Bulk Topology Research Articles

Vortex superlattice induced second-order topological mid-gap states in first-order topological insulators and superconductors

Topological defects such as vortex and dislocations, support zero-energy localized states as a reflection of the bulk topology, in first-order topological insulators and superconductors. Furthermore, emergent first-order topological mid-gap states have been discovered driven by the magnetic vortex superlattice. However, whether the higher-order topological mid-gap states would emerge from the first-order topological insulators and superconductors with the vortex superlattice remains elusive. In this work, we propose vortex superlattice could induce second-order topological mid-gap states with staggered lattice spacings for vortices in first-order topological insulators and superconductors. These higher-order topological mid-gap states originate from the staggered tunneling between vortex-induced bound states and the emergent π flux on vortex superlattices, as an intrinsic exhibition of the interplay between vortices and bulk topology for the first-order topological states. Our work uncovers higher-topological characteristics of topological-defect superlattice in first-order topological states, and develops a controllable environment for the creation and exploration of higher-order topological states.

Bulk-entanglement spectrum correspondence in PT - and PC -symmetric topological insulators and superconductors

In this study, we discuss a type of bulk-boundary correspondence, which holds for topological insulators and superconductors when the parity-time (PT) and/or parity-particle-hole (PC) symmetry are present. In these systems, even when the bulk topology is nontrivial, the edge spectrum is generally gapped, and thus the conventional bulk-boundary correspondence does not hold. We find that, instead of the edge spectrum, the single-particle entanglement spectrum becomes gapless when the bulk topology is nontrivial: i.e., the holds in PT- and/or PC-symmetric topological insulators and superconductors. After showing the correspondence using K-theoretic approach, we provide concrete models for each symmetry class up to three dimensions where nontrivial topology because of PT and/or PC is expected. An implication of our results is that, when the bulk topology under PT and/or PC symmetry is nontrivial, the noninteracting many-body entanglement spectrum is multiply degenerate in one dimension and is gapless in two or higher dimensions. Published by the American Physical Society 2024

Emerging topological multiferroics from the 2D Rice-Mele model

We introduce a two-dimensional dimerized lattice model that reveals a remarkable feature: the emergence of a complex, non-trivial topological multiferroic phase marked by zero Berry curvature and a significant Berry connection that influences the model’s bulk topology. This model extends the one-dimensional Rice-Mele Hamiltonian model to explore polarization-dependent topological properties in a 2D Su-Schrieffer-Heeger lattice, providing a detailed framework for studying the impact of symmetry-breaking and spatially varying potentials on electronic and spin properties. The findings are particularly relevant for spintronics, offering a foundation for topologically robust and electrically controlled spin-conducting edge states, with implications for developing advanced spin-dependent transport devices.

Phase- and temperature-driven chiral topological superfluids on a honeycomb lattice

The correlated spinful Haldane model exhibits rich topological phases consisting of chiral topological superfluids (TSFs) and topological spin density waves. However, most of previous studies mainly focus on the case with the fixed hopping phase or at zero temperature. In this paper, we study the attractive spinful Haldane model with arbitrary phase at finite temperature. The chiral TSFs with Chern number C = 2 and 4 emerge driven by the phase and temperature. In particular, the temperature can drive a C = 2 topological superfluid from a trivial normal insulator phase at an appropriate interaction. The bulk topology of all TSFs is uncovered by the Wilson loop method, and confirmed by the responses of edge dislocations.

Higher-order topological transport protected by boundary Chern number in phononic crystals

Topological pumps enable robust transports of topological states when the system parameters are varied in a cyclic process. The reported topological pumps are protected by the bulk topology. However, the exploration of topological pump protected by other mechanism remains elusive. Here we report our prediction and observation of higher-order topological pumps linked to the boundary topology, i.e., boundary Chern number. Based on such topological pump, the higher-order transports between the topological states of different dimensions (e.g., corner-edge-corner) are directly observed by spatial scanning of the sound field, and their topological robustness is observed in the paths with defects. Furthermore, modulated by the fundamental corner-edge-corner topological transport, topological splitting effects are unambiguously observed in our acoustic experiments. Our findings not only advance the research of the higher-order topological transports, but also offer good platforms to design unconventional devices.

Topological modes and spectral flows in inhomogeneous PT-symmetric continuous media

In classical Hermitian continuous media, the spectral-flow index of topological modes is linked to the bulk topology via index theorem. However, the interface between two bulks is usually non-Hermitian due to the inhomogeneities of system parameters. We show that the connection between topological modes and bulk topology still exists despite the non-Hermiticity at the interface if the system is endowed with PT symmetry. The theoretical framework developed is applied to the Hall magnetohydrodynamic model to identify a topological mode called topological Alfvén sound wave in magnetized plasmas. Published by the American Physical Society 2024

Crystalline finite-size topology

Topological phases stabilized by crystalline point group symmetry protection are a large class of symmetry-protected topological phases subjected to considerable experimental scrutiny. Here, we show that the canonical three-dimensional (3D) crystalline topological insulator protected by time-reversal symmetry T and fourfold rotation symmetry C4 individually or the product symmetry C4T, generically realizes finite-size crystalline topological phases in thin film geometry [a quasi-(3−1)-dimensional, or q(3−1)D, geometry]: response signatures of the 3D bulk topology coexist with topologically protected, quasi-(3−2)D, and quasi-(3−3)D boundary modes within the energy gap resulting from strong hybridization of the Dirac cone surface states of the underlying 3D crystalline topological phase. Importantly, we find qualitative distinctions between these gapless boundary modes and those of strictly 2D crystalline topological states with the same symmetry protection and develop a low-energy, analytical theory of the finite-size topological magnetoelectric response. Published by the American Physical Society 2024

Topological disclination mode in photonic Chern insulators

Topological defects in topological materials offer novel routes for creating topological modes and probing bulk topology. Disclination, a class of topological defects, has been recently shown to host fractional charges in topological crystalline insulators with well-defined Wannier centers. Here, we study the effects of disclinations in gyromagnetic photonic crystals with non-zero Chern numbers that prohibit the Wannier center picture. We find the emergence of topological disclination modes carrying orbital angular momentum from the interplay between the Chern-type topology and the effective flux induced by the disclination. When the Chern number changes its sign, the chirality of the disclination mode also flips, revealing the bulk-disclination correspondence. Furthermore, we perform numerical experiments to probe the disclination mode. Our results expand the study of disclination physics in photonic crystals to time-reversal-broken systems.

Entanglement spectroscopy of anomalous surface states

We study the entanglement spectra of surface states of symmetry-protected topological phases. The topological bulk imprints the surface with an anomaly that does not permit it to form a trivial “vacuum” state that is gapped, unfractionalized, and symmetry preserving. Any surface wave function encodes the topology of the underlying bulk in addition to a specific surface phase. We show that the real-space entanglement spectrum of such surface wave functions are dominated by the bulk topology and do not readily permit identifying the surface phase. We thus use a modified form of entanglement spectra that incorporates the anomaly and argue that they correspond to physical edge states between different surface states. We support these arguments by explicit analytical and numerical calculations for free and interacting surfaces of three-dimensional topological insulators of electrons. Published by the American Physical Society 2024

Andreev reflection in Euler materials

Many previous studies of Andreev reflection have demonstrated that unusual effects can occur in media which have a nontrivial bulk topology. Following this line of investigation, we study Andreev reflection by analysing a simple model of a bulk node with a generic winding number n > 0, where the even cases directly relate to topological Euler materials. We find that the magnitudes of the resultant reflection coefficients depend strongly on whether the winding is even or odd. Moreover this parity dependence is reflected in the differential conductance curves, which are highly suppressed for n even but not n odd. This gives a possible route through which the recently discovered Euler topology could be probed experimentally.

Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles

Topology plays a crucial role in many physical systems, leading to interesting states at the surface. A paradigmatic example is the Chern number defined in the Brillouin zone that leads to robust gapless edge states. Here we introduce the reduced Chern number, defined in the subregions of Brillouin zone, and construct a family of Chern dartboard insulators with quantized reduced Chern numbers but with trivial bulk topology. Chern dartboard insulators are protected by the mirror symmetries and exhibit distinct pseudospin textures, including (anti)skyrmions, inside the sub-Brillouin zone. These Chern dartboard insulators host exotic gapless edge states, such as Möbius fermions and midgap corner states, and can be realized in the photonic crystals. Our work opens up new possibilities for exploring sub-Brillouin zone topology and nontrivial surface responses in topological systems.

3D quantum Hall effect in a topological nodal-ring semimetal

A quantized Hall conductance (not conductivity) in three dimensions has been searched for more than 30 years. Here we explore it in 3D topological nodal-ring semimetals, by employing a minimal model describing the essential physics. In particular, the bulk topology can be captured by a momentum-dependent winding number, which confines the drumhead surface states in a specific momentum region. This confinement leads to a surface quantum Hall conductance in a specific energy window in this 3D system. The winding number for the drumhead surface states and Chern number for their quantum Hall effect form a two-fold topological hierarchy. We demonstrate the one-to-one correspondence between the momentum-dependent winding number and wavefunction of the drumhead surface states. More importantly, we stress that breaking chiral symmetry is necessary for the quantum Hall effect of the drumhead surface states. The analytic theory can be verified numerically by the Kubo formula for the Hall conductance. We propose an experimental setup to distinguish the surface and bulk quantum Hall effects. The theory will be useful for ongoing explorations on nodal-ring semimetals.

Bulk-local-density-of-state correspondence in topological insulators

In the quest to connect bulk topological quantum numbers to measurable parameters in real materials, current established approaches often necessitate specific conditions, limiting their applicability. Here we propose and demonstrate an approach to link the non-trivial hierarchical bulk topology to the multidimensional partition of local density of states (LDOS), denoted as the bulk-LDOS correspondence. In finite-size topologically nontrivial photonic crystals, we observe the LDOS partitioned into three distinct regions: a two-dimensional interior bulk area, a one-dimensional edge region, and zero-dimensional corner sites. Contrarily, topologically trivial cases exhibit uniform LDOS distribution across the entire two-dimensional bulk area. Our findings provide a general framework for distinguishing topological insulators and uncovering novel aspects of topological directional band-gap materials, even in the absence of in-gap states.

Photonic topological insulators induced by non-Hermitian disorders in a coupled-cavity array

Recent studies of disorder or non-Hermiticity induced topological insulators inject new ingredients for engineering topological matter. Here, we consider the effect of purely non-Hermitian disorders, a combination of these two ingredients, in a 1D coupled-cavity array with disordered gain and loss. Topological photonic states can be induced by increasing gain-loss disorder strength with topological invariants carried by localized states in the complex bulk spectra. The system showcases rich phase diagrams and distinct topological states from Hermitian disorders. The non-Hermitian critical behavior is characterized by the biorthogonal localization length of zero-energy edge modes, which diverges at the critical transition point and establishes the bulk-edge correspondence. Furthermore, we show that the bulk topology may be experimentally accessed by measuring the biorthogonal chiral displacement, which can be extracted from a proper Ramsey interferometer that works in both clean and disordered regions. The proposed coupled-cavity photonic setup relies on techniques that have been experimentally demonstrated and, thus, provides a feasible route toward exploring such non-Hermitian disorder driven topological insulators.

Unsupervised Learning of non‐Hermitian Photonic Bulk Topology

AbstractMachine‐learning has proven useful in distinguishing topological phases. However, there is still a lack of relevant research in the non‐Hermitian community, especially from the perspective of the momentum‐space. Here, an unsupervised machine‐learning method, diffusion maps, is used to study non‐Hermitian topologies in the momentum‐space. Choosing proper topological descriptors as input datasets, topological phases are successfully distinguished in several prototypical cases, including a line‐gapped tight‐binding model, a line‐gapped Floquet model, and a point‐gapped tight‐binding model. The datasets can be further reduced when certain symmetries exist. A mixed diffusion kernel method is proposed and developed, which could study several topologies at the same time and give hierarchical clustering results. As an application, a novel phase transition process is discovered in a non‐Hermitian honeycomb lattice without tedious numerical calculations. This study characterizes band properties without any prior knowledge, which provides a convenient and powerful way to study topology in non‐Hermitian systems.

Absence of Topological Protection of the Interface States in Z_{2} Photonic Crystals.

Inspired from electronic systems, topological photonics aims to engineer new optical devices with robust properties. In many cases, the ideas from topological phases protected by internal symmetries in fermionic systems are extended to those protected by crystalline symmetries. One such popular photonic crystal model was proposed by Wu and Hu in 2015 for realizing a bosonic Z_{2} topological crystalline insulator with robust topological edge states, which led to intense theoretical and experimental studies. However, a rigorous relationship between the bulk topology and edge properties for this model, which is central to evaluating its advantage over traditional photonic designs, has never been established. In this Letter, we revisit the expanded and shrunken honeycomb lattice structures proposed by Wu and Hu and show that they are topologically trivial in the sense that symmetric, localized Wannier functions can be constructed. We show that the Z and Z_{2} type classifications of the Wu-Hu model are equivalent to the C_{2}T protected Euler class and the second Stiefel-Whitney class, respectively, with the latter characterizing the full valence bands of the Wu-Hu model, indicating only a higher order topological insulator. Additionally, we show that the Wu-Hu interface states can be gapped by a uniform topology preserving C_{6} and T symmetric perturbation, which demonstrates the trivial nature of the interface. Our result reveals that topology is not a necessary condition for the reported helical edge states in many photonics systems and opens new possibilities for interface engineering that may not be constrained by topological considerations.

Topological Microlaser with a Non-Hermitian Topological Bulk.

Bulk-edge correspondence, with quantized bulk topology leading to protected edge states, is a hallmark of topological states of matter and has been experimentally observed in electronic, atomic, photonic, and many other systems. While bulk-edge correspondence has been extensively studied in Hermitian systems, a non-Hermitian bulk could drastically modify the Hermitian topological band theory due to the interplay between non-Hermiticity and topology, and its effect on bulk-edge correspondence is still an ongoing pursuit. Importantly, including non-Hermicity can significantly expand the horizon of topological states of matter and lead to a plethora of unique properties and device applications, an example of which is a topological laser. However, the bulk topology, and thereby the bulk-edge correspondence, in existing topological edge-mode lasers is not well defined. Here, we propose and experimentally probe topological edge-mode lasing with a well-defined non-Hermitian bulk topology in a one-dimensional (1D) array of coupled ring resonators. By modeling the Hamiltonian with an additional degree of freedom (referred to as synthetic dimension), our 1D structure is equivalent to a 2D non-Hermitian Chern insulator with precise mapping. Our Letter may open a new pathway for probing non-Hermitian topological effects and exploring non-Hermitian topological device applications.

Nonadiabatic dynamical characterization in arbitrary quenching processes of two-dimensional Chern insulators and three-dimensional chiral topological insulators

Recently, dynamical characterization of bulk topology has been experimentally realized under nonadiabatic sudden quench dynamics. However, it has been shown that only the topology of final phase can be characterized when the system is quenched from initial topologically trivial phase. In this paper, we make a thorough investigation of different types of quenching processes under nonadiabatic slow quench dynamics, and study not only the processes between nontrivial phase and trivial phase, but also between the phases with different topological invariants. We find that, under slow quench dynamics, both the initial and final topological phase can be characterized and the topological invariant can be captured by time-averaged spin polarization. Moreover, different types of processes can be distinguished from the special regions where the time-averaged spin polarization vanishes. Our findings are not only restricted to two-dimensional Chern insulators, but also are valid for three-dimensional chiral topological insulators. All the dynamical characterization schemes are entirely based on the experimentally measurable quantity time-averaged spin polarization, and thus one can expect our findings may provide reference for future experiments.

Exact solution of slow quench dynamics and nonadiabatic characterization of topological phases

Previous studies have shown that the bulk topology of single-particle systems can be captured by the band inversion surface or by the spin inversion surface emerging on the time-averaged spin polarization. Most of the studies, however, are based on the single-particle picture even though the systems are fermionic and multi-bands. Here, we study the slow quench dynamics of topological systems with all the valence bands fully occupied, and show that the concepts of band inversion surface and spin inversion surface are still valid. More importantly, the many-particle nonadiabatic quench dynamics is shown to be reduced to a new and nontrivial three-level Landau–Zener model. This nontrivial three-level Landau–Zener problem is then solved analytically by applying the integrability condition and symmetry considerations, and thus adds a new member to the few models that are exactly solvable. Based on the analytical results, the topological spin texture revealed by the time-averaged spin polarization can be applied to characterize the bulk topology and thus provides a direct comparison for future experiments.

Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators

Topological crystalline insulators (TCIs) can host anomalous surface states which inherits the characteristics of crystalline symmetry that protects the bulk topology. Especially, the diversity of magnetic crystalline symmetries indicates the potential for novel magnetic TCIs with distinct surface characteristics. Here, we propose a topological magnetic Dirac insulator (TMDI), whose two-dimensional surface hosts fourfold-degenerate Dirac fermions protected by either the {p}_{c}^{{prime} }4mm or p{4}^{{prime} }{g}^{{prime} }m magnetic wallpaper group. The bulk topology of TMDIs is protected by diagonal mirror symmetries, which give chiral dispersion of surface Dirac fermions and mirror-protected hinge modes. We propose candidate materials for TMDIs including Nd4Te8Cl4O20 and DyB4 based on first-principles calculations, and construct a general scheme for searching TMDIs using the space group of paramagnetic parent states. Our theoretical discovery of TMDIs will facilitate future research on magnetic TCIs and illustrate a distinct way to achieve anomalous surface states in magnetic crystals.