermions and time-reversal symmetry is respected. The topological zero modes are mathematically similar to fractional vortices in p-wave superconductors. They correspond to a twist in the phase in the mass of the Dirac fermions, akin to cosmic strings in particle physics.     

New constraints on the 3-3-1 model with right-handed neutrinos

In the framework of a 3-3-1 model with right-handed neutrinos and three scalar triplets we consider different spontaneous symmetry breaking patterns seeking for a non-linear realization of accidental symmetries of the model, which will produce physical Nambu–Goldstone (NG) bosons in the neutral scalar spectrum. We make a detailed study of the safety of the model concerning the NG boson emission in energy-loss processes which could affect the standard evolution of astrophysical objects. We consider the model with a \(\mathbb {Z}_2\) symmetry, conventionally used in the literature, finding that in all of the symmetry breaking patterns the model is excluded. Additionally, looking for solutions for that problem, we introduce soft \(\mathbb {Z}_2\)-breaking terms in the scalar potential in order to remove the extra accidental symmetries and at the same time maintain the model as simple as possible. We find that there is only one soft \(\mathbb {Z}_2\)-breaking term that enables us to get rid of the problematic NG bosons.     

New family of Dirac and Weyl semimetals in XAuTe (X = Na,K, Rb) ternary honeycomb compounds

We propose a new family of 3D Dirac semimetals based on XAuTe(X = K, Na, Rb) ternary honeycomb compounds, determined based on first-principles calculations, which are shown to be topological Dirac semimetals in which the Dirac points are induced by band inversion. Dirac points with four-fold degeneracy that are protected by C3 rotation symmetry and located on the Γ-A high-symmetry path are found. Through spatial-inversion symmetry breaking, a K(Au0.5 Hg0.5)(Te0.5As0.5) superlattice structure composed of KHgAs and KAuTe compounds is proven to be a Weyl semimetal with type-II Weyl points, which connect electronand hole-like bands. In this superlattice structure, the six pairs of Weyl nodes are distributed along the K-Γ high-symmetry path on the kz = 0 plane. Our research expands the family of topological Dirac and type-II Weyl semimetals.     

On the influence of a Rashba-type coupling induced by Lorentz-violating effects on a Landau system for a neutral particle

We study a possible scenario of the Lorentz symmetry violation background that allows us to build an analogue of the Landau system for a nonrelativistic Dirac neutral particle interacting with a field configuration of crossed electric and magnetic fields. We also discuss the arising of analogues of the Rashba coupling, the Zeeman term and the Darwin term from the Lorentz symmetry breaking effects, and the influence of these terms on the analogue of the Landau system confined to a two-dimensional quantum ring. Finally, we show that this analogy with the Landau system confined to a two-dimensional quantum ring allows us to establish an upper bound for the Lorentz symmetry breaking parameters.     

Dirac光子晶体

Dirac费米子作为粒子物理中的基本粒子之一,其理论在近年来蓬勃发展的拓扑电子理论领域中被广泛提及并用来刻画具有Dirac费米子性质的电子态.这种特殊的能态通常被称为Dirac点,在能谱上表现为两条不同能带之间的线性交叉点.由于Dirac点往往是发生拓扑相变的转变点,因而也被视为实现各种拓扑态的重要母态.作为可与拓扑电子体系类比的拓扑光子晶体因其独特的潜在应用价值也受到人们的广泛关注,实现包含Dirac点的光子能带已成为研究拓扑光子晶体的核心课题.本文基于电子的拓扑理论,简要地回顾了Dirac点在光子系统中的研究进展,特别介绍了如何在光子晶体中利用不同晶格对称性实现在高对称点/线上的Dirac点,以及由Dirac点衍生的Weyl点.     

Topological Invariants of Edge States for Periodic Two-Dimensional Models

Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott–Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a ${\mathbb Z}_2$ -invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators.     

Atomic-Ordering-Induced Quantum Phase Transition between Topological Crystalline Insulator and Z_2 Topological Insulator

Topological phase transition in a single material usually refers to transitions between a trivial band insulator and a topological Dirac phase, and the transition may also occur between different classes of topological Dirac phases.It is a fundamental challenge to realize quantum transition between Z_2 nontrivial topological insulator(TI) and topological crystalline insulator(TCI) in one material because Z_2 TI and TCI have different requirements on the number of band inversions. The Z_2 TIs must have an odd number of band inversions over all the time-reversal invariant momenta, whereas the newly discovered TCIs, as a distinct class of the topological Dirac materials protected by the underlying crystalline symmetry, owns an even number of band inversions. Taking PbSnTe_2 alloy as an example, here we demonstrate that the atomic-ordering is an effective way to tune the symmetry of the alloy so that we can electrically switch between TCI phase and Z_2 TI phase in a single material. Our results suggest that the atomic-ordering provides a new platform towards the realization of reversibly switching between different topological phases to explore novel applications.     

Detecting topological order through a continuous quantum phase transition

We study a continuous quantum phase transition that breaks a Z2 symmetry. We show that the transition is described by a new critical point which does not belong to the Ising universality class, despite the presence of well-defined symmetry-breaking order parameter. The new critical point arises since the transition not only breaks the Z2 symmetry, it also changes the topological or quantum order in the two phases across the transition. We show that the new critical point can be identified in experiments by measuring critical exponents. So measuring critical exponents and identifying new critical points is a way to detect new topological phases and a way to measure topological or quantum orders in those phases.     

Quantum phase transitions across a p-wave Feshbach resonance

We study a single-species polarized Fermi gas tuned across a narrow p-wave Feshbach resonance. We show that in the course of a Bose-Einstein condensation (BEC)-BCS crossover, the system can undergo a magnetic-field-tuned quantum phase transition from a px-wave to a px+ipy-wave superfluid. The latter state, that spontaneously breaks time-reversal symmetry, furthermore undergoes a topological px+ipy to px+ipy transition at zero chemical potential mu. In two dimensions, for mu > 0 it is characterized by a Pfaffian ground state exhibiting topological order and non-Abelian excitations familiar from fractional quantum Hall systems.     

A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal

We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrödinger equation in a double well potential. The Landau levels obey the general scaling law epsilon_n ∝B^2/3 f_n(Δ/B^2/3), and they evolve continuously from a \(\sqrt{n B}\) to a linear (n+1/2)B dependence, with a [(n+1/2)B]^2/3 dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameteris varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice.     

Quantum criticality and minimal conductivity in graphene with long-range disorder

We consider the conductivity sigma of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class sigma model including a topological term with theta=pi. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of e(2)/h. When the effective time-reversal symmetry is broken, the symmetry class becomes unitary, and sigma acquires the value characteristic for the quantum Hall transition.