Chern Insulator Research Articles

Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number

As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the robust properties of condensed-matter systems immune to the existence of disorder. The bulk–boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.

Multiple Brillouin Zone Winding of Topological Chiral Edge States for Slow Light Applications.

Photonic Chern insulators are known for their topological chiral edge states (CESs), whose absolute existence is determined by the bulk band topology, but concrete dispersion can be engineered to exhibit various properties. For example, the previous theory suggested that the edge dispersion can wind many times around the Brillouin zone to slow down light, which can potentially overcome fundamental limitations in conventional slow-light devices: narrow bandwidth and keen sensitivity to fabrication imperfection. Here, we report the first experimental demonstration of this idea, achieved by coupling CESs with resonance-induced nearly flat bands. We show that the backscattering-immune hybridized CESs are significantly slowed down over a relatively broad bandwidth. Our work thus paves an avenue to broadband topological slow-light devices.

Intrinsically topological second Chern insulator via synthetic dimensions.

Exploring Hidden Dimensions: Unveiling Topological Crystals in a 4D Space.

Field induced Chern insulating states in twisted monolayer–bilayer graphene

Unraveling the mechanism underlying topological phases, notably the Chern insulators (ChIs) in strong correlated systems at the microscopy scale, has captivated significant research interest. Nonetheless, ChIs harboring topological information have not always manifested themselves, owing to the constraints imposed by displacement fields in certain experimental configurations. In this study, we employ density-tuned scanning tunneling microscopy (DT-STM) to investigate the ChIs in twisted monolayer–bilayer graphene (tMBG). At zero magnetic field, we observe correlated metallic states. While under a magnetic field, a metal–insulator transition happens and an integer ChI is formed emanating from the filling index s = 3 with a Chern number C = 1. Our results underscore the pivotal role of magnetic fields as a powerful probe for elucidating topological phases in twisted Van der Waals heterostructures.

Bound states and local topological phase diagram of classical impurity spins coupled to a Chern insulator

Bound states and local topological phase diagram of classical impurity spins coupled to a Chern insulator

Gate-Tunable Antiferromagnetic Chern Insulator in Twisted Bilayer Transition Metal Dichalcogenides.

A series of recent experimental works on twisted MoTe_{2} homobilayers have unveiled an abundance of exotic states in this system. Valley-polarized quantum anomalous Hall states have been identified at hole doping of ν=-1, and the fractional quantum anomalous Hall effect is observed at ν=-2/3 and ν=-3/5. In this Letter, we investigate the electronic properties of AA-stacked twisted bilayer MoTe_{2} at ν=-2 by k-space Hartree-Fock calculations. We identify a series of phases, among which a noteworthy phase is the antiferromagnetic Chern insulator, stabilized by an external electric field. We attribute the existence of this Chern insulator to an antiferromagnetic instability at a topological phase transition between the quantum spin hall phase and a band insulator phase. Our research proposes the potential of realizing a Chern insulator beyond ν=-1, and contributes fresh perspectives on the interplay between band topology and electron-electron correlations in moiré superlattices.

Measurement of the Chern number for non-Hermitian Chern insulators

Measurement of the Chern number for non-Hermitian Chern insulators

Realization of a quadrupole topological insulator phase in a gyromagnetic photonic crystal

ABSTRACT The field of topological photonics was initiated with the realization of a Chern insulator phase in a gyromagnetic photonic crystal (PhC) with broken time-reversal symmetry (T), hosting chiral edge states that are topologically protected propagating modes. Along a separate line of research, a quadrupole topological insulator was the first higher-order topological phase supporting localized corner states, but has been so far limited to T-invariant systems, as T is a key ingredient in early models. Here we report the realization of a quadrupole topological insulator phase in a gyromagnetic PhC, as a consequence of topological phase transition from the previously demonstrated Chern insulator phase. The phase transition has been demonstrated with microwave measurements, which characterize the evolution from propagating chiral edge states to localized corner states. We also demonstrate the migration of topological boundary states into the continuum, when the gyromagnetic PhC is magnetically tuned. These results extend the quadrupole topological insulator phase into T-broken systems, and integrate topologically protected propagating and localized modes in a magnetically tunable photonic crystal platform.

Fundamental Bound on Topological Gap

We provide a universal tight bound on the energy gap of topological insulators by exploring relationships between topology, quantum geometry, and optical absorption. Applications of our theory to infrared absorption near topological band inversion, magnetic circular dichroism in Chern insulators, and topological gap in moiré materials are demonstrated. Published by the American Physical Society 2024

Evidence of the fractional quantum spin Hall effect in moiré MoTe2.

Quantum spin Hall (QSH) insulators are two-dimensional electronic materials that have a bulk band gap similar to an ordinary insulator but have topologically protected pairs of edge modes of opposite chiralities1-6. So far, experimental studies have found only integer QSH insulators with counter-propagating up-spins and down-spins at each edge leading to a quantized conductance G0 = e2/h (with e and h denoting the electron charge and Planck's constant, respectively)7-14. Here we report transport evidence of a fractional QSH insulator in 2.1° twisted bilayer MoTe2, which supports spin-Sz conservation and flat spin-contrasting Chern bands15,16. At filling factor ν = 3 of the moiré valence bands, each edge contributes a conductance with zero anomalous Hall conductivity. The state is probably a time-reversal pair of the even-denominator 3/2-fractional Chern insulators. Furthermore, at ν = 2, 4 and 6, we observe a single, double and triple QSH insulator with each edge contributing a conductance G0, 2G0 and 3G0, respectively. Our results open up the possibility of realizing time-reversal symmetric non-abelian anyons and other unexpected topological phases in highly tunable moiré materials17-19.

Far-field thermal radiation of layered ferromagnetic topological materials

High Chern number topological insulators can be obtained in a film of layered magnetic block system theoretically and experimentally. With nonzero Chern numbers, Chern insulators become valuable for fundamental topological physics and for improving next-generation electronic devices. We study energy and angular momentum radiation from layered topological insulators using the Dirac Fermion approach and by Green’s function method. We make a connection between radiation magnitude and topological phase transitions. We find that the magnetic exchange field, intra-layer coupling, and inter-layer interaction are efficient measures to modify the energy radiation of layered topological materials. Moreover, the magnetic exchange field is indispensable for emitting angular momentum due to the need for breaking time-reversal symmetry.

Dual quantum spin Hall insulator by density-tuned correlations in TaIrTe4.

The convergence of topology and correlations represents a highly coveted realm in the pursuit of new quantum states of matter1. Introducing electron correlations to a quantum spin Hall (QSH) insulator can lead to the emergence of a fractional topological insulator and other exotic time-reversal-symmetric topological order2-8, not possible in quantum Hall and Chern insulator systems. Here we report a new dual QSH insulator within the intrinsic monolayer crystal of TaIrTe4, arising from the interplay of its single-particle topology and density-tuned electron correlations. At charge neutrality, monolayer TaIrTe4 demonstrates the QSH insulator, manifesting enhanced nonlocal transport and quantized helical edge conductance. After introducing electrons from charge neutrality, TaIrTe4 shows metallic behaviour in onlya small range of charge densities but quickly goes into a new insulating state, entirely unexpected on the basis of the single-particle band structure of TaIrTe4. This insulating state could arise from a strong electronic instability near the van Hove singularities, probably leading to a charge density wave (CDW). Remarkably, within this correlated insulating gap, we observe a resurgence of the QSH state. The observation of helical edge conduction in a CDW gap could bridge spin physics and charge orders. The discovery of a dual QSH insulator introduces a new method for creating topological flat minibands through CDW superlattices, which offer a promising platform for exploring time-reversal-symmetric fractional phases and electromagnetism2-4,9,10.

Tautomerization Makes Topological Phase Transition in 2D Organometallic Lattices

AbstractThe investigation into approaches for achieving reversible control of topological phase transitions has attracted substantial research interest. So far, several system‐dependent physical control methods are developed, while a chemical strategy for efficient regulation of topological properties is still lack. Here, this study presents a chemical means to achieve reversible topological control in 2D organometallic lattices by introducing the lactam‐lactim tautomerization modules in the organic linkers. The strategy is validated in the theoretically designed 2D hexagonal framework with planar tricoordinated Co as nodes and 2,2′‐bithiazole derivatives as organic linkers, which transforms from a ferrimagnetic half metal with a local gap featuring Chern number of +1 to a ferromagnetic Chern insulator possessing Chern number of −1 after lactam‐lactim tautomerization. Simultaneously, a flip in the spin polarization of charge carriers between the two lattice tautomers is also observed.

Polarization and Weak Topology in Chern Insulators.

Chern insulators, and more broadly, topological insulators, present an obstruction to the construction of exponentially localized electronic Wannier functions. This implies a fundamental difficulty in determining whether such insulators exhibit electric polarization. Here, we show that these insulators can indeed exhibit bound charges and adiabatic currents consistent with changes in bulk polarization over space and time, respectively. We also show that the change in polarization across crystalline domains within these strong topological insulators is quantized in the presence of crystalline symmetries.

Band mixing in the quantum anomalous Hall regime of twisted semiconductor bilayers

Remarkable recent experiments have observed fractional quantum anomalous Hall effects at zero field and unusually high temperatures in twisted semiconductor bilayer tMoTe2, hence realizing the first genuine fractional Chern insulators. Intriguing observations in these experiments, such as the absence of integer Hall effects at twist angles where a fractional Hall effect is observed, do however remain unexplained. The experimental phase diagram as a function of twist angle remains to be established. By a comprehensive numerical study, including entanglement spectroscopy, we show that band mixing has large qualitative and quantitative effects on the energetics of competing states and their energy gaps throughout the twist angle range θ≤4∘. This lays the foundation for a detailed realistic study of a rich variety of strongly correlated moiré superlattices and an understanding of the phase diagram of these fascinating systems. Published by the American Physical Society 2024

Spectroscopy of edge and bulk collective modes in fractional Chern insulators

The exploration of atomic fractional quantum Hall (FQH) states is now within reach in optical-lattice experiments. While ground-state signatures have been observed in a system realizing the Hofstadter-Bose-Hubbard model in a box [Léonard , ], how to access hallmark low-energy collective modes remains a central open question in this context. We introduce a spectroscopic scheme based on two interfering Laguerre-Gaussian beams, which transfer a controlled angular momentum and energy to the system. The edge and bulk responses to the probe are detected through local density measurements by tracking the transfer of atoms between the bulk and the edge of the FQH droplet. This detection scheme is shown to simultaneously reveal two specific signatures of FQH states: their chiral edge branch and their bulk magnetoroton mode. We numerically benchmark our method by considering few bosons in the ν=1/2 Laughlin ground state of the Hofstadter-Bose-Hubbard model, and demonstrate that these signatures are already detectable in realistic systems of two bosons, provided that the box potential is sufficiently large compared to the droplet. Our paper paves the way for the detection of fractional statistics in cold atoms through edge signatures. Published by the American Physical Society 2024

Dynamic mass generation and topological order in overscreened Kondo lattices

Multichannel Kondo lattice models are examples of strongly correlated electronic systems that exhibit non-Fermi-liquid behavior due to the presence of a continuous channel symmetry. Mean-field analyses have predicted that these systems undergo channel symmetry breaking at low temperature. We use the dynamical large-N technique to study temporal and spatial fluctuations of the multichannel Kondo model on a honeycomb lattice and find that this prediction is not generally true. Rather, we find a (2+1)-dimensional conformally invariant fixed point, governed by critical exponents that are found numerically. When we break time-reversal symmetry by adding a Haldane mass to the conduction electrons, three phases, separated by continuous transitions, are discernible: one characterized by dynamic mass generation and spontaneous breaking of the channel symmetry, one where topological defects restore channel symmetry but preserve the gap, and one with a Kondo-coupled chiral spin liquid. We argue that the last phase is a fractional Chern insulator with anyonic excitations. Published by the American Physical Society 2024

Cn -symmetric quasi-periodic Chern insulators

Chern insulators have been recently extended to quasicrystal systems, namely the quasi-periodic Chern insulators (QCIs). Here we study the topological properties and topological response of QCIs on two-dimensional singular surfaces. Such singular QCIs with arbitrary n− fold rotational symmetry (i.e. C n -symmetric QCIs) can be constructed by ‘cutting and gluing’ unit sectors on the Dürer’s pentagonal tiling. Chiral edge states and real-space Chern number can well characterize the topological properties of C n -symmetric QCIs. Intriguingly, we numerically identify the emergence of charge fractionalization in unit of e/10 around the singular center in Cn -symmetric QCIs though their bulk densities are inhomogeneous. In addition, we explore the phase transitions of these C n -symmetric QCIs.

Magic Angles and Fractional Chern Insulators in Twisted Homobilayer Transition Metal Dichalcogenides.

We explain the appearance of magic angles and fractional Chern insulators in twisted K-valley homobilayer transition metal dichalcogenides by mapping their continuum model to a Landau level problem. Our approach relies on an adiabatic approximation for the quantum mechanics of valence band holes in a layer-pseudospin field that is valid for sufficiently small twist angles and on a lowest Landau level approximation that is valid for sufficiently large twist angles. It provides a simple qualitative explanation for the nearly ideal quantum geometry of the lowest moiré miniband at particular twist angles, predicts that topological flat bands occur only when the valley-dependent moiré potential is sufficiently strong compared to the interlayer tunneling amplitude, and provides a convenient starting point for the study of interactions.

Visualizing the Local Twist Angle Variation within and between Domains of Twisted Bilayer Graphene

Moiré superlattices in twisted two-dimensional materials have emerged as ideal platforms for engineering quantum phenomena, which are highly sensitive to twist angles, including both the global value and the spatial inhomogeneity. However, only a few methods provide spatial-resolved information for characterizing local twist angle distribution. Here we directly visualize the variations of local twist angles and angle-dependent evolutions of the quantum states in twisted bilayer graphene by scanning microwave impedance microscopy (sMIM). Spatially resolved sMIM measurements reveal a pronounced alteration in the local twist angle, approximately 0.3° over several micrometers in some cases. The variation occurs not only when crossing domain boundaries but also occasionally within individual domains. Additionally, the full-filling density of the flat band experiences a change of over 2 × 1011 cm−2 when crossing domain boundaries, aligning consistently with the twist angle inhomogeneity. Moreover, the correlated Chern insulators undergo variations in accordance with the twist angle, gradually weakening and eventually disappearing as the deviation from the magic angle increases. Our findings signify the crucial role of twist angles in shaping the distribution and existence of quantum states, establishing a foundational cornerstone for advancing the study of twisted two-dimensional materials.