Nontrivial Topological Phase Research Articles

Critical and noncritical non-Hermitian topological phase transitions in one-dimensional chains

In this work we investigate non-Hermitian topological phase transitions using real-space edge states as a paradigmatic tool. We focus on the simplest non-Hermitian variant of the Su-Schrieffer-Heeger model, including a parameter that denotes the degree of non-Hermiticity of the system. We study the behavior of the zero-energy edge states in the nontrivial topological phases with integer and semi-integer topological winding numbers, according to the distance to the critical point. We find that, depending on the parameters of the model, the edge states may penetrate into the bulk, as expected in Hermitian topological phase transitions. We also show that, using the topological characterization of the exceptional points, we can describe the intricate chiral behavior of the edge states across the whole phase diagram. Moreover, we characterize the criticality of the model by determining the correlation length critical exponent directly from numerical calculations of the penetration length of the zero-mode edge states.

Stacking-induced Chern insulator

Graphene can be turned into a semimetal with broken time-reversal symmetry by adding a valley-dependent pseudo-scalar potential that shifts the Dirac point energies in opposite directions, as in the modified Haldane model. We consider a bilayer obtained by stacking two time-reversed copies of the modified Haldane model, where conduction and valence bands cross to give rise to a nodal line in each valleys. In the AB stacking, the interlayer hopping lifts the degeneracy of the nodal lines and induces a band repulsion, leading surprisingly to a chiral insulator with a Chern number $C=\pm2$. As a consequence a pair of chiral edge states appears at the boundaries of the ribbon bilayer geometry. In contrast, the AA stacking does not show nontrivial topological phases. We discuss possible experimental implementations of our results.

Progress on the antiferromagnetic topological insulator MnBi2Te4.

Topological materials, which feature robust surface and/or edge states, have now been a research focus in condensed matter physics. They represent a new class of materials exhibiting nontrivial topological phases, and provide a platform for exploring exotic transport phenomena, such as the quantum anomalous Hall effect and the quantum spin Hall effect. Recently, magnetic topological materials have attracted considerable interests due to the possibility to study the interplay between topological and magnetic orders. In particular, the quantum anomalous Hall and axion insulator phases can be realized in topological insulators with magnetic order. MnBi2Te4, as the first intrinsic antiferromagnetic topological insulator discovered, allows the examination of existing theoretical predictions; it has been extensively studied, and many new discoveries have been made. Here we review the progress made on MnBi2Te4 from both experimental and theoretical aspects. The bulk crystal and magnetic structures are surveyed first, followed by a review of theoretical calculations and experimental probes on the band structure and surface states, and a discussion of various exotic phases that can be realized in MnBi2Te4. The properties of MnBi2Te4 thin films and the corresponding transport studies are then reviewed, with an emphasis on the edge state transport. Possible future research directions in this field are also discussed.

Non-volatile control of topological phase transition in an asymmetric ferroelectric In2Te2S monolayer.

The coupling of topological electronic states and ferroelectricity is highly desired due to their abundant physical phenomenon and potential applications in multifunctional devices. However, it is difficult to achieve such a phenomenon in a single ferroelectric (FE) monolayer because the two polarized states are topologically equivalent. Here, we demonstrate that the symmetry of polarized states can be broken by constructing a Janus structure in a FE monolayer. We illustrate such a general idea by replacing a layer of Te atoms in the In2Te3 monolayer with S atoms. Using first-principles calculations, we show that the In2Te2S monolayer has two asymmetric polarized states, which are characterized by a metal and semiconductor, respectively. Importantly, as the spin-orbit coupling is included, a band gap (50.4 meV) is created in the metallic state, resulting in a non-trivial topological phase. Thus, it proves to be a feasible method to engineer non-volatile FE control of topological order in a single-phase system. We also demonstrate the underlying physical mechanism of topological phase transition, which is unveiled to be related to the weakened intrinsic electric field resulting from charge transfer. These interesting results provide a general way to design asymmetric FE materials and shed light on their potential application in non-volatile multifunctional devices.

Charge instability of topological Fermi arcs in chiral crystal CoSi

Topological boundary states emerged at the spatial boundary between topological non-trivial and trivial phases, are usually gapless, or commonly referred as metallic states. For example, the surface state of a topological insulator is a gapless Dirac state. These metallic topological boundary states are typically well described by non-interacting fermions. However, the behavior of topological boundary states with significant electron–electron interactions, which could turn the gapless boundary states into gapped ordered states, e.g., density wave states or superconducting states, is of great interest theoretically, but is still lacking evidence experimentally. Here, we report the observation of incommensurable charge density wave (CDW) formed on the topological boundary states driven by the electron–electron interactions on the (001) surface of CoSi. The wavevector of CDW varies as the temperature changes, which coincides with the evolution of topological surface Fermi arcs with temperature. The orientation of the CDW phase is determined by the chirality of the Fermi arcs, which indicates a direct association between CDW and Fermi arcs. Our finding will stimulate the search of more interactions-driven ordered states, such as superconductivity and magnetism, on the boundaries of topological materials.

Topological phase transition and skyrmions in a Janus MnSbBiSe2Te2 monolayer.

Using first-principles calculations and micro-magnetic simulations, we propose a Janus MnSbBiSe2Te2 (MSBST) monolayer derived from the MnBi2Te4 (MBT) ferromagnet and investigate the influence of biaxial strain on the electronic structures, topological characteristics and spin textures. Different from pristine MBT with an out-of-plane easy axis, the anisotropy of MSBST prefers an in-plane direction. Intriguingly, switching the easy axis direction of MSBST will significantly modify the band structure. Topological phase transition can be achieved by applying a compressive strain, making MSBST become a topological insulator with . Moreover, due to the inherent inversion asymmetry of Janus MSBST, a large Dzyaloshinskii-Moriya interaction (DMI) is induced for generating and stabilizing skyrmions. By micro-magnetic simulations, the results of spin textures show that the skyrmions phase can be achieved in MSBST with an external magnetic field of 0.8 T. Our findings provide guidelines for the development and application of spintronic devices with nontrivial topological properties and a large DMI.

Thermal Hall conductivity with sign change in the Heisenberg–Kitaev kagome magnet

The Heisenberg–Kitaev (HK) model on various lattices has attracted a lot of attention because it may lead to exotic states such as quantum spin liquid and topological orders. The rare-earth-based kagome lattice (KL) compounds and have q = 0, 120° order and canted ferromagnetic (CFM) order, respectively. Interestingly, the HK model on the KL has the same ground state long-range orders. In the theoretical phase diagram, the CFM phase resides in a continuous parameter region and there is no phase change across special parameter points, such as the Kitaev ferromagnetic (KFM) point, the ferromagnetic (FM) point and its dual FM point. However, a ground state property cannot distinguish a system with or without topological nontrivial excitations and related phase transitions. Here, we study the topological magnon excitations and related thermal Hall conductivity in the HK model on the KL with CFM order. The CFM phase can be divided into two regions related by the Klein duality, with the self dual KFM point as their boundary. We find that the scalar spin chirality, which is intrinsic in the CFM order, changes sign across the KFM point. This leads to the opposite Chern numbers of corresponding magnon bands in the two regions, and also the sign change of the magnon thermal Hall conductivity.

Detecting subsystem symmetry protected topological order through strange correlators

We employ strange correlators to detect 2D subsystem symmetry-protected topological (SSPT) phases which are nontrivial topological phases protected by subsystem symmetries. Specifically, we analytically construct efficient strange correlators in the 2D cluster model in the presence of a uniform magnetic field and then perform the projector Quantum Monte Carlo simulation within the quantum annealing scheme. We find that strange correlators show the long-range correlation in the SSPT phase, from which we define strange order parameters to characterize the topological phase transition between the SSPT phase at low fields and the trivial paramagnetic phase at high fields. Thus, the detection of the fully localized zero modes on the 1D physical boundary of SSPT phase has been transformed into the bulk correlation measurement about the local operators with the periodic boundary condition. We also find interesting spatial anisotropy of a strange correlator, which can be intrinsically traced back to the nature of spatial anisotropy of subsystem symmetries that protect SSPT order in the 2D cluster model. By simulating strange correlators, we, therefore, provide the first unbiased large-scale quantum Monte Carlo simulation on the easy and efficient detection in the SSPT phase and open the avenue of the investigation of the subtle yet fundamental nature of the novel interacting topological phases.

Spin direction tunable topological transition in two-dimensional frustrate antiferromagnetic triangular lattice T-FeO2 monolayer

Recently, numerous two-dimensional (2D) magnetic materials are predicted with various promising properties, whereas noncollinear antiferromagnetic 2D materials are rarely reported. In this paper, we predicted a stable 2D noncollinear antiferromagnetic triangular lattice T-FeO2. The ground state is 120° antiferromagnetic with stronger next nearest neighbor exchange coupling than that of nearest neighbor exchange coupling because of the RKKY interaction. Our Monte Carlo simulations reveal that the magnetic phase transition is from a 120° antiferromagnetic state to a stripy state and then to a paramagnetic state with increasing temperature. In addition, by tuning the spin direction from an in plane antiferromagnetic state to a canted weak ferromagnetic state, a nontrivial topological phase transition could be induced. Our investigation about magnetic property and nontrivial topological phase transition is very promising for antiferromagnetic spintronics.

Prediction of van Hove singularities, excellent thermoelectric performance, and non-trivial topology in monolayer rhenium dichalcogenides

Two-dimensional (2D) thermoelectric materials are gaining more intense attention with the ever-increasing global demand for energy. Recently, the search for compounds exhibiting excellent thermoelectric and topological characteristics has also attracted research exploration. In this study, a systematic investigation of Re-based transition metal dichalcogenides (TMDs) (ReX 2 , X = S, Se, and Te) through first-principles calculations identified the stability and electronic properties of the 1T dp (1 T double prime), 1 T’, 1 T, and 2H phases for the pristine bulk and monolayer ReX 2 . Formation energy and phonon dispersion calculations showed that the bulk and monolayer structures are only stable in the 1T dp structure. The calculated bandgaps of bulk under the hybrid functional approach are 1.567 eV, 1.429 eV, and 0.745 eV, while those of the monolayer phases are 1.902 eV, 1.658 eV, and 1.323 eV for ReS 2 , ReSe 2 , and ReTe 2 , respectively. Moreover, van Hove singularities (vHss) are observed in monolayer ReX 2 suggesting possible superconductivity. Remarkably, the calculated figure of merits (ZT) of bulk and monolayer ReX 2 with values up to 2.30 presents them as excellent materials for thermoelectric (TE) applications since good TE materials have ZT > 0.4. In addition, the effect of one- (1 h) and two-sided (2 h) hydrogenation on the structural, electronic, magnetic, and topological properties of monolayer ReX 2 were also investigated. Two-sided hydrogenation caused a stable structural phase transition from 1 T dp to 1T. A ferromagnetic phase transition was also observed upon the two-sided hydrogenation of ReS 2 (2h-ReS 2 ) and the one - sided hydrogenation of ReSe 2 (1h-ReSe 2 ). Finally, one-sided hydrogenation of monolayer ReSe 2 and ReTe 2 (1h-ReSe 2 and 1h-ReTe 2 ) resulted in non-trivial topological phases as confirmed by the calculated Z 2 and Chern topological invariant numbers. Our findings show that the 2D Re-based TMDs exhibit highly tunable properties which have the potential for thermoelectric and spintronics applications. • Monolayer ReX2 is thermodynamically stable in the distorted 1 T phase (1Tdp) • van Hove singularities are present in monolayer ReX2 suggesting superconductivity • Excellent thermoelectric properties are observed in bulk and monolayer ReX2 • Stable structural and magnetic phase transition occurred upon hydrogenation of ReX2 • Hydrogenation of monolayer ReX2 induced non-trivial topological phases

Electric-field switching of the antiferromagnetic topological state in a multiferroic heterobilayer

Coupling nontrivial topological physics to ferroelectricity in two-dimensional lattice is highly desirable in both fundamental research and devices applications. Here, using first-principles calculations, we report that in a multiferroic heterobilayer consisting of a antiferromagnetic layer MnSe and a ferroelectric layer ${\mathrm{In}}_{2}{\mathrm{S}}_{3}$, the typical type-III band alignment can be realized. Upon introduction of spin-orbit coupling, a band gap is created, giving rise to a nontrivial antiferromagnetic topological phase. By reversing ferroelectric polarization, the nontrivial antiferromagnetic topology of $\mathrm{MnSe}/{\mathrm{In}}_{2}{\mathrm{S}}_{3}$ can be annihilated, yielding a wide-gap antiferromagnetic semiconductor with trivial physics. It thus proves to be a feasible approach to realize purely electric-field control of antiferromagnetic topological physics in this heterobilayer. The physical mechanism of such phenomenon is further unveiled to be related to the interlayer charge transfer between the two layers. These findings shed light on the design and control of antiferromagnetic topological physics in two dimensions.

Thermodynamic signature of topological quantum phase transition in a two-leg Kitaev ladder

We study the topological quantum phase transitions (QPTs) and quantum criticality of a two-leg Kitaev ladder. In the absence of superconducting phase difference, the thermal Drude weight portrays well the gapped topological quantum phases (Dth=0) and gapless quantum criticality (Dth>0), which is further manifested by the critical energy spectrum with different gapless linear relation modes. The self-dual quantum criticality is demonstrated by the Grüneisen ratio (GR) Γ= constant irrelevant of temperature. Tuning the superconducting phase difference Φ=π, the system undergoes a gapped topological nontrivial phase transition into a gapless trivial one, manifested by the quadratic contact point at the center of Brillouin zone. Meanwhile, it is identified by the temperature dependent power-law divergence: Γ∼±T−1 at low temperature, which is similar to the general QPT. In particular, with the chemical potential vanishing, one can obtain full flat bands, giving rise to the discontinuous topological QPT. Furthermore, it is worth noting that the scaling of GR provides a new thermodynamic means to check the order of topological QPTs, which falls on a universal curve or not, indicating the continuous or discontinuous transition.

Anomalous π modes by Floquet engineering in optical lattices with long-range coupling.

Photonic Floquet topological insulators provide a powerful tool to manipulate the optical fields, which have been extensively studied with only nearest-neighbor coupling. Here, we demonstrate that nontrivial Floquet topological phase and photonic π modes are brought from long-range coupling in a one-dimensional periodically driven optical lattice. Interestingly, the long-range coupling is found to give rise to new Floquet π modes that do not exist in the traditional Floquet lattices. We interpret the underlying physics by analyzing the replica bands, which shows quasienergies band crossing and reopening of new nontrivial π gaps due to the long-range coupling. Our results provide a new route in manipulating optical topological modes by Floquet engineering with long-range coupling.

Dynamic generation of nonequilibrium superconducting states in a Kitaev chain

Non-equilibrium state can exhibit the same macroscopic properties, such as conductivity or superconductivity, as a static state when they share the identical average of an observable over a period of time. We investigate the quench dynamics of a Kitaev chain by introducing two kinds of order parameters which relate to two channels of pairing, local pair in real space and BCS-like pair in momentum space. Based on exact solutions, we find that two order parameters are identical for the ground state, indicating the balance between two kinds of pairing channels, and can identify the quantum phase diagram. However, for a non-equilibrium state obtained by the time evolution from initially prepared vacuum state, the two are different but both can still clearly identify the phase diagram. In the topologically non-trivial region, the non-equilibrium states prefer the BCS-like pairing state. Our finding provides an alternative way to dynamically generate a superconducting state from a trivial empty state and sheds light on the mechanism of pairing.

Is the continuum SSH model topological?

The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698–1701 (1979)] is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tight-binding asymptotic parameter λ.

Quantum Anomalous Hall Phase and Time‐Reversal‐Symmetry‐Broken Quantum Spin Hall Phase on the Square–Hexagon Lattice

The topological properties of the square–hexagon lattice are studied when the time‐reversal (TR) symmetry is broken. Based on the Chern number and spin Chern number, the topological trivial and nontrivial phases can be determined. When the spin‐dependent and ‐independent staggered potentials are taken into consideration, rich phase diagrams are obtained. A number of topological nontrivial phases, including the TR‐symmetry‐broken quantum spin Hall phase and quantum anomalous Hall (QAH) phase, can be achieved by tuning the spin‐dependent staggered potential. When the spin‐independent staggered potential is varied, it is possible to observe the topological phase transition from the metal phase to the QAH phase. The evolutions of the bulk bandgap and spin spectrum gap are also presented together. To support the identification of the topological phases, helical and chiral edge states are given numerically. Moreover, the corresponding density distribution and spin polarization are analyzed and discussed.

Spontaneous generation and active manipulation of real-space optical vortices.

Optical vortices are beams of light that carry orbital angularmomentum1, which represents an extra degree of freedom that can be generated and manipulated for photonic applications2-8. Unlike vortices in other physical entities, the generation of optical vortices requires structural singularities9-12, but this affects their quasiparticle nature and hampers the possibility of altering their dynamics or making them interacting13-17. Here we report a platform that allows the spontaneous generation and active manipulation of an optical vortex-antivortex pair using an external field. An aluminium/silicon dioxide/nickel/silicon dioxide multilayer structure realizes a gradient-thickness optical cavity, where the magneto-optic effects of the nickel layer affect the transition between a trivial and anon-trivialtopological phase. Rather than a structural singularity, the vortex-antivortex pairs present in the light reflected by our device are generated through mathematical singularities in the generalized parameter space of the top and bottom silicon dioxide layers, which can be mapped onto real space and exhibit polarization-dependent and topology-dependent dynamics driven by external magnetic fields. We expect that the field-induced engineering of optical vortices that we report will facilitate the study of topological photonic interactions and inspire further efforts to bestow quasiparticle-like properties to various topological photonic textures such as toroidal vortices, polarization and vortex knots, and optical skyrmions.

Observation of the topological phase transition from the spatial correlation of a biphoton in a one-dimensional topological photonic waveguide array.

We propose a simple method, using the first singular value (FSV) of the spatial correlation of biphotons, to characterize topological phase transitions (TPTs) in one-dimensional (1D) topological photonic waveguide arrays (PWAs). After analyzing the spatial correlation of biphotons using the singular value decomposition, we found that the FSV of the spatial correlation of biphotons in real space can characterize TPTs and distinguish between the topological trivial and nontrivial phases in PWAs based on the Su-Schrieffer-Heeger model. The analytical simulation results were demonstrated by applying the coupled-mode theory to biphotons and were found to be in good agreement with those of the numerical simulation. Moreover, the numerical simulation of the FSV (of the spatial correlation of biphotons) successfully characterized the TPT in a PWA based on the Aubry-André-Harper and Rice-Mele models, demonstrating the universality of this method for 1D topological PWAs. Our method provides biphotons with the possibility of acquiring information regarding TPTs directly from the spatial correlation in real space, and their potential applications in quantum topological photonics and topological quantum computing as quantum simulators and information carriers.

Observation of Anderson phase in a topological photonic circuit

Disordered systems play a central role in condensed matter physics, quantum transport, and topological photonics. It is commonly believed that a topological nontrivial phase would turn into a trivial phase where the transport vanishes under the effect of Anderson localization. Recent studies predict a counterintuitive result, that adding disorder to the trivial band structure triggers the emergence of protected edge states, the so-called topological Anderson phase. Here, we experimentally observe such a topological Anderson phase in a CMOS-compatible nanophotonic circuit, which implements the Su-Schrieffer-Heeger (SSH) model with incommensurate disorder in the intercell coupling amplitudes. The existence of the Anderson phase is verified by the spectral method, based on the continuous detection of the nanoscale light dynamics at the edge. Our results demonstrate the inverse transition between distinct topological phases in the presence of disorder, as well as offering a single-shot measurement technique to study the light dynamics in nanophotonic systems.

Interface-induced topological phase and doping-modulated bandgap of two-dimensioanl graphene-like networks

Biphenylene is a new topological material that has attracted much attention recently. By amplifying its size of unit cell, we construct a series of planar structures as homogeneous carbon allotropes in the form of polyphenylene networks. We first use the low-energy effective model to prove the topological three periodicity for these allotropes. Then, through first-principles calculations, we show that the topological phase has the Dirac point. As the size of per unit cell increases, the influence of the quaternary rings decreases, leading to a reduction in the anisotropy of the system, and the Dirac cone undergoes a transition from type II to type I. We confirm that there are two kinds of non-trivial topological phases with gapless and gapped bulk dispersion. Furthermore, we add a built-in electric field to the gapless system by doping with B and N atoms, which opens a gap for the bulk dispersion. Finally, by manipulating the built-in electric field, the dispersion relations of the edge modes will be transformed into a linear type. These findings provide a hopeful approach for designing the topological carbon-based materials with controllable properties of edge states.