Topological phases and edge modes of an uneven ladder

We investigate the topological properties of a two-chain quantum ladder with uneven legs,i.e.,the two chains differ in their periods by a factor of 2.Such an uneven ladder presents rich band structures classified by the closure of either direct or indirect bandgaps.It also provides opportunities to explore fundamental concepts concerning band topology and edge modes,including the difference of intracellular and intercellular Zak phases,and the role of the inversion symmetry(IS).We calculate the Zak phases of the two kinds and find excellent agreement with the dipole moment and extra charge accumulation.We also find that configurations with IS feature a pair of degenerate two-side edge modes emerging as the closure of the direct bandgap,while configurations without IS feature one-side edge modes emerging as not only the closure of both direct and indirect bandgaps but also within the band continuum.Furthermore,by projecting to the two sublattices,we find that the effective Bloch Hamiltonian corresponds to that of a generalized Su–Schrieffer–Heeger model or the Rice–Mele model whose hopping amplitudes depend on the quasimomentum.In this way,the topological phases can be efficiently extracted through winding numbers.We propose that uneven ladders can be realized by spin-dependent optical lattices and their rich topological characteristics can be examined by near future experiments.

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.

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.

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.

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.

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.

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.

Finite-size-induced non-Hermitian phase transitions in real space

While non-Hermiticity provokes intriguing phenomena without Hermitian counterparts, e.g., the skin effect and the breakdown of bulk-boundary correspondence, attracting extensive attention both in fundamental physics and device engineering, the role of finite sizes therein remains elusive. Here, we propose a class of finite-size-induced non-Hermitian phase transitions, relying upon higher-order topological invariants associated with real-space wave functions. The phase diagrams for general non-Hermitian chiral models are further acquired to demonstrate our topological definition. Such phase transitions are elucidated qualitatively by an effective intercell coupling alteration that depends on finite sizes in respective directions. Besides, we mimic these phenomena by analogizing the circuit Laplacian in finite-size electric circuits with nonreciprocal couplings. The resultant admittance spectra agree with our theoretical predictions. Our findings shed light on the finite-size mechanism of non-Hermitian topological phase transitions and pave the way for applications in switching and sensing.

What Can Topology Bring to Chemistry?

The unique mathematical perspectives and principal concepts of topology have not only promoted the development of other branches of mathematics but have also deeply influenced other subjects,particularly physics and chemistry.This minireview aims to elucidate the substantial influence of topology on chemistry by critically examining both the contributions of topology to current chemical science and its potential future impacts.We will discuss the topology of molecular structures and assemblies across various scales—from small molecules and mechanically interlocked molecules to polymers,biomacromolecules,and supramolecular networks.This discussion will include an exploration of cutting-edge techniques for characterizing topological features,underscoring the role of topology as a novel paradigm for the design and synthesis of new molecular assemblies.Furthermore,a critical analysis of topology’s role in the development of functional materials—such as photonic materials,polymers,biomacromolecules,and chiral materials—will demonstrate its emerging significance as a crucial parameter in unveiling novel properties and functionalities.Given topology’s formidable potential,it is anticipated to be used increasingly to address pivotal challenges within chemistry and adjacent disciplines.

Simulation and detection of the topological properties of a modulated Rice-Mele model in a one-dimensional circuit-QED lattice

Here, we propose a simple scheme to realize a one-dimensional （1D） modulated Rice-Mele model （RMM） and investigate its topological properties with a 1D circuit quantum electrodynamics （QED） lattice. The system can be mapped into a Chern insulator model by introducing a period parameter. Interestingly and surprisingly, we found that the circuit-QED lattice system always exhibits topologically nonttrvial phases if both the nearest-neighbor hopping strength between two resonators and the qubitassisted on-site potentials are alternately changed in the direction of the lattice. The numerical results show that the topological phases can be obtained by introducing an additional modulation parameter and both the edge state and topological invariant can be unambiguously seen with the existence of decay and disorders, even with few resonators in the lattice.

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.

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.

High-throughput screening for Weyl semimetals with S_(4) symmetry

Based on irreducible representations(or symmetry eigenvalues) and compatibility relations(CR), a material can be predicted to be a topological/trivial insulator(satisfying CR) or a topological semimetal(violating CR). However, Weyl semimetals(WSMs) usually go beyond this symmetry-based strategy. In other words, Weyl nodes could emerge in a material, no matter if its occupied bands satisfy CR, or if the symmetry indicators are zero. In this work, we propose a new topological invariant v for the systems with S4 symmetry(i.e., the improper rotation S_(4)(≡IC_(4z)) is a proper fourfold rotation(C_(4z)) followed by inversion(I)), which can be used to diagnose the WSM phase. Moreover, v can be easily computed through the onedimensional Wilson-loop technique. By applying this method to the high-throughput screening in our first-principles calculations, we predict a lot of WSMs in both nonmagnetic and magnetic compounds.Various interesting properties(e.g., magnetic frustration effects, superconductivity and spin-glass order,etc.) are found in predicted WSMs, which provide realistic platforms for future experimental study of the interplay between Weyl fermions and other exotic states.

Algebraic equations and non-integer orders of fractal operators abstracted from biomechanics

The previous study implies that the fractional operator may stem from the motion in fractal space or the motion of fractal structure abstracted from biological systems.This study is devoted to answering another essential question,i.e.,what determines the order of the fractional operator in fractal structure?This paper generalizes the concept of the fractal cell defined in the previous paper,explores the tree-like and net-like fractal structures with higher-order topology,abstracts two classes of higherorder fractal operators,and derives the algebraic equations satisfied by the fractal operators to answer this question.It is proved that the solutions of the algebraic equations for fractal operators are deterministically related to the fractional-time operators that are usually of fractional orders.By the Vieta theorem,the relation between the solutions of algebraic equations for fractal operators and the physical-component operators is clarified,and the duality constraints between them are revealed.The solutions of the fractal operators show that the topological invariants of the fractal cells are one of the essential factors in determining the fractional orders.A conjecture on the specific order of the fractional-time operator in fractal structure is proposed.

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.

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.