Chern Insulator States Research Articles

Topological phase transitions and flat bands on a square-octagon lattice

We construct a square-octagon lattice model by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-nearest-neighbor (NNN) hoppings, where zero-Chern-number topological insulator (ZCNTI) phases emerge. At 1/4 filling, by tuning the staggered fluxes and NNN hopping potential, this model supports the phase transition from a Chern insulator (CI) with Chern number C = 2 to a ZCNTI phase. At half filling, we observe that staggered magnetic fluxes can induce a higher-order topological insulator (HOTI) state. Interestingly, the ZCNTI appears at 3/4 filling when considering the NN and NNN hoppings in the absence of staggered fluxes, which has been rarely reported in previous work. Contrary to conventional HOTIs, this ZCNTI phase hosts both robust corner states and gapless edge states which can be identified based on the quantized dipole and quadrupole moments. Additionally, a topological flat band (TFB) with flatness ratio about 24 appears. We further investigate ν = 1/2 fractional Chern insulator (FCI) state when hard-core bosons fill into this TFB model.

Quantum Geometry and Stabilization of Fractional Chern Insulators Far from the Ideal Limit.

In the presence of strong electronic interactions, a partially filled Chern band may stabilize a fractional Chern insulator (FCI) state, the zero-field analog of the fractional quantum Hall phase. While FCIs have long been hypothesized, feasible solid-state realizations only recently emerged, largely due to the rise of moiré materials. In these systems, the quantum geometry of the electronic bands plays a critical role in stabilizing the FCI in the presence of competing correlated phases. In the limit of "ideal" quantum geometry, where the quantum geometry is identical to that of Landau levels, this role is well understood. However, in more realistic scenarios only empiric numerical evidence exists, accentuating the need for a clear understanding of the mechanism by which the FCI deteriorates, moving further away from these ideal conditions. We introduce and analyze an anisotropic model of a |C|=1 Chern insulator, whereupon partial filling of its bands, an FCI phase is stabilized over a certain parameter regime. We incorporate strong electronic interaction analytically by employing a coupled-wires approach, studying the FCI stability and its relation to the quantum metric. We identify an unusual anti-FCI phase benefiting from nonideal geometry, generically subdominant to the FCI. However, its presence hinders the formation of FCI in favor of other competitive phases at fractional fillings, such as the charge density wave. Though quite peculiar, this anti-FCI phase may have already been observed in experiments at high magnetic fields. This establishes a direct link between quantum geometry and FCI stability in a tractable model far from any ideal band conditions, and illuminates a unique mechanism of FCI deterioration.

Fractional Chern insulator states in multilayer graphene moiré superlattices

Fractional Chern insulator states in multilayer graphene moiré superlattices

Even-Odd Layer-Dependent Exchange Bias Effect in MnBi2Te4 Chern Insulator Devices.

Magnetic topological materials with coexisting magnetism and nontrivial band structures exhibit many novel quantum phenomena, including the quantum anomalous Hall effect, the axion insulator state, and the Weyl semimetal phase. As a stoichiometric layered antiferromagnetic topological insulator, thin films of MnBi2Te4 show fascinating even-odd layer-dependent physics. In this work, we fabricate a series of thin-flake MnBi2Te4 devices using stencil masks and observe the Chern insulator state at high magnetic fields. Upon magnetic field training, a large exchange bias effect is observed in odd but not in even septuple layer (SL) devices. Through theoretical calculations, we attribute the even-odd layer-dependent exchange bias effect to the contrasting surface and bulk magnetic properties of MnBi2Te4 devices. Our findings reveal the microscopic magnetic configuration of MnBi2Te4 thin flakes and highlight the challenges in replicating the zero magnetic field quantum anomalous Hall effect in odd SL MnBi2Te4 devices.

Chern Insulator States with Tunable Chern Numbers in a Graphene Moiré Superlattice.

Moiré superlattices, constituted by two-dimensional materials, demonstrate a variety of strongly correlated and topological phenomena including correlated insulators, superconductivity, and integer/fractional Chern insulators. In the realm of topological nontrivial Chern insulators within specific moiré superlattices, previous studies usually observe a single Chern number at a given filling factor in a device. Here we present the observation of gate-tunable Chern numbers within the Chern insulator state of an ABC-stacked trilayer graphene/hexagonal boron nitride moiré superlattice device. Near quarter filling, the moiré superlattice exhibits spontaneous valley polarization and distinct ferromagnetism associated with the Chern insulator states over a range of the displacement field. Surprisingly we find a transition of the Chern number from C = 3 to 4 as the displacement field is increased. Our observation of gate-tunable correlated Chern insulators suggests new ways to control and manipulate topological states in a moiré superlattice device.

Symmetry-Broken Chern Insulators in Twisted Double Bilayer Graphene.

Twisted double bilayer graphene (tDBG) has emerged as a rich platform for studying strongly correlated and topological states, as its flat bands can be continuously tuned by both a perpendicular displacement field and a twist angle. Here, we construct a phase diagram representing the correlated and topological states as a function of these parameters, based on measurements of over a dozen tDBG devices encompassing two distinct stacking configurations. We find a hierarchy of symmetry-broken states that emerge sequentially as the twist angle approaches an apparent optimal value of θ ≈ 1.34°. Nearby this angle, we discover a symmetry-broken Chern insulator (SBCI) state associated with a band filling of 7/2 as well as an incipient SBCI state associated with 11/3 filling. We further observe an anomalous Hall effect at zero field in all samples supporting SBCI states, indicating spontaneous time-reversal symmetry breaking and possible moiré unit cell enlargement at zero magnetic field.

Magnetic breakdown and spin-zero effect in quantum oscillations in kagome metal CsV3Sb5

In the recently discovered kagome metal CsV3Sb5, an intriguing proposal invoking a doped Chern insulator state suggests the presence of small Chern Fermi pockets hosting spontaneous orbital-currents and large orbital magnetic moments. While the net thermodynamic magnetization is nearly insensitive to these moments, due to their antiferromagnetic alignment, their presence can be revealed by the Zeeman effect, which shifts electron energies in magnetic fields with a proportionality given by the effective g−factor. Here, we determine the g-factor using the spin-zero effect in magnetic quantum oscillations. A large g-factor enhancement is visible only in magnetic breakdown orbits between conventional and concentrated Berry curvature Fermi pockets that host large orbital moments. Such Berry-curvature-generated large orbital moments are almost always concealed by other effects. In this system, however, magnetic breakdown orbits due to the proximity to a conventional Fermi-surface section allow them to be visibly manifested in magnetic quantum oscillations. Our results provide a remarkable example of the interplay between electronic correlations and more conventional electronic bands in quantum materials.

Floquet fractional Chern insulators and competing phases in twisted bilayer graphene

We study the many-body physics in twisted bilayer graphene coupled to periodic driving of a circularly polarized light when electron-electron interactions are taken into account. In the limit of high driving frequency \OmegaΩ, we use Floquet theory to formulate the system by an effective static Hamiltonian truncated to the order of \Omega^{-2}Ω−2, which consists of a single-electron part and the screened Coulomb interaction. We numerically simulate this effective Hamiltonian by extensive exact diagonalization in the parameter space spanned by the twist angle and the driving strength. Remarkably, in a wide region of the parameter space, we identify Floquet fractional Chern insulator states in the partially filled Floquet valence bands. We characterize these topologically ordered states by ground-state degeneracy, spectral flow, and entanglement spectrum. In regions of the parameter space where fractional Chern insulator states are absent, we find topologically trivial charge density waves and band-dispersion-induced Fermi liquids which strongly compete with fractional Chern insulator states.

Correlated insulator and Chern insulators in pentalayer rhombohedral-stacked graphene.

Rhombohedral-stacked multilayer graphene hosts a pair of flat bands touching at zero energy, which should give rise to correlated electron phenomena that can be tuned further by an electric field. Moreover, when electron correlation breaks the isospin symmetry, the valley-dependent Berry phase at zero energy may give rise to topologically non-trivial states. Here we measure electron transport through hexagonal boron nitride-encapsulated pentalayer graphene down to 100 mK. We observed a correlated insulating state with resistance at the megaohm level or greater at charge density n = 0 and displacement field D = 0. Tight-binding calculations predict a metallic ground state under these conditions. By increasing D, we observed a Chern insulator state with C = -5 and two other states with C = -3 at a magnetic field of around 1 T. At high D and n, we observed isospin-polarized quarter- and half-metals. Hence, rhombohedral pentalayer graphene exhibits two different types of Fermi-surface instability, one driven by a pair of flat bands touching at zero energy, and one induced by the Stoner mechanism in a single flat band. Our results establish rhombohedral multilayer graphene as a suitable system for exploring intertwined electron correlation and topology phenomena in natural graphitic materials without the need for moiré superlattice engineering.

Phononic Weyl pair, phononic Weyl complex, phononic real Chern insulator state, and phononic corner modes in 2D Kekulé-order graphene

The conceptual framework of topological states has recently been extended to bosonic systems, particularly phononic systems. In this work, we chose the recently experimentally prepared two-dimensional (2D) Kekulé-order graphene as a target to propose the coexistence of gapless and gapped topological phonon states in its phonon curves. This is the first work to investigate rich gapped and gapless topological phonon states in experimentally feasible 2D materials. For the gapped topological phonons, 2D Kekulé-order graphene hosts phononic real Chern insulator states, i.e., second-order topological states, and corner vibrational modes inside frequency gaps at 27.96 and 37.065 THz. For the gapless topological phonons, 2D Kekulé-order graphene hosts a phononic Weyl pair [comprising two linear Weyl points (LWPs)] and a phononic Weyl complex [comprising one quadratic nodal point (QNP) and two LWPs] around 7.54 and 47.3 THz (39.2 THz), respectively. Moreover, the difference between the phononic Weyl pair and the phononic Weyl complex was investigated in detail. Our study not only promotes 2D Kekulé-order graphene as a concrete material platform for exploring the intriguing physics of phononic second-order topology but also proposes the coexistence of different categories of Weyl phonons, i.e., a Weyl complex that comprises two LWPs and one QNP, in two dimensions. Our work paves the way for new advancements in topological phononics comprising gapless and gapped topological phonons.

Magnetic Real Chern Insulator in 2D Metal-Organic Frameworks.

Real Chern insulators have attracted great interest, but so far, their material realization is limited to nonmagnetic crystals and systems without spin-orbit coupling. Here, we reveal the magnetic real Chern insulator (MRCI) state in a recently synthesized metal-organic framework material Co3(HITP)2. Its ground state with in-plane ferromagnetic ordering hosts a nontrivial real Chern number, enabled by the C2zT symmetry and robustness against spin-orbit coupling. Distinct from previous nonmagnetic examples, the topological corner zero modes of MRCIs are spin-polarized. Furthermore, under small tensile strains, the material undergoes a topological phase transition from the MRCI to a magnetic double-Weyl semimetal phase, via a pseudospin-1 critical state. Similar physics can also be found in closely related materials Mn3(HITP)2 and Fe3(HITP)2, which also exist. Possible experimental detections and implications of an emerging magnetic flat band in the system are discussed.

Flat bands with high Chern numbers and multiple flat bands in multifold staggered-flux models

Topological flat bands (TFBs) have been proposed in Chern-insulator lattice models. Some two-dimensional monolayer systems can host TFBs with the Chern number $|\mathcal{C}|=1$, and some TFBs with higher Chern numbers are achieved in multilayer or multicomponent systems. In this work, we construct a series of TFBs with higher Chern numbers in the monolayer star lattice by adding several distant-neighbor hopping terms with staggered magnetic fluxes. By considering the third and the fourth nearest-neighbor hopping terms, we find the $|\mathcal{C}|=2$ and $|\mathcal{C}|=3$ TFBs with large flatness ratios, respectively. We also obtain the $|\mathcal{C}|=4$ and $|\mathcal{C}|=5$ TFBs when the fifth and the sixth nearest-neighbor hopping terms are added. Intriguingly, we also observe multiple flat bands with various Chern numbers by tuning the distant-neighbor hopping amplitudes and the staggered-flux parameters. Furthermore, a series of fractional Chern insulator (FCI) states have been numerically observed when hard-core bosons are filled into these TFB models, with fractional fillings $\ensuremath{\nu}=1/3, 2/5, 3/7$ and $\ensuremath{\nu}=1/4, 2/7$ in the $|\mathcal{C}|=2$ and $|\mathcal{C}|=3$ TFBs, respectively. Our findings provide an approach to search for more TFBs with higher Chern numbers and further investigate more interesting FCIs.

Collective excitations of the Chern-insulator states in commensurate double moiré superlattices of twisted bilayer graphene on hexagonal boron nitride

We study the collective excitation modes of the Chern insulator states in magic-angle twisted bilayer graphene aligned with hexagonal boron nitride (TBG/BN) at odd integer fillings ($\nu$) of the flat bands. For the $1 \times 1$ commensurate double moir\'{e} superlattices in TBG/BN at three twist angles ($\theta'$) between BN and graphene, self-consistent Hartree-Fock calculations show that the electron-electron interaction and the broken $C_{2z}$ symmetry lead to the Chern-insulator ground states with valley-spin flavor polarized HF bands at odd $\nu$. In the active-band approximation, the HF bands in the same flavor of TBG/BN are much more separated than those of the pristine TBG with TBG/BN having a larger intra-flavor band gap so that the energies of the lowest intra-flavor exciton modes of TBG/BN computed within the time-dependent HF method are much higher than those of TBG and reach about 20 meV, and the exciton wavefunctions of TBG/BN become less localized than those of TBG. The inter-flavor valley-wave modes in TBG/BN have excitation energies higher than 2.5 meV which is also much larger than that of TBG, while the spin-wave modes all have zero excitation gap. In contrast to TBG with particle-hole symmetric excitation modes for positive and negative $\nu$, the excitation spectrums and gaps of TBG/BN at positive $\nu$ are rather different from those at negative $\nu$. The quantitative behavior of the excitation spectrum of TBG/BN also varies with $\theta'$. Full HF calculations demonstrate that more HF bands besides the two central bands can have rather large contributions from the single-particle flat-band states, then the lowest exciton modes that determine the optical properties of the Chern insulator states in TBG/BN are generally the ones between the remote and flat-like bands, while the valley-wave modes have similar energies as those in the active-band approximation.

Monolayer and bilayer PtCl3 : Energetics, magnetism, and band topology

Two-dimensional (2D) magnetic materials hosting nontrivial topological states are interesting for fundamental research as well as practical applications. Recently, the topological state of 2D Weyl half-semimetal (WHS) was proposed, which hosts fully spin polarized Weyl points robust against spin-orbit coupling in a 2D ferromagnetic system, and single-layer ${\mathrm{PtCl}}_{3}$ was predicted as a platform for realizing this state. Here, we perform an extensive search of 2D ${\mathrm{PtCl}}_{3}$ structures, by using the particle swarm optimization technique and density-functional theory calculation. We show that the desired ${\mathrm{PtCl}}_{3}$ phase corresponds to the most stable one at its stoichiometry. The 2D structure also possesses good thermal stability up to 600 K. We suggest ${\mathrm{SnS}}_{2}$ as a substrate for the growth of 2D ${\mathrm{PtCl}}_{3}$, which has excellent lattice matching and preserves the WHS state in ${\mathrm{PtCl}}_{3}$. We find that uniaxial strains along the zigzag direction maintain the WHS state, whereas small strains along the armchair direction drives a topological phase transition from the WHS to a quantum anomalous Hall (QAH) insulator phase. Furthermore, we study bilayer ${\mathrm{PtCl}}_{3}$ and show that the stacking configuration has strong impact on the magnetism and the electronic band structure. Particularly, the $A{A}^{\ensuremath{'}}$ stacked bilayer ${\mathrm{PtCl}}_{3}$ realizes an interesting topological state---the 2D antiferromagnetic mirror Chern insulator, which has a pair of topological gapless edge bands. Our work provides guidance for the experimental realization of 2D ${\mathrm{PtCl}}_{3}$ and will facilitate the study of 2D magnetic topological states, including WHS, QAH insulator, and magnetic mirror Chern insulator states.

Isospin polarized Chern insulator state of <i>C</i> = 4 in twisted double bilayer graphene

A flat band with nearly zero dispersion can be created by twisting the relative orientation of van der Waals materials, leading to a series of strongly correlated states, such as unconventional superconductivity, correlated insulating state, and orbital magnetism. The bandwidth and topological property of electronic band structure in a twisted double bilayer graphene are tunable by an external displacement field. This system can be an excellent quantum simulator to study the interplay between topological phase transition and strong electron correlation. Theoretical calculation shows that the <inline-formula><tex-math id="M4">\begin{document}$ {C}_{2x} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M4.png"/></alternatives></inline-formula> symmetry in twisted double bilayer graphene (TDBG) can be broken by an electric displacement field, leading the lowest conduction and valence band near charge neutrality to obtain a finite Chern number. The topological properties of the band and the symmetry breaking driven by the strong interaction make it possible to realize and regulate the old insulation state at low magnetic fields. Hence Chern insulator may emerge from this topological non-trivial flat band under strong electron interaction. Here, we observe Chern insulator state with Chern number 4 at filling factor <inline-formula><tex-math id="M5">\begin{document}$ \nu =1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M5.png"/></alternatives></inline-formula> under a small magnetic field on twisted double bilayer graphene with twist angle 1.48°. Moreover, the longitudinal resistance shows a peak under a parallel magnetic field and increases with temperature or field rising, which is similar to the Pomeranchuk effect in <sup>3</sup>He. This phenomenon indicates that Chern insulator at <inline-formula><tex-math id="M6">\begin{document}$ \nu =1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M6.png"/></alternatives></inline-formula> may originate from isospin polarization.

Two-dimensional semimetal states in transition metal trichlorides: A first-principles study

The two-dimensional (2D) transition metal trihalide (TMX3, X = Cl, Br, I) family has attracted considerable attention in recent years due to the realization of CrCl3, CrBr3, and CrI3 monolayers. Up to now, the main focus of the theoretically predicted TMX3 monolayers has been on the Chern insulator states, which can realize the quantum anomalous Hall effect. Here, using first-principles calculations, we theoretically demonstrate that the stable OsCl3 monolayer has a ferromagnetic ground state and a spin-polarized Dirac point without spin–orbit coupling (SOC), which disappears in the band structure of a Janus OsBr1.5Cl1.5 monolayer. We find that OsCl3 exhibits in-plane magnetization when SOC is included. By manipulating the magnetization direction along the C2 symmetry axis of the OsCl3 structure, a gapless half-Dirac semimetal state with SOC can be achieved, which is different from the gapped Chern insulator state. Both semimetal states of OsCl3 monolayer without and with SOC exhibit a linear half-Dirac point (twofold degenerate) with high Fermi velocities. The achievement of the 2D semimetal state with SOC is expected to be found in other TMX3 monolayers, and we confirm it in a TiCl3 monolayer. This provides a different perspective to study the band structure with SOC of the 2D TMX3 family.

Chern mosaic and Berry-curvature magnetism in magic-angle graphene

Charge carriers in magic angle graphene come in eight flavors described by a combination of their spin, valley, and sublattice polarizations. When the inversion and time reversal symmetries are broken by the substrate or by strong interactions, the degeneracy of the flavors can be lifted and their corresponding bands can be filled sequentially. Due to their non-trivial band topology and Berry curvature, each of the bands is classified by a topological Chern number, leading to the quantum anomalous Hall and Chern insulator states at integer fillings $\nu$ of the bands. It has been recently predicted, however, that depending on the local atomic-scale arrangements of the graphene and the encapsulating hBN lattices, rather than being a global topological invariant, the Chern number C may become position dependent, altering transport and magnetic properties of the itinerant electrons. Using a SQUID-on-tip, we directly image the nanoscale Berry-curvature-induced equilibrium orbital magnetism, the polarity of which is governed by the local Chern number, and detect its two constituent components associated with the drift and the self-rotation of the electronic wave packets. At $\nu=1$, we observe local zero-field valley-polarized Chern insulators forming a mosaic of microscopic patches of C=-1, 0, or 1, governed by the local sublattice polarization, consistent with predictions. Upon further filling, we find a first-order phase transition due to recondensation of electrons from valley K to K', which leads to irreversible flips of the local Chern number and the magnetization, and to the formation of valley domain walls giving rise to hysteretic global anomalous Hall resistance. The findings shed new light on the structure and dynamics of topological phases and call for exploration of the controllable formation of flavor domain walls and their utilization in twistronic devices.

Topological states in a dimerized system with staggered magnetic fluxes

The bulk-boundary correspondence is a generic feature of topological states of matter, reflecting the intrinsic relation between topological bulk and boundary states. For example, robust edge states propagate along the edges and corner states gather at corners in the two-dimensional first-order and second-order topological insulators, respectively. Here, we report two kinds of topological states hosting anomalous bulk-boundary correspondence in the extended two-dimensional dimerized lattice with staggered flux threading. At $\frac{1}{2}$-filling, we observe isolated corner states with no fractional charge as well as metallic near-edge states in the $\mathcal{C}=2$ Chern insulator states. At $\frac{1}{4}$-filling, we find a $\mathcal{C}=0$ topologically nontrivial state, where the robust edge states are well localized along edges but bypass corners. These robust topological insulating states significantly differ from both conventional Chern insulators and usual high-order topological insulators.

Magnetization-tuned topological quantum phase transition in MnBi2Te4 devices

Recently, the intrinsic magnetic topological insulator MnBi2Te4 has attracted enormous research interest due to the great success in realizing exotic topological quantum states, such as the quantum anomalous Hall effect (QAHE), axion insulator state, high-Chern-number and high-temperature Chern insulator states. One key issue in this field is to effectively manipulate these states and control topological phase transitions. Here, by systematic angle-dependent transport measurements, we reveal a magnetization-tuned topological quantum phase transition from Chern insulator to magnetic insulator with gapped Dirac surface states in MnBi2Te4 devices. Specifically, as the magnetic field is tilted away from the out-of-plane direction by around 40-60 degrees, the Hall resistance deviates from the quantization value and a colossal, anisotropic magnetoresistance is detected. The theoretical analyses based on modified Landauer-Buttiker formalism show that the field-tilt-driven switching from ferromagnetic state to canted antiferromagnetic state induces a topological quantum phase transition from Chern insulator to magnetic insulator with gapped Dirac surface states in MnBi2Te4 devices. Our work provides an efficient means for modulating topological quantum states and topological quantum phase transitions.

Topological lattice models with constant Berry curvature

Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.