Topological phase transition in compressed van der Waals superlattice heterostructure BiTeCl/HfTe_(2)

Based on first-principles calculations,we investigate the electronic band structures and topological properties of heterostructure BiTeCl/HfTe_(2) under c-direction strain.In the primitive structure,this material undergoes a phase transition from an insulator with a narrow indirect gap to a metal by strong spin-orbital coupling.When strain effect is considered,band inversion at time-reversal invariant point Z is responsible for the topological phase transition.These nontrivial topologies are caused by two different types of band crossings.The observable topological surface states in(110)surface also support that this material experiences topological phase transition twice.The layered heterostructure with van der Waals force provides us with a new desirable platform upon which to control topological phase transition and construct topological superconductors.

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 phase transitions driven by next-nearest-neighbor hopping in noncentrosymmetric cold Fermi gases

We investigate the topological phase marked by the Thouless–Kohmoto–Nightingale–Nijs（TKNN） number and the phase transitions driven by the next nearest neighbor（NNN） hopping in noncentrosymmetric cold Fermi gases, both spinsinglet pairing and spin-triplet pairing are considered. There exists a critical t＇c for the NNN hopping, at which the quantum phase transition occurs, and the system changes from an Abelian（non-Abelian） phase to a non-Abelian（Abelian） one. By numerically diagonalizing the Hamiltonian in the real space, the energy spectra with edge states for different topological phases and the Majorana zero modes are discussed. Although the spin-triplet pairing does not contribute to the gap closing and the phase diagram, it induces gapless states in the presence of a magnetic field, and the TKNN number in this region is still zero.

Characterization of topological phase of superlattices in superconducting circuits

The recent experimental observation of topological magnon insulator states in a superconducting circuit chain marks a breakthrough for topological physics with qubits, in which a dimerized qubit chain has been realized. Here, we extend such a dimer lattice to superlattice with arbitrary number of qubits in each unit cell in superconducting circuits, which exhibits rich topological properties. Specifically, by considering a quadrimeric superlattice, we show that the topological invariant(winding number) can be effectively characterized by the dynamics of the single-excitation quantum state through time-dependent quantities. Moreover, we explore the appearance and detection of the topological protected edge states in such a multiband qubit system. Finally, we also demonstrate the stable Bloch-like-oscillation of multiple interface states induced by the interference of them. Our proposal can be readily realized in experiment and may pave the way towards the investigation of topological quantum phases and topologically protected quantum information processing.

Electronic properties and topological phases of ThXY(X= Pb, Au, Pt and Y= Sb, Bi, Sn) compounds

The electronic properties and topological phases of ThXY （X = Pb, Au, Pt, Pd and Y = Sb, Bi, Sn） compounds in the presence of spin-orbit coupling, using density functional theory are investigated. The ThPtSn compound is stable in the ferromagnetic phase and the other ThXY compounds are stable in nonmagnetic phases. Band structures of these compounds in topological phases （insulator or metal） and normal phases within generalized gradient approximation （GGA） and Engel- Vosko generalized gradient approximation （GGA_EV） are compared. The ThPtSn, ThPtBi, ThPtSb, ThPdBi, and ThAuBi compounds have topological phases and the other ThXY compounds have normal phases. Band inversion strengths and topological phases of these compounds at different pressure are studied. It is seen that the band inversion strengths of these compounds are sensitive to pressure and for each compound a second-order polynomial fitted on the band inversion strengths-pressure curves.

Acoustic topological phase transition induced by band inversion of high-order compound modes and robust pseudospin-dependent transport

A simple two-dimensional phononic crystal hosting topologically protected edge states is proposed to emulate the quantum spin Hall effect in electronic systems, whose phononic topological phase can be reconfigured through the rotation of scatters. In particular, the band inversion occurs between two pairs of high-order compound states, resulting in topological phase transition from trivial to nontrivial over a relatively broad high-frequency range. This is further evidenced by an effective Hamiltonian derived by the k·p perturbation theory. The phononic topology is related to a pseudo-timereversal symmetry constructed by the point group symmetry of two doubly degenerate eigenstates. Numerical simulations unambiguously demonstrate robust helical edge states whose pseudospin indices are locked to the propagation direction along the interface between topologically trivial and nontrivial phononic crystals. Our designed phononic systems provide potential applications in robust acoustic signal transport along any desired path over a high-frequency range.

Topological phase transition in a ladder of the dimerized Kitaev superconductor chains

We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains.The topological class of the system is determined by the relative phase θ between the inter-and intra-chain superconducting pairing.One topological class is the class BDI characterized by the Z index,and the other is the class D characterized by the Z;index.For the two different topological classes,the topological phase diagrams of the system are presented by calculating two different topological numbers,i.e.,the Z index winding number W and the Z;index Majorana number M,respectively.In the case of θ=0,the topological class belongs to the class BDI,multiple topological phase transitions accompanying the variation of the number of Majorana zero modes are observed.In the case of θ = π/2 it belongs to the class D.Our results show that for the given value of dimerization,the topologically nontrivial and trivial phases alternate with the variation of chemical potential.

Topological phase transition in cavity optomechanical system with periodical modulation

We investigate the topological phase transition and the enhanced topological effect in a cavity optomechanical system with periodical modulation.By calculating the steady-state equations of the system,the steady-state conditions of cavity fields and the restricted conditions of effective optomechanical couplings are demonstrated.It is found that the cavity optomechanical system can be modulated to different topological Su–Schrieffer–Heeger(SSH)phases via designing the optomechanical couplings legitimately.Meanwhile,combining the effective optomechanical couplings and the probability distributions of gap states,we reveal the topological phase transition between trivial SSH phase and nontrivial SSH phase via adjusting the decay rates of cavity fields.Moreover,we find that the enhanced topological effect of gap states can be achieved by enlarging the size of system and adjusting the decay rates of cavity fields.

Topological phase diagrams and Majorana zero modes of the Kitaev ladder and tube

In this paper,we study two quasi-one-dimensional（1 D） Kitaev models with ladder-like and tube-like spatial structures,respectively.Our results provide the phase diagrams and explicit expressions of the Majorana zero modes.The topological phase diagrams are obtained by decomposing the topological invariants and the topological conditions for topologically nontrivial phases are given precisely.For systems which belongs to topological class BDI,we obtain the regions in the phase diagrams where the topological numbers show even-odd effect.For the Kitaev tube model a phase factor induced by the magnetic flux in the axial direction of the tube is introduced to alter the classification of the tube Hamiltonian from class BDI to D.The Kitaev tube of class D is characterized by the Z2 index when the number of chains is odd while 0,1,2 when the number of chains is even.The phase diagrams show periodic behaviors with respect to the magnetic flux.The bulk-boundary correspondence is demonstrated by the observations that the topological conditions for the bulk topological invariant to take nontrivial values are precisely those for the existence of the Majorana zero modes.

Effect of disorders on topological phases in one-dimensional optical superlattices

In a recent paper, Lang et al. proposed that edge states and topological phases can be observed in one-dimensional optical superlattices. They showed that the topological phases can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions. However, disorders are not considered in their model. To study the effect of disorders on the topological phases, we introduce random potentials to the model for optical superlattcies.Our calculations show that edge states are robust against the disorders. We find the edge states are very sensitive to the number of the sites in the optical superlattice and we propose a simple rule to describe the relationship between the edge states and the number of sites. The density plateaus are also robust against weak disorders provided that the average density is calculated over a long interval. The widths of the plateaus are proportional to the widths of the bulk energy gaps when there are disorders. The disorders can diminish the bulk energy gaps. So the widths of the plateaus decrease with the increase of disorders and the density plateaus disappear when disorders are too strong. The results in our paper can be used to guide the experimental detection of topological phases in one-dimensional systems.

Topological phase in one-dimensional Rashba wire

We study the possible topological phase in a one-dimensional（1D） quantum wire with an oscillating Rashba spin–orbital coupling in real space. It is shown that there are a pair of particle–hole symmetric gaps forming in the bulk energy band and fractional boundary states residing in the gap when the system has an inversion symmetry. These states are topologically nontrivial and can be characterized by a quantized Berry phase ±π or nonzero Chern number through dimensional extension. When the Rashba spin–orbital coupling varies slowly with time, the system can pump out 2 charges in a pumping cycle because of the spin flip effect. This quantized pumping is protected by topology and is robust against moderate disorders as long as the disorder strength does not exceed the opened energy gap.

Emergent topological ordered phase for the Ising-XY model revealed by cluster-updating Monte Carlo method

The two-component cold atom systems with anisotropic hopping amplitudes can be phenomenologically described by a two-dimensional Ising-XY coupled model with spatial anisotropy.At low temperatures,theoretical predictions[Phys.Rev.A 72053604(2005)]and[arXiv:0706.1609]indicate the existence of a topological ordered phase characterized by Ising and XY disorder but with 2XY ordering.However,due to ergodic difficulties faced by Monte Carlo methods at low temperatures,this topological phase has not been numerically explored.We propose a linear cluster updating Monte Carlo method,which flips spins without rejection in the anisotropy limit but does not change the energy.Using this scheme and conventional Monte Carlo methods,we succeed in revealing the nature of topological phases with half-vortices and domain walls.In the constructed global phase diagram,Ising and XY-type transitions are very close to each other and differ significantly from the schematic phase diagram reported earlier.We also propose and explore a wide range of quantities,including magnetism,superfluidity,specific heat,susceptibility,and even percolation susceptibility,and obtain consistent and reliable results.Furthermore,we observed first-order transitions characterized by common intersection points in magnetizations for different system sizes,as opposed to the conventional phase transition where Binder cumulants of various sizes share common intersections.The critical exponents of different types of phase transitions are reasonably fitted.The results are useful to help cold atom experiments explore the half-vortex topological phase.

Spin‑controlled topological phase transition in non‑Euclidean space

Modulation of topological phase transition has been pursued by researchers in both condensed matter and optics research fields,and has been realized in Euclidean systems,such as topological photonic crystals,topological metamaterials,and coupled resonator arrays.However,the spin-controlled topological phase transition in non-Euclidean space has not yet been explored.Here,we propose a non-Euclidean configuration based on Mobius rings,and we demonstrate the spin-controlled transition between the topological edge state and the bulk state.The Mobius ring,which is designed to have an 8πperiod,has a square cross section at the twist beginning and the length/width evolves adiabatically along the loop,accompanied by conversion from transverse electric to transverse magnetic modes resulting from the spin-locked effect.The 8πperiod Mobius rings are used to construct Su–Schrieffer–Heeger configuration,and the configuration can support the topological edge states excited by circularly polarized light,and meanwhile a transition from the topological edge state to the bulk state can be realized by controlling circular polarization.In addition,the spin-controlled topological phase transition in non-Euclidean space is feasible for both Hermitian and non-Hermitian cases in 2D systems.This work provides a new degree of polarization to control topological photonic states based on the spin of Mobius rings and opens a way to tune the topological phase in non-Euclidean space.

Topological phase transitions and Majorana zero modes induced by the periodic potential in an antiferromagnetic chain

We investigate the topological properties of an antiferromagnetic(AFM)chain with an on-site periodic potential,considering the intrinsic spin–orbit coupling and an external Zeeman field along with the nanowire.Our results indicate that Majorana zero modes(MZMs)can be observed by adjusting the strength of the periodic potential.We have calculated the energy spectrum,the wave-function and transport properties,and all these results support the existence of MZMs in the AFM chain.Additionally,multiple topological phase transitions occur as the strength of the periodic potential changes,and several regions support MZMs.

Various topological phases and their abnormal effects of topological acoustic metamaterials

The last 20 years have witnessed growing impacts of the topological concept on the branches of physics,including materials,electronics,photonics,and acoustics.Topology describes objects with some global invariant property under continuous deformation,which in mathematics could date back to the 17th century and mature in the 20th century.In physics,it successfully underpinned the physics of the Quantum Hall effect in 1984.To date,topology has been extensively applied to describe topological phases in acoustic metamaterials.As artificial structures,acoustic metamaterials could be well theoretically analyzed,on-demand designed,and easily fabricated by modern techniques,such as three-dimensional printing.Some new theoretical topological models were first discovered in acoustic metamaterials analogous to electronic counterparts,associated with novel effects for acoustics closer to applications.In this review,we focused on the concept of topology and its realization in airborne acoustic crystals,solid elastic phononic crystals,and surface acoustic wave systems.We also introduced emerging concepts of non-Hermitian,higher-order,and Floquet topological insulators in acoustics.It has been shown that the topology theory has such a powerful generality that among the disciplines from electron to photon and phonon,from electronic to photonics and acoustics,from acoustic topological theory to acoustic devices,could interact and be analogous to fertilize fantastic new ideas and prototype devices,which might find applications in acoustic engineering and noisevibration control engineering in the near future.

RKKY interaction in helical higher-order topological insulators

We theoretically investigate the Ruderman–Kittel–Kasuya–Yosida(RKKY) interaction in helical higher-order topological insulators(HOTIs), revealing distinct behaviors mediated by hinge and Dirac-type bulk carriers. Our findings show that hinge-mediated interactions consist of Heisenberg, Ising, and Dzyaloshinskii–Moriya(DM) terms, exhibiting a decay with impurity spacing z and oscillations with Fermi energy εF. These interactions demonstrate ferromagnetic behaviors for the Heisenberg and Ising terms and alternating behavior for the DM term. In contrast, bulk-mediated interactions include Heisenberg, twisted Ising, and DM terms, with a conventional cubic oscillating decay. This study highlights the nuanced interplay between hinge and bulk RKKY interactions in HOTIs, offering insights into designs of next-generation quantum devices based on HOTIs.

Topological edge and corner states of valley photonic crystals with zipper-like boundary conditions

We present a stable valley photonic crystal(VPC)unit cell with C_(3v)symmetric quasi-ring-shaped dielectric columns and realize its topological phase transition by breaking mirror symmetry.Based on this unit cell structure,topological edge states(TESs)and topological corner states(TCSs)are realized.We obtain a new type of wave transmission mode based on photonic crystal zipper-like boundaries and apply it to a beam splitter assembled from rectangular photonic crystals(PCs).The constructed beam splitter structure is compact and possesses frequency separation functions.In addition,we construct a box-shaped triangular PC structures with zipper-like boundaries and discover phenomena of TCSs in the corners,comparing its corner states with those formed by other boundaries.Based on this,we explore the regularities of the electric field patterns of TESs and TCSs,explain the connection between the characteristic frequencies and locality of TCSs,which helps better control photons and ensures low power consumption of the system.

Interacting topological magnons in a checkerboard ferromagnet

This work is devoted to studying the magnon-magnon interaction effect in a two-dimensional checkerboard ferromagnet with the Dzyaloshinskii-Moriya interaction.Using a first-order Green function method,we analyze the influence of magnon-magnon interaction on the magnon band topology.We find that Chern numbers of two renormalized magnon bands are different above and below the critical temperature,which means that the magnon band gap-closing phenomenon is an indicator for one topological phase transition of the checkerboard ferromagnet.Our results show that the checkerboard ferromagnet possesses two topological phases,and its topological phase can be controlled either via the temperature or the applied magnetic field due to magnon-magnon interactions.Interestingly,it is found that the topological phase transition can occur twice with the increase in the temperature,which is different from the results of the honeycomb ferromagnet.

Two-dimensional Sb net generated nontrivial topological states in SmAgSb_(2) probed by quantum oscillations

The REAgSb_(2)(RE = rare earth and Y) family has drawn considerable research interest because the two-dimensional Sb net in their crystal structures hosts topological fermions and hence rich topological properties. We report herein the magnetization and magnetotransport measurements of SmAgSb_(2) single crystal, which unveil very large magnetoresistance and high carrier mobility up to 6.2 × 10^(3)% and 5.58 × 10^(3)cm^(2)·V^(-1)·s^(-1), respectively. The analysis of both Shubnikov–de Haas and de Haas–van Alphen quantum oscillations indicates nontrivial Berry phases in the paramagnetic state while trivial Berry curvature in the antiferromagnetic state, indicating a topological phase transition induced by the antiferromagnetic order. It is also supported by the first-principles calculations. The results not only provide a new interesting topological material but also offer valuable insights into the correlation between magnetism and nontrivial topological states.

Simulation of higher-order topological phases and related topological phase transitions in a superconducting qubit

Higher-order topological phases give rise to new bulk and boundary physics,as well as new classes of topological phase transitions.While the realization of higher-order topological phases has been confirmed in many platforms by detecting the existence of gapless boundary modes,a direct determination of the higher-order topology and related topological phase transitions through the bulk in experiments has still been lacking.To bridge the gap,in this work we carry out the simulation of a twodimensional second-order topological phase in a superconducting qubit.Owing to the great flexibility and controllability of the quantum simulator,we observe the realization of higher-order topology directly through the measurement of the pseudo-spin texture in momentum space of the bulk for the first time,in sharp contrast to previous experiments based on the detection of gapless boundary modes in real space.Also through the measurement of the evolution of pseudo-spin texture with parameters,we further observe novel topological phase transitions from the second-order topological phase to the trivial phase,as well as to the first-order topological phase with nonzero Chern number.Our work sheds new light on the study of higher-order topological phases and topological phase transitions.