Complex energy plane and topological invariant in non-Hermitian systems

Non-Hermitian systems as theoretical models of open or dissipative systems exhibit rich novel physical properties and fundamental issues in condensed matter physics. We propose a generalized local–global correspondence between the pseudo-boundary states in the complex energy plane and topological invariants of quantum states. We find that the patterns of the pseudo-boundary states in the complex energy plane mapped to the Brillouin zone are topological invariants against the parameter deformation. We demonstrate this approach by the non-Hermitian Chern insulator model. We give the consistent topological phases obtained from the Chern number and vorticity. We also find some novel topological invariants embedded in the topological phases of the Chern insulator model, which enrich the phase diagram of the non-Hermitian Chern insulators model beyond that predicted by the Chern number and vorticity. We also propose a generalized vorticity and its flipping index to understand physics behind this novel local–global correspondence and discuss the relationships between the local–global correspondence and the Chern number as well as the transformation between the Brillouin zone and the complex energy plane. These novel approaches provide insights to how topological invariants may be obtained from local information as well as the global property of quantum states, which is expected to be applicable in more generic non-Hermitian systems.     

Topological Phase Transition with Larger Winding Number in the Non-Hermitian Dimerized Lattice

The topological phase transitions among normal insulator phase, two kinds of topological insulator phases, and topological semimetal phase are shown based on the non-Hermitian dimerized Su–Schrieffer–Heeger (SSH) model with the nonreciprocal intercell and long-range hopping. In contrast to the previous work, it is found that the topological insulator phase in the present SSH model can hold the larger non-Bloch winding number accompanied by exceptional winding of the generalized Brillouin zone around the gap-closing points. Compared with the usual topological insulator phase in non-Hermitian SSH model, the topological insulator with the larger winding number owns two pairs of zero energy modes with a distinct form of edge localization in the gap. The physical mechanism of the distinct edge localization for zero energy modes via a equivalent Hermitian version of the non-Hermitian SSH model is revealed. Additionally, the process of the phase transition is visualized among normal insulator phase, topological insulator phases, and topological semimetal phase in detail via the evolution of the gap-closing points on the plane of generalized Brillouin zone. This work further verifies the non-Bloch theory and enrich the investigation about the topologically nontrivial phase with the larger topological invariant in the non-Hermitian SSH model.     

Non-Hermitian topological Anderson insulators

Non-Hermitian systems can exhibit exotic topological and localization properties.Here we elucidate the non-Hermitian effects on disordered topological systems using a nonreciprocal disordered Su-Schrieffer-Heeger model.We show that the non-Hermiticity can enhance the topological phase against disorders by increasing bulk gaps.Moreover,we uncover a topological phase which emerges under both moderate non-Hermiticity and disorders,and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes.Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators.We reveal that the system has unique non-monotonous localization behavior and the topological transition is accompanied by an Anderson transition.These properties are general in other non-Hermitian models.     

Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk-boundary correspondence

We investigate novel features of three-dimensional non-Hermitian Weyl semimetals,paying special attention to the unconventional bulk-boundary correspondence.We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram,which is also confirmed by the energy spectra from our numerical results.It is shown that,in sharp contrast to Hermitian systems,the conventional(Bloch)bulk-boundary correspondence breaks down in non-Hermitian topological semimetals,which is caused by the non-Hermitian skin effect.We establish the non-Bloch bulk-boundary correspondence for non-Hermitian Weyl semimetals:the topological edge modes are determined by the non-Bloch Chern number of the bulk bands.Moreover,these topological edge modes can manifest as the unidirectional edge motion,and their signatures are consistent with the non-Bloch bulk-boundary correspondence.Our work establishes the non-Bloch bulk-boundary correspondence for non-Hermitian topological semimetals.     

Z2 topological order and the quantum spin Hall effect

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems.     

Topological superfluid in a fermionic bilayer optical lattice

In this paper, a topological superfluid phase with Chern number ?? = ±1, possessing gapless edge states and non-Abelian anyonsis designed in a ?? = ±1 topological insulator proximity to ans-wave superfluid on an optical lattice with the effective gauge fieldand layer-dependent Zeeman field coupled to ultracold fermionic atoms’ pseudo spin. Wealso study its topological properties and calculate the phase stiffness by using therandom-phase-approximation approach. Finally we derive the temperature of theKosterlitz-Thouless transition by means of renormalized group theory. Owning to theexistence of non-Abelian anyons, this ?? = ±1 topological superfluid may be a possible candidate fortopological quantum computation.     

Non-Hermitian Kitaev chain with complex periodic and quasiperiodic potentials

We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials. We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant. The phase transition points are given analytically. The Majorana zero modes in the topological nontrivial regimes are obtained. Focusing on the quasiperiodic potential, we obtain the phase transition from the topological superconducting phase to the Anderson localization, which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system. We also find that the topological regime can be reduced by increasing the non-Hermiticity.     

外尔半金属及其输运特性

外尔半金属是继石墨烯以及拓扑绝缘体之后的又一个研究热点。相比于后两者,外尔半金 属独特的三维无能隙线性色散能带结构使得它有很多奇特的性质,如：手性反常、手性磁效应、 反弱局域化、手性朗道能级和负磁阻效应等。实际样品中无序总是不可避免的,所以考虑无序对 体系的影响是很有必要的。我们回顾了无序下第一类以及第二类外尔半金属的相变特性,并获得 了完整的相图,这些无序诱导的相变丰富了拓扑安德森绝缘体和安德森金属绝缘体相变的物理内 涵。我们同样回顾了长程短程无序影响下的第一类外尔半金属体系的输运,发现了一种不能采用 玻尔兹曼输运方程来描述的输运过程。我们介绍Imbert-Fedorov 位移这一光学中的效应在外尔 半金属中的实现,这为更好地应用外尔半金属提供了更多的可能性,随后采用波包散射,我们解 释了外尔半金属中的超高载流子迁移率问题的原因,最后我们给出一个简要的总结。     

Coupling-matrix approach to the Chern number calculation in disordered systems

The Chern number is often used to distinguish different topological phases of matter in two-dimensional electron systems. A fast and efficient coupling-matrix method is designed to calculate the Chern number in finite crystalline and disordered systems. To show its effectiveness, we apply the approach to the Haldane model and the lattice Hofstadter model, and obtain the correct quantized Chern numbers. The disorder-induced topological phase transition is well reproduced, when the disorder strength is increased beyond the critical value. We expect the method to be widely applicable to the study of topological quantum numbers.     

Effects of interaction on quantum spin Hall insulators

We study the S(z)-conserving quantum spin Hall insulator in the presence of Hubbard U from a field theory point of view. The main findings are the following. (1) For arbitrarily small U the edges possess power-law correlated antiferromagnetic XY local moments. Gapless charge excitations arise from the Goldstone-Wilczek mechanism. (2) Electron tunneling between opposite edges allows vortex instantons to proliferate when K, the XY stiffness constant, satisfies 4πK+(4πK)(-1)<4. When the preceding inequality is violated, the edge modes remain gapless despite the sample width being finite. (3) The phase transition from the topological insulator to the large U antiferromagnetic insulator is triggered by the condensation of magnetic excitons. (4) In the large U antiferromagnetic insulating phase the magnetic vortices carry charges proportional to the square magnitude of the antiferromagnetic order parameter.     

Quantum criticality between topological and band insulators in 3+1 dimensions

Four-component massive and massless Dirac fermions in the presence of long range Coulomb interaction and chemical potential disorder exhibit striking fermionic quantum criticality. For an odd number of flavors of Dirac fermions, the sign of the Dirac mass distinguishes the topological and the trivial band insulator phases, and the gapless semimetallic phase corresponds to the quantum critical point that separates the two. Up to a critical strength of disorder, the semimetallic phase remains stable, and the universality class of the direct phase transition between two insulating phases is unchanged. Beyond the critical strength of disorder the semimetallic phase undergoes a phase transition into a disorder controlled diffusive metallic phase, and there is no longer a direct phase transition between the two types of insulating phases.     

声子晶体中的多重拓扑相

研究了圆环型波导依照蜂窝结构排列的声子晶体系统中的拓扑相变.利用晶格结构的点群对称性实现赝自旋,并在圆环中引入旋转气流来打破时间反演对称性.通过紧束缚近似模型计算的解析结果表明,没有引入气流时,调节几何参数,系统存在普通绝缘体和量子自旋霍尔效应绝缘体两个相;引入气流后,可以实现新的时间反演对称性破缺的量子自旋霍尔效应相,而增大气流强度,则可以实现量子反常霍尔效应相.这三个拓扑相可以通过自旋陈数来分类.通过有限元软件模拟了多个系统中边界态的传播,发现不同于量子自旋霍尔效应相,量子反常霍尔相系统的表面只支持一种自旋的边界态,并且它无需时间反演对称性保护.     

Rashba自旋轨道耦合下square-octagon晶格的拓扑相变

本文研究了各向同性square-octagon晶格在内禀自旋轨道耦合、Rashba自旋轨道耦合和交换场作用下的拓扑相变,同时引入陈数和自旋陈数对系统进行拓扑分类.系统在自旋轨道耦合和交换场的影响下会出现许多拓扑非平庸态,包括时间反演对称破缺的量子自旋霍尔态和量子反常霍尔态.特别的是,在时间反演对称破缺的量子自旋霍尔效应中,无能隙螺旋边缘态依然能够完好存在.调节交换场或者填充因子的大小会导致系统发生从时间反演对称破缺的量子自旋霍尔态到自旋过滤的量子反常霍尔态的拓扑相变.边缘态能谱和自旋谱的性质与陈数和自旋陈数的拓扑刻画完全一致.这些研究成果为自旋量子操控提供了一个有趣的途径.     

Floquet topological phase transition in two-dimensional quadratic band crossing system

We investigate the Hall effects of quadratic band crossing(QBC) fermions in a square optical lattice with spin–orbit coupling and orbital Zeeman term. We find that the orbital Zeeman term and shaking play critical roles in the systems,which can drive a topological transition from spin Hall phases to anomalous Hall phase with nonvanishing(spin) Chern numbers. Due to the interplay among the orbital Zeeman term, spin–orbit coupling, and the shaking, the phase diagram of the system exhibits rich phases, which are characterized by Chern number.     

Signatures of bulk topology in the non-linear optical spectra of Dirac-Weyl materials

Graphene, topological insulators (TI) and the Weyl semimetal are shown to be well-characterized using the phenomenon of anomalous Rabi oscillation (ARO). These oscillations occur far from conventional resonance and Floquet theory shows them to be unique to these systems. Of particular interest is the bulk topological insulator (TI) where the wave-vector dependent frequency of the ARO is seen to be gapped in topologically trivial situations and gapless when there is a non-vanishing Chern number. It is shown that the Chern number may be directly inferred by performing a pump-probe experiment in the bulk without referring to surface states. A simpler alternative to the Lindblad method is invoked in order to incorporate dephasing effects that despite leading to a non-unitary time evolution of the wave-function, is nevertheless, probability conserving. The differential transmission coefficient versus the pump pulse duration (when all else is held fixed) has the form of a sinusoidal function with an amplitude that decays as a power law in the pump duration (alternatively, the “area” of the pump pulse). The exponent of this power law decay is indicative of the Chern number of the bulk in case of TI and more generally indicative of the particular member of the family of materials that may be collectively referred to as – Dirac-Weyl materials.     

Optical tuning of Berry phase effect in topological insulators

We theoretically discuss the influence of driving laser field on the topological nature, one of the manifestation of the electron Berry phase effect, in two-dimensional electronic systems. Adiabatic change of the laser amplitude with circular polarization alters the “order parameter”, termed the Chern number, in topological insulator with broken time-reversal symmetry, resulting in photo-induced phase transition. The finding is an optical analog of the integer quantum Hall effect, that is triggered by the laser field instead of magnetic field. This parallelism suggests the similarity of effects to electron dynamics between circularly polarized light and magnetic field.     

Majorana modes in time-reversal invariant s-wave topological superconductors

We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes. The multiband character of the model together with spin-orbit coupling are key to realizing such a topological superconductor. We characterize the topological phase diagram by using a partial Chern number sum, and show that the latter is physically related to the parity of the fermion number of the time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap into account, we also establish the possibility of a direct topological superconductor-to-topological insulator quantum phase transition.     

Effect of interactions, disorder and magnetic field in the Hubbard model in two dimensions

The effects of both interactions and Zeeman magnetic field in disordered electronic systems are explored in the Hubbard model on a square lattice. We investigate the thermodynamic (density, magnetization, density of states) and transport (conductivity) properties using determinantal quantum Monte Carlo and inhomogeneous Hartree Fock techniques. We find that at half filling there is a novel metallic phase at intermediate disorder that is sandwiched between a Mott insulator and an Anderson insulator. The metallic phase is highly inhomogeneous and coexists with antiferromagnetic long-range order. At quarter filling also the combined effects of disorder and interactions produce a conducting state which can be destroyed by applying a Zeeman field, resulting in a magnetic field-driven transition. We discuss the implication of our results for experiments.     

Photoinduced Weyl semimetal phase and anomalous Hall effect in a three-dimensional topological insulator

Motivated by the fact that Weyl fermions can emerge in a three-dimensional topological insulator on breaking either time-reversal or inversion symmetries, we propose that a topological quantum phase transition to a Weyl semimetal phase occurs under the off-resonant circularly polarized light, in a three-dimensional topological insulator, when the intensity of the incident light exceeds a critical value. The circularly polarized light effectively generates a Zeeman exchange field and a renormalized Dirac mass, which are highly controllable. The phase transition can be exactly characterized by the first Chern number. A tunable anomalous Hall conductivity emerges, which is fully determined by the location of the Weyl nodes in momentum space, even in the doping regime. Our predictions are experimentally realizable through pump-probe angle-resolved photoemission spectroscopy and raise a new way for realizing Weyl semimetals and quantum anomalous Hall effects.     

Quantum Anomalous Hall Multilayers Grown by Molecular Beam Epitaxy

Quantum anomalous Hall(QAH) effect is a quantum Hall effect that occurs without the need of external magnetic field. A system composed of multiple parallel QAH layers is an effective high Chern number QAH insulator and the key to the applications of the dissipationless chiral edge channels in low energy consumption electronics. Such a QAH multilayer can also be engineered into other exotic topological phases such as a magnetic Weyl semimetal with only one pair of Weyl points. This work reports the first experimental realization of QAH multilayers in the superlattices composed of magnetically doped(Bi,Sb)_2Te_3 topological insulator and Cd Se normal insulator layers grown by molecular beam epitaxy. The obtained multilayer samples show quantized Hall resistance h/N_e~2, where h is Planck's constant, e is the elementary charge and N is the number of the magnetic topological insulator layers, resembling a high Chern number QAH insulator. The QAH multilayers provide an excellent platform to study various topological states of matter.