Probe of topological invariants using quantum walks of a trapped ion in coherent state space

We present a protocol to realize topological discrete-time quantum walks,which comprise a sequence of spindependent flipping displacement operations and quantum coin tossing operations,with a single trapped ion.It is demonstrated that the information of bulk topological invariants can be extracted by measuring the average projective phonon number when the walk takes place in coherent state space.Interestingly,the specific chiral symmetry owned by our discrete-time quantum walks simplifies the measuring process.Furthermore,we prove the robustness of such bulk topological invariants by introducing dynamical disorder and decoherence.Our work provides a simple method to measure bulk topological features in discrete-time quantum walks,which can be experimentally realized in the system of single trapped ions.

A new way to construct topological invariants of non-Hermitian systems with the non-Hermitian skin effect

The non-Hermitian skin effect breaks the conventional bulk–boundary correspondence and leads to non-Bloch topological invariants.Inspired by the fact that the topological protected zero modes are immune to perturbations,we construct a partner of a non-Hermitian system by getting rid of the non-Hermitian skin effect.Through adjusting the imbalance hopping,we find that the existence of zero-energy boundary states still dictate the bulk topological invariants based on the band-theory framework.Two non-Hermitian Su–Schrieffer–Heeger(SSH)models are used to illuminate the ideas.Specially,we obtain the winding numbers in analytical form without the introduction of the generalized Brillouin zone.The work gives an alternative method to calculate the topological invariants of non-Hermitian systems.

Emergent Z2 topological invariant and robust helical edge states in two-dimensional topological metals

In this work,we study the effects of disorder on topological metals that support a pair of helical edge modes deeply embedded inside the gapless bulk states.Strikingly,we predict that a quantum spin Hall(QSH)phase can be obtained from such topological metals without opening a global band gap.To be specific,disorder can lead to a pair of robust helical edge states which is protected by an emergent Z2 topological invariant,giving rise to a quantized conductance plateau in transport measurements.These results are instructive for solving puzzles in various transport experiments on QSH materials that are intrinsically metallic.This work also will inspire experimental realization of the QSH effect in disordered topological metals.

Complex energy plane and topological invariant in non-Hermitian system

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.

Fractal growth kinematics abstracted from snowflakes:topological evolution

Based on the kinematic viewpoint, the concept of proportional movement is abstracted to explain the strange behaviors of fractal snowflakes. A transformation group for proportional movement is defined. Under the proportional movement transformation groups, necessary and sufficient conditions for self-similarity of multilevel structures are presented. The characteristic topology of snowflake-like fractal patterns, identical to the topology of ternary-segment fractal line, is proved. Moreover, the topological evolution of N-segment line is explored. The concepts of limit growth and infinite growth are clarified,and the corresponding growth conditions are derived. The topological invariant properties of N-segment line are exposed. In addition, the proposition that the topological evolution of the N-segment line is mainly controlled by the topological invariant N is verified.

Topologically protected edge gap solitons of interacting Bosons in one-dimensional superlattices

We comprehensively investigate the nontrivial states of an interacting Bose system in a cosine potential under the open boundary condition. Our results show that there exists a kind of stable localized state： edge gap solitons. We argue that the states originate from the eigenstates of independent edge parabolas. In particular, the edge gap solitons exhibit a nonzero topological-invariant behavior. The topological nature is due to the connection of the present model to the quantized adiabatic particle transport problem. In addition, the composition relations between the gap solitons and the extended states are also discussed.

Static and dynamic properties of polymer brush with topological ring structures: Molecular dynamic simulation

By defining a topological constraint value（rn）,the static and dynamic properties of a polymer brush composed of moderate or short chains with different topological ring structures are studied using molecular dynamics simulation,and a comparison with those of linear polymer brush is also made.For the center-of-mass height of the ring polymer brush scaled by chain length h;,there is no significant difference of exponent from that of a linear brush in the small topological constraint regime.However,as the topological constraint becomes stronger,one obtains a smaller exponent.It is found that there exists a master scaling power law of the total stretching energy scaled by chain length N for moderate chain length regime,F;p;,for ring polymer brushes,but with a larger exponent v than 5/6,indicating an influence of topological constraint to the dynamic properties of the system.A topological invariant of free energy scaled by（c）;is found.

Topological nodal line semimetals

We review the recent,mainly theoretical,progress in the study of topological nodal line semimetals in three dimensions.In these semimetals,the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone,and any perturbation that preserves a certain symmetry group（generated by either spatial symmetries or time-reversal symmetry） cannot remove this crossing line and open a full direct gap between the two bands.The nodal line（s） is hence topologically protected by the symmetry group,and can be associated with a topological invariant.In this review,（ⅰ） we enumerate the symmetry groups that may protect a topological nodal line;（ⅱ） we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface,establishing a topological classification;（ⅲ） for certain classes,we review the proposals for the realization of these semimetals in real materials;（ⅳ） we discuss different scenarios that when the protecting symmetry is broken,how a topological nodal line semimetal becomes Weyl semimetals,Dirac semimetals,and other topological phases;and（ⅴ） we discuss the possible physical effects accessible to experimental probes in these materials.

Multiple surface states,nontrivial band topology,and antiferromagnetism in GdAuAl_(4)Ge_(2)

Magnetic topological states of matter provide a fertile playground for emerging topological physics and phenomena.The current main focus is on materials whose magnetism stems from 3d magnetic transition elements,e.g.,MnBi_(2)Te_(4),Fe_(3)Sn_(2),and Co_(3)Sn_(2)S_(2).In contrast,topological materials with the magnetism from rare earth elements remain largely unexplored.Here we report rare earth antiferromagnet GdAuAl_(4)Ge_(2)as a candidate magnetic topological metal.Angle resolved photoemission spectroscopy(ARPES)and first-principles calculations have revealed multiple bulk bands crossing the Fermi level and pairs of low energy surface states.According to the parity and Wannier charge center analyses,these bulk bands possess nontrivial Z2 topology,establishing a strong topological insulator state in the nonmagnetic phase.Furthermore,the surface band pairs exhibit strong termination dependence which provides insight into their origin.Our results suggest GdAuAl_(4)Ge_(2)as a rare earth platform to explore the interplay between band topology,magnetism and f electron correlation,calling for further study targeting on its magnetic structure,magnetic topology state,transport behavior,and microscopic properties.

ELECTRICALLY CHARGED SOLITONS IN GAUGE FIELD THEORY

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.

Quantum spin Hall insulators in chemically functionalized As(110)and Sb(110)films

We propose a new type of quantum spin Hall （QSH） insulator in chemically functionalized As （110） and Sb （110） film. According to first-principles calculations, we find that metallic As （110） and Sb （110） films become QSH insulators after being chemically functionalized by hydrogen （H） or halogen （C1 and Br） atoms. The energy gaps of the functionalized films range from 0.121 eV to 0.304 eV, which are sufficiently large for practical applications at room temperature. The energy gaps originate from the spin-orbit coupling （SOC）. The energy gap increases linearly with the increase of the SOC strength λ/λ0. The Z2 invariant and the penetration depth of the edge states are also calculated and studied for the functionalized films.

生物力学中分形算子的代数方程和非整数阶

以往的研究表明,分形算子可能源于分形空间中的运动,也可能源于从生物系统中抽象出来的分形结构的运动.这项研究致力于回答另一个基本问题,即什么决定分形结构中分数算子的顺序?为了回答这个问题,本文推广了前人定义的分形单元的概念,探索了具有高阶拓扑结构的树状和网状分形结构,抽象了两类高阶分形算子,并推导了分形算子满足的代数方程.证明了分形算子代数方程组的解与通常为分数阶的分数时间算子有确定关系.利用Vieta定理,阐明了分形算子代数方程解与物理分量算子解之间的关系,揭示了它们之间的对偶约束.分形算子的解表明,分形单元的拓扑不变量是决定分数阶数的重要因素之一.提出了分形结构中分数时间算子具体阶数的一个猜想.

SOME PROBLEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN FIELD THEORY

This paper is a brief introduction to Yang-Mills-Higgs model, MaxwellHiggs model, Einstein’s vacuum model, Yang-Baxter model, Chern-SimonsHiggs model and a discussion of the associated partial differential equation problems.

Some Characterizations of the Properties ■~∞ and LB_∞

The aim of this paper is to give some properties of the linear topological invariant LB^-∞ Using these results we show that a nuclear Fréchet space F has the property LB∞ if and only if every separately holomorphic function on an open subset U × V of E × F^* has a local Dirichlet representation,where E is a nuclear Fréchet space with the property LB^-∞ having a basis.