Distinct Topological Phases Research Articles

Type-II Dirac points and edge transports in phononic crystal plates

The accidentally degenerate type-II Dirac points in sonic crystal has been realized recently. However, elastic phononic crystals with type-II Dirac points have not yet been explored. In this work, we design a two-dimensional phononic crystal plate in square lattice with type-II Dirac points for elastic waves. The type-II Dirac points, different from the type-I counterparts, have the tiled dispersions and thus the iso-frequency contours become crossed lines. By tuning structures to break the mirror symmetry, the degeneracies of the type-II Dirac points are lifted, leading to a band inversion. In order to have a further explanation, we also calculate the Berry curvatures of phononic crystals with opposite structure parameters, and it turns out that these two crystals hold opposite signs around the valley. The phononic crystal plates before and after the band inversion belong to different topological valley phases, whose direct consequence is that the topologically protected gapless interface states exist between two distinct topological phases. Topologically protected interface states are found by calculating the projected band structures of a supercell that contains two kinds of interfaces between two topological phases. Robustness of the interface transport is verified by comparing the transmission rate for perfect interface with that for defective interface. Moreover, owing to the special stress field distributions of the elastic plate waves, the boundaries of a single phononic crystal phase can similarly host the gapless boundary states, which is found by calculating the projected band structures of a supercell with a single phase, thus having two free boundaries on the edges. This paper extends the two-dimensional Dirac points and valley states in graphene-like systems to the type-II cases, and obtains in the same structure the gapless interface and boundary propagations. Owing to the simple design scheme of the structure, the phononic crystal plates can be fabricated and scaled to a small size. Our system provides a feasible way of constructing high-frequency elastic wave devices.

Observation of Time-Reversal Invariant Helical Edge-Modes in Bilayer Graphene/WSe2 Heterostructure

Topological insulators, along with Chern insulators and quantum Hall insulator phases, are considered as paradigms for symmetry protected topological phases of matter. This article reports the experimental realization of the time-reversal invariant helical edge-modes in bilayer graphene/monolayer WSe2-based heterostructures-a phase generally considered as a precursor to the field of generic topological insulators. Our observation of this elusive phase depended crucially on our ability to create mesoscopic devices comprising both a moiré superlattice potential and strong spin-orbit coupling; this resulted in materials whose electronic band structure could be tuned from trivial to topological by an external displacement field. We find that the topological phase is characterized by a bulk bandgap and by helical edge-modes with electrical conductance quantized exactly to 2e2/h in zero external magnetic field. We put the helical edge-modes on firm ground through supporting experiments, including the verification of predictions of the Landauer-Büttiker model for quantum transport in multiterminal mesoscopic devices. Our nonlocal transport properties measurements show that the helical edge-modes are dissipationless and equilibrate at the contact probes. We achieved the tunability of the different topological phases with electric and magnetic fields, which allowed us to achieve topological phase transitions between trivial and multiple, distinct topological phases. We also present results of a theoretical study of a realistic model which, in addition to replicating our experimental results, explains the origin of the topological insulating bulk and helical edge-modes. Our experimental and theoretical results establish a viable route to realizing the time-reversal invariant topological phase of matter.

Proximity‐Induced Magnetism Enhancement Emerged in Chiral Magnet MnSi/Topological Insulator Bi2Se3 Bilayer

AbstractA proximity effect between magnetic materials and topological surface states can generate and modulate the localized spins without complicated material structures, but its origin is not clearly verified. MnSi single layer and MnSi/Bi2Se3 bilayer on Al2O3(001) substrates are fabricated by magnetron co‐sputtering and molecular beam epitaxy systems, in which a large proximity effect between the chiral magnetic structure and the topological surface states is manifested. The magnetic and electronic properties of both samples are meticulously compared and the proximity‐induced magnetism enhancement in the MnSi/Bi2Se3 bilayer is found. Interestingly, this effect persists up to temperatures above 300 K. Furthermore, for the MnSi/Bi2Se3 bilayer, the increase of charge carrier density and the decrease of carrier mobility near the Curie temperature TC = 40 K are observed, which can mediate the ferromagnetic exchange interaction enhancing the magnetization. The finding provides insight into a new platform to consist of materials with distinct topological phases for future spintronic devices.

Mesoscopic conductance fluctuations and noise in disordered Majorana wires

Superconducting wires with broken time-reversal and spin-rotational symmetries can exhibit two distinct topological gapped phases and host bound Majorana states at the phase boundaries. When the wire is tuned to the transition between these two phases and the gap is closed, Majorana states become delocalized leading to a peculiar critical state of the system. We study transport properties of this critical state as a function of the length $L$ of a disordered multichannel wire. Applying a non-linear supersymmetric sigma model of symmetry class D with two replicas, we identify the average conductance, its variance and the third cumulant in the whole range of $L$ from the Ohmic limit of short wires to the regime of a broad conductance distribution when $L$ exceeds the correlation length of the system. In addition, we calculate the average shot noise power and variance of the topological index for arbitrary $L$. The general approach developed in the paper can also be applied to study combined effects of disorder and topology in wires of other symmetries.

Chern insulator transitions with Wilson fermions on a hyperrectangular lattice

A $U(1)$ gauge theory coupled to a Wilson fermion on a $2+1$ dimensional cubic lattice is known to exhibit Chern insulator like topological transitions as a function of the the ratio $M/R$ where $M$ is the fermion mass and $R$ is the Wilson parameter. I show that, with $M$ and $R$ held fixed, a rectangular lattice with anisotropic lattice spacing can exhibit distinct topological phases as a function of the lattice anisotropy. As a consequence, a $2+1$ dimensional lattice theory without any domain wall in the fermion mass can still exhibit chiral edge modes on a $1+1$ dimensional defect across which lattice spacing changes abruptly. Likewise, a domain wall in the fermion mass on a uniform rectangular lattice can exhibit discrete changes in the number and chirality of zero modes as a function of lattice anisotropy. The construction presented in this paper can be generalized to higher dimensional space-time lattices.

First-principles study of the topological surface states of α-Sn(111)

α-Sn is on the boundary of a couple of distinct topological phases. It will transform into a topological insulator under a suitable strain. However, a clear picture of its topological surface states (TSSs) is still lacking. Here we perform first-principles calculations on the electronic structure of α-Sn(111) surface to identify its TSSs and reveal their properties. The results show that the presence of valence band reorganizes the TSSs in the inverted sp gap into two Dirac cones. The lower one is in the valence band continuum; the upper one resides in the gap between the valence and conduction bands. We also demonstrate the transformation of the surface states by switching on or off of strain and/or spin-orbit coupling. Without spin-orbit coupling, only the TSSs associated with the lower Dirac cone survive, and they are spin unpolarized. The results are useful for understanding and engineering the topological properties of α-Sn.

Fragile phonon topology on the honeycomb lattice with time-reversal symmetry

We use the methods of topological quantum chemistry to explore the topology of phonons on time-reversal symmetric crystals with the structure of the planar honeycomb (layer group $p6/mmm$). This approach is not tied to a particular model of atomic vibrations, but is applied to the most general dynamical matrix constrained only by the symmetries of the system. We show that four distinct fragile topological phonon phases are generically possible. Truncating the dynamical matrix to third nearest neighbors yields a model that realizes the different phonon topologies, characterized by the existence of phononic edge and corner modes and by Wilson loops with winding numbers one and two. Fitting the dynamical matrix to the DFT phonon bands shows that graphene is not very far from a topologically nontrivial phonon phase.

Interfaces and the extended Hilbert space of Chern-Simons theory

The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling.

The combination of spin-orbit coupling with interactions results in many exotic phases of matter. In this Letter, we investigate the superconducting pairing instability of the two-dimensional extended Hubbard model with both Rashba and Dresselhaus spin-orbit coupling within the mean-field level at both zero and finite temperature. We find that both first- and second-order time-reversal symmetry breaking topological gapped phases can be achieved under appropriate parameters and temperature regimes due to the presence of a favored even-parity s+id-wave pairing even in the absence of an external magnetic field or intrinsic magnetism. This results in two branches of chiral Majorana edge states on each edge or a single zero-energy Majorana corner state at each corner of the sample. Interestingly, we also find that not only does tuning the doping level lead to a direct topological phase transition between these two distinct topological gapped phases, but also using the temperature as a highly controllable and reversible tuning knob leads to different direct temperature-driven topological phase transitions between gapped and gapless topological superconducting phases. Our findings suggest new possibilities in interacting spin-orbit coupled systems by unifying both first- and higher-order topological superconductors in a simple but realistic microscopic model.

Topological States Due to Third‐Neighbor Coupling in Diatomic Linear Elastic Chains

The vibrations of a diatomic linear chain exhibit two bands, the acoustic and the optical one. In the case of equal masses and nearest‐neighbor spring constants, no gap exists. The effects of nearest, next‐nearest, and third‐nearest neighbor coupling with different spring constants are analyzed. Nearest neighbor coupling leads to two topologically different bulk states with winding numbers ν = 0 and −1. Next‐nearest neighbor coupling does not change the topological properties and introduces only a trivial “Semenov” gap. Third‐nearest neighbor coupling leads to four distinct topological bulk phases with winding numbers .

Magnetic-field-induced topological phase transition in Fe-doped (Bi,Sb)2Se3 heterostructures

Three-dimensional topological insulators (3D-TIs) possess a specific topological order of electronic bands, resulting in gapless surface states via bulk-edge correspondence. Exotic phenomena have been realized in ferromagnetic TIs, such as the quantum anomalous Hall (QAH) effect with a chiral edge conduction and a quantized value of the Hall resistance ${R_{yx}}$. Here, we report on the emergence of distinct topological phases in paramagnetic Fe-doped (Bi,Sb)${_2}$Se${_3}$ heterostructures with varying structure architecture, doping, and magnetic and electric fields. Starting from a 3D-TI, a two-dimensional insulator appears at layer thicknesses below a critical value, which turns into an Anderson insulator for Fe concentrations sufficiently large to produce localization by magnetic disorder. With applying a magnetic field, a topological transition from the Anderson insulator to the QAH state occurs, which is driven by the formation of an exchange gap owing to a giant Zeeman splitting and reduced magnetic disorder. Topological phase diagram of (Bi,Sb)${_2}$Se${_3}$ allows exploration of intricate interplay of topological protection, magnetic disorder, and exchange splitting.

Robustness of entanglement as an indicator of topological phases in quantum walks

How to reveal topological phases and their boundaries is an intriguing issue in various systems. Entanglement, which plays a fundamental role in quantum information, has been found profoundly related to topological phases. However, experimentally exploring this relation is precluded by the limited ability to obtain entanglement in many-body systems. In this work, we propose and experimentally demonstrate that the robustness of entanglement, quantified by the von Neumann entropy, can be used to reveal the topological phase with winding number W = 1 and topological phase with W = 0 in quantum walks. With the different robustness of entanglement against perturbations of a parameter, the phase boundaries between the distinct topological phases can be further determined. As a result, our work not only offers a new perspective for quantum walks but also exhibits the deep connection between entanglement and topological physics.

Fractional quantum transport and staggered topological transition in a lossy trimerized lattice

Transport in a non-Hermitian trimerized lattice with one lossy site per unit cell exhibits a quantized mean displacement before decay characterized by a two-step phase transition from zero to unity through a fractional value. Such a peculiar behavior, sensitive to both the lattice parameters and the initial state, is shown to be closely related to the presence of two nondegenerate dark states decoupled from the lossy sites and correspond to two staggered band closing points. We further check that distinct topological phases can also be characterized by the total Zak phase and by the number of midgap edge states. Moreover, we demonstrate that the edge states are topologically protected and their localization can be controlled via the couplings imbalance. Our results are expected to be relevant for implementations of topological photonics such as those based on arrays of evanescently coupled waveguides.

Fixed Points and Dynamic Topological Phenomena in a Parity-Time-Symmetric Quantum Quench.

SummaryWe identify dynamic topological phenomena such as dynamic Chern numbers and dynamic quantum phase transitions in quantum quenches of the non-Hermitian Su-Schrieffer-Heeger Hamiltonian with parity-time symmetry. Their occurrences in the non-unitary dynamics are intimately connected with fixed points in the Brillouin zone, where the density matrices do not evolve in time. Based on our theoretical formalism characterizing topological properties of non-unitary dynamics, we prove the existence of fixed points for quenches between distinct static topological phases in the -symmetry-preserving regime, thus unveiling the interplay between dynamic topological phenomena and symmetry. Interestingly, non-Hermiticity of the driving Hamiltonian gives rise to rich dynamic topological phenomena which are different, either qualitatively or quantitatively, from their counterparts in unitary dynamics. Our work sheds light on dynamic topological phenomena in open systems and is readily accessible in experiments.

Topologically protected edge states of phoxonic crystals

In this paper, we designed a phoxonic crystal (PxC) which simultaneously exhibit the topological states of sound and light. By modulating the impedance ratio between the cylinder scatterers and the matrix in a honeycomb lattice, we found that the inevitable Dirac cones and the accidental double Dirac cone of the phononic mode could be respectively degenerated at the corner and center of the Brillouin zone (BZ). For the photonic mode, we can find that the transverse-magnetic (TM) bands degenerate to Dirac cones at the corner of the BZ and the transverse-electric (TE) bands degenerate to a position-varying Dirac cone along the M-Γ line of the BZ. When the point group symmetry of the PxC is reduced from C6v to C3v, the double Dirac cone of the phononic mode is lifted and the topological pseudospin-dependent edge states of sound are induced when we construct a ribbon-shaped supercell with zigzag interface. Under the same operation, the topological valley spin state of the TM mode is excited and the edge state of the TE mode is also realized at the same time. Based on the all full-wave simulations, we simultaneously observed the topologically protected edge propagations of sound and light along the interfaces of two PxCs with the distinct topological phases. Our PxC provides a platform to design the topological optomechanical devices.

Fractional charged edge states in ladder topological insulators

We propose a model of two-leg ladder topological insulator in which the spin–orbit couplings are presented in both intra-chain and inter-chain interactions. The topological phase supports four fractional charged edge states localized at opposite ends of the ladder, which belongs to the chiral symplectic (CII) class protected by time-reversal symmetry and chiral symmetry. In our model, the presence of time-reversal and chiral symmetry generates fourfold degeneracy for the edge states, and the two edge states with same chirality at one end of the ladder each carries half charge. In contrast to the two edge states spatially localized at one end of the ladder being not distinguished, these two edge states can be detected by the momentum density. The experimental scheme for realizing our model with cold atoms in optical lattice is discussed. By introducing a magnetic field in the x direction, the system is driven from CII class to AIII class. In AIII class, there exist two distinct topological phases that exhibit four degenerate edge states and two degenerate edge states in the gap, respectively. As same as the system in CII class, each edge state carries a half charge in AIII class.

Simulating Dynamic Quantum Phase Transitions in Photonic Quantum Walks.

Signaled by nonanalyticities in the time evolution of physical observables, dynamic quantum phase transitions (DQPTs) emerge in quench dynamics of topological systems and possess an interesting geometric origin captured by dynamic topological order parameters (DTOPs). In this Letter, we report the experimental study of DQPTs using discrete-time quantum walks of single photons. We simulate quench dynamics between distinct Floquet topological phases using quantum-walk dynamics and experimentally characterize DQPTs and the underlying DTOPs through interference-based measurements. The versatile photonic quantum-walk platform further allows us to experimentally investigate DQPTs for mixed states and in parity-time-symmetric nonunitary dynamics for the first time. Our experiment directly confirms the relation between DQPTs and DTOPs in quench dynamics of topological systems and opens up the avenue of simulating emergent topological phenomena using discrete-time quantum-walk dynamics.

Interacting Hofstadter Interface.

Two-dimensional topological insulators possess conducting edge states at their boundary while being insulating in the bulk. We investigate the edge state emergent at a smooth topological phase boundary of interacting fermions within a full real-space analysis of the time-reversal invariant Hofstadter-Hubbard model. We characterize the localization of the edge state and the topological phase boundary by means of the local compressibility, the spectral density, a generalized local spin Chern marker as well as the Hall response and find good agreement between all these quantities. Computing the edge state spectra at the interface we observe robustness of the edge state against fermionic two-body interactions and conclude that interactions only shift its position. Hence the bulk-boundary correspondence for the interacting system is confirmed. Since experimental probing of edge states remains a challenge in ultracold atom setups, we propose the detection of the local compressibility by measuring correlations with a quantum gas microscope.

Complete characterization of spin chains with two Ising symmetries

Spin chains with two Ising symmetries are the Jordan-Wigner duals of one-dimensional interacting fermions with particle-hole and time-reversal symmetry. From earlier works on Majorana chains, it is known that this class of models has 10 distinct topological phases. In this paper, we analyze the physical properties of the correspondent 10 phases of the spin model. In particular, thanks to a set of two non-commuting dualities, we determine the local and non-local order parameters of the phases. We find that 4 phases are topologically protected by the Ising symmetries, while the other 6 break at least one symmetry. Our study highlights the non-trivial relation between the topological classifications of interacting bosons and fermions.

Ideal intersecting nodal-ring phonons in bcc C8

Carbon, a basic versatile element in our universe, exhibits rich varieties of allotropic phases, most of which possess promising nontrivial topological fermions. In this work, we identify a distinct topological phonon phase in a realistic carbon allotrope with a body-centered cubic structure, termed bcc-C$_{8}$. We show by symmetry arguments and effective model analysis that there are three intersecting phonon nodal rings perpendicular to each other in different planes. The intersecting phonon nodal rings are protected by time-reversal and inversion symmetries, which quantize the corresponding Berry phase into integer multiples of $\pi$. Unlike the electron systems, the phonon nodal rings in bcc-C$_{8}$ are guaranteed to remain gapless due to the lack of spin-orbital coupling. The nearly flat drumhead surface states projected on semi-infinite (001) and (110) surfaces of bcc-C$_{8}$ are clearly visible. Our findings not only discover promising nodal ring phonons in a carbon allotrope, but also provide emergent avenues for exploring topological phonons beyond fermionic electrons in carbon-allotropic structures with attractive features.