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 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.

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.

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.

New approach to measuring topological phase transitions utilizing Floquet technology

The Floquet technique provides a novel anomalous topological phase for non-equilibrium phase transitions.Based on the high symmetry of the quantum anomalous Hall model,the findings suggest a one-to-one correspondence between the average spin texture and the Floquet quasi-energy spectrum.A new approach is proposed to directly measure the quasienergy spectrum,replacing previous measurements of the average spin texture.Finally,we proposed a reliable experimental scheme based on ion trap platforms.This scheme markedly reduces the measurement workload,improves the measurement fidelity,and is applicable to multiple platforms such as cold atoms and nuclear magnetic resonance.

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.

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.

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.

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.

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.

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.

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.

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.

Topological properties of tetratomic Su-Schrieffer-Heeger chains with hierarchical long-range hopping

We propose a new generalized Su–Schrieffer–Heeger model with hierarchical long-range hopping based on a onedimensional tetratomic chain. The properties of the topological states and phase transition, which depend on the cointeraction of the intracell and intercell hoppings, are investigated using the phase diagram of the winding number. It is shown that topological states with large positive/negative winding numbers can readily be generated in this system. The properties of the topological states can be verified by the ring-type structures in the trajectory diagram of the complex plane. The topological phase transition is strongly related to the opening(closure) of an energy bandgap at the center(boundaries) of the Brillouin zone. Finally, the non-zero-energy edge states at the ends of the finite system are revealed and matched with the bulk–boundary correspondence.

Design of sign-reversible Berry phase effect in 2D magneto-valley material

Manipulating sign-reversible Berry phase effects is both fundamentally intriguing and practically appealing for searching for exotic topological quantum states.However,the realization of multiple Berry phases in the magneto-valley lattice is rather challenging due to the complex interactions from spin-orbit coupling(SOC),band topology,and magnetic ordering.Here,taking single-layer spin-valley RuCl_(2)as an example,we find that sign-reversible Berry phase transitions from ferrovalley(FV)to half-valley semimetal(HVS)to quantum anomalous valley Hall effect(QAVHE)can be achieved via tuning electronic correlation effect or biaxial strains.Remarkably,QAVHE phase,which combines both the features of quantum anomalous Hall and anomalous Hall valley effect,is introduced by sign-reversible Berry curvature or band inversion of d_(xy)/d_(x^(2)-y^(2))and d_(z^(2))orbitals at only one of the K/K′valleys of single-layer RuCl_(2).And the boundary of QAVHE phase is the HVS state,which can achieve 100%intrinsically valley polarization.Further,a k·p model unveiled the valleycontrollable sign-reversible Berry phase effects.These discoveries establish RuCl_(2)as a promising candidate to explore exotic quantum states at the confluence of nontrivial topology,electronic correlation,and valley degree of freedom.

Spin direction dependent quantum anomalous Hall effect in two-dimensional ferromagnetic materials

We propose a scheme for realizing the spin direction-dependent quantum anomalous Hall effect(QAHE)driven by spin-orbit couplings(SOC)in two-dimensional(2D)materials.Based on the sp^(3)tight-binding(TB)model,we find that these systems can exhibit a QAHE with out-of-plane and in-plane magnetization for the weak and strong SOC,respectively,in which the mechanism of quantum transition is mainly driven by the band inversion of p_(x,y)/p_(z)orbitals.As a concrete example,based on first-principles calculations,we realize a real material of monolayer 1T-SnN_(2)/PbN_(2)exhibiting the QAHE with in-plane/out-of-plane magnetization characterized by the nonzero Chern number C and topological edge states.These findings provide useful guidance for the pursuit of a spin direction-dependent QAHE and hence stimulate immediate experimental interest.