Chern Number Research Articles

Chiral edge transport along domain walls in magnetic topological insulator nanoribbons

Quantum anomalous Hall insulators are topologically characterized by non-zero integer Chern numbers, the sign of which depends on the direction of the exchange field that breaks time-reversal symmetry. This feature allows the manipulation of the conducting chiral edge states present at the interface of two magnetic domains with opposite magnetization and opposite Chern numbers. Motivated by this broad understanding, the present study investigates the quantum transport properties of a magnetized Bi2Se3 topological insulator nanoribbon with a domain wall (DW) oriented either parallel or perpendicular to the transport direction. Employing an atomistic tight-binding model and a non-equilibrium Green’s function formalism, we calculate the quantum conductance and explore the nature of the edge states. We elucidate the conditions leading to exact conductance quantization and identify the origin of deviations from this behavior. Our analysis shows that although the conductance is quantized in the presence of the horizontal DW, the quantization is absent in the perpendicular DW case. Furthermore, the investigation of the spin character of the edge modes confirms that the conductance in the horizontal DW configuration is spin polarized. This finding underscores the potential of our system as a simple three dimensional spin-filter device.

Time-reversal soliton pairs in even spin Chern number higher-order topological insulators

Time-reversal soliton pairs in even spin Chern number higher-order topological insulators

Flat bands in three-dimensional lattice models with non-trivial Hopf index

We report the presence of exactly and nearly flat bands with non-trivial topology in three-dimensional (3D) lattice models. We first show that an exactly flat band can be realized in a 3D lattice model characterized by a 3D topological invariant, namely Hopf invariant. In contrast, we find another distinct 3D model, exhibiting both 2D Chern and 3D Hopf invariant, namely Hopf-Chern insulator, that can host nearly or perfect flat bands across different 2D planes. Such a Hopf-Chern model can be constructed by introducing specific hopping along the orthogonal direction of a simple two-orbital 2D Chern insulator in the presence of in-plane nearest-neighbor and next-nearest hopping among different orbitals. While the Chern planes host nearly perfect flat bands, the orthogonal planes can host both perfect or nearly perfect flat bands with zero Chern number at some special parameter values. Interestingly, such a 3D lattice construction from 2D allows finite Hopf invariant too. Finally, we show that higher Chern models can also be constructed in the same lattice setup with only nearest and next-nearest hopping, but the appearance of flat bands along high-symmetric path in the Brillouin zone requires longer-range hopping. We close with a discussion on possible experimental platforms to realize the models.

Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a nontrivial Z2 index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, Cs=(C↑−C↓)/2. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar Z2 index vanishes. Identification of solid-state systems hosting an odd, quantized Cs and trivial Z2 index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial Z2 index and odd Cs. We demonstrate the potential of one such system, Ti2CO2, to support Majorana corner modes via the superconducting proximity effect. Published by the American Physical Society 2024

Holographic entanglement renormalisation for fermionic quantum matter

We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov–Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.

Topological heavy fermions in magnetic field

The recently introduced topological heavy fermion model (THFM) provides a means for interpreting the low-energy electronic degrees of freedom of the magic angle twisted bilayer graphene as hybridization amidst highly dispersing topological conduction and weakly dispersing localized heavy fermions. In order to understand the Landau quantization of the ensuing electronic spectrum, a generalization of THFM to include the magnetic field B is desired, but currently missing. Here we provide a systematic derivation of the THFM in B and solve the resulting model to obtain the interacting Hofstadter spectra for single particle charged excitations. While naive minimal substitution within THFM fails to correctly account for the total number of magnetic subbands within the narrow band i.e., its total Chern number, our method—based on projecting the light and heavy fermions onto the irreducible representations of the magnetic translation group— reproduces the correct total Chern number. Analytical results presented here offer an intuitive understanding of the nature of the (strongly interacting) Hofstadter bands.

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.

Edge-dependent anomalous topology in synthetic photonic lattices subject to discrete step walks

Anomalous topological phases, where edge states coexist with topologically trivial Chern bands, can only appear in periodically driven lattices. When the driving is smooth and continuous, the bulk-edge correspondence is guaranteed by the existence of a bulk invariant known as the winding number. However, in lattices subject to periodic discrete step walks the existence of edge states does not only depend on bulk invariants but also on the boundary. This is a consequence of the absence of an intrinsic time dependence or micromotion in discrete step walks. We report the observation of edge states and a simultaneous measurement of the bulk invariants in anomalous topological phases in a two-dimensional synthetic photonic lattice subject to discrete step walks. The lattice is implemented using time multiplexing of light pulses in two coupled fibre rings, in which one of the dimensions displays real-space dynamics and the other one is parametric. The presence of edge states is inherent to the periodic driving and depends on the properties of the boundary in the implemented two-band model with zero Chern number. We provide a suitable expression for the topological invariants whose calculation does not rely on micromotion dynamics. Published by the American Physical Society 2024

Utilizing topological invariants for encoding and manipulating chiral phonon devices

As a fundamental degree of freedom, phonon chirality is expected to promote the development of quantum information technology just like electron spin. Currently, central to this area is the realization of efficient transmission and control of chiral information. In this paper, we propose an approach by integrating topological theory, leveraging topologically invariant Chern numbers, to encode hexagonal lattice systems. Our investigation reveals the presence of topologically protected chiral interface states within the shared band gaps of both trivial and non-trivial system units. By precisely modulating the magnetic field distribution within the encoding system, we can effectively manipulate the topological pathways. Building upon this framework, we design and implement a chiral phonon three-port device. Through dynamic calculations, we demonstrate the transmission process of chiral information, showcasing the chiral phonon switching effect and logical OR operation. Our findings not only establish a fundamental mechanism for the manipulation and control of phonon chiral information but also provide a promising direction for research in harnessing chirality degrees of freedom in practical applications.

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.

Rules for dissipationless topotronics.

Topological systems hosting gapless boundary states have attracted huge attention as promising components for next-generation information processing, attributed to their capacity for dissipationless electronics. Nevertheless, recent theoretical and experimental inquiries have revealed the emergence of energy dissipation in precisely quantized electrical transport. Here, we present a criterion for the realization of truly no-dissipation design, characterized as Nin = Ntunl + Nbs, where Nin, Ntunl, and Nbs represent the number of modes participating in injecting, tunneling, and backscattering processes, respectively. The key lies in matching the number of injecting, tunneling, and backscattering modes, ensuring the equilibrium among all engaged modes inside the device. Among all the topological materials, we advocate for the indispensability of Chern insulators exhibiting higher Chern numbers to achieve functional devices and uphold the no-dissipation rule simultaneously. Furthermore, we design the topological current divider and collector, evading dissipation upon fulfilling the established criterion. Our work paves the path for developing the prospective topotronics.

Quasi-localization and Wannier obstruction in partially flat bands

The localized nature of a flat band is understood by the existence of a compact localized eigenstate. However, the localization properties of a partially flat band, ubiquitous in surface modes of topological semimetals, have been unknown. We show that the partially flat band is characterized by a non-normalizable quasi-compact localized state (Q-CLS), which is compactly localized along several directions but extended in at least one direction. The partially flat band develops at momenta where normalizable Bloch wave functions can be obtained from a linear combination of the non-normalizable Q-CLSs. Outside this momentum region, a ghost flat band, unseen from the band structure, is introduced based on a counting argument. Then, we demonstrate that the Wannier function corresponding to the partially flat band exhibits an algebraic decay behavior. Namely, one can have the Wannier obstruction in a band with a vanishing Chern number if it is partially flat. Finally, we develop the construction scheme of a tight-binding model for a topological semimetal by designing a Q-CLS.

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.

Near-Field Spin Chern Number Quantized by Real-Space Topology of Optical Structures.

The Chern number has been widely used to describe the topological properties of periodic structures in momentum space. Here, we introduce a real-space spin Chern number for the optical near fields of finite-sized structures. This new spin Chern number is intrinsically quantized and equal to the structure's Euler characteristic. The relationship is robust against continuous deformation of the structure's geometry and is irrelevant to the specific material constituents or external excitation. Our Letter enriches topological physics by extending the Chern number to real space, opening exciting possibilities for exploring the real-space topological properties of light.

Engineering Plateau Phase Transition in Quantum Anomalous Hall Multilayers.

The plateau phase transition in quantum anomalous Hall (QAH) insulators corresponds to a quantum state wherein a single magnetic domain gives way to multiple domains and then reconverges back to a single magnetic domain. The layer structure of the sample provides an external knob for adjusting the Chern number C of the QAH insulators. Here, we employ molecular beam epitaxy to grow magnetic topological insulator multilayers and realize the magnetic field-driven plateau phase transition between two QAH states with odd Chern number change ΔC. We find that critical exponents extracted for the plateau phase transitions with ΔC = 1 and ΔC = 3 in QAH insulators are nearly identical. We construct a four-layer Chalker-Coddington network model to understand the consistent critical exponents for the plateau phase transitions with ΔC = 1 and ΔC = 3. This work will motivate further investigations into the critical behaviors of plateau phase transitions with different ΔC in QAH insulators.

Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli

In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems it is crucial to establish several isomorphisms between different deformation functors, that we expect that will be useful in further work as well.

Topological charge pumping in tetramerized Kitaev chains with nodal loop in parameter space

We investigated the topological pumping charge of a tetramerized Kitaev chain with spatially modulated chemical potential, which hosts nodal loops in parameter space and violates particle number conservation. In the simplest case, with alternatively assigned hopping and pairing terms, we show that the model can be mapped into the Rice–Mele model by a partial particle-hole transformation and subsequently supports topological charge pumping as a demonstration of the Chern number for the ground state. Beyond this special case, analytic analysis shows that the nodal loops are conic curves. Numerical simulation of a finite-size chain indicates that the pumping charge is zero for a quasiadiabatic loop within the nodal loop and is ±1 for a quasiadiabatic passage enclosing the nodal loop. Our findings unveil the topology of Kitaev chains in parameter space.

In-plane magnetization orientation driven topological phase transition in OsCl3 monolayer

The quantum anomalous Hall effect resulting from the in-plane magnetization in the OsCl3 monolayer is shown to exhibit different electronic topological phases determined by the crystal symmetries and magnetism. In this Chern insulator, the Os-atoms form a two dimensional planar honeycomb structure with an easy-plane ferromagnetic configuration and the required non-adiabatic paths to tune the topology of electronic structure exist for specific magnetic orientations based on mirror symmetries of the system. Using density functional theory (DFT) calculations, these tunable phases are identified by changing the orientation of the magnetic moments. We argue that in contrast to the buckled system, here the Cl-ligands bring non-trivial topology into the system by breaking the in-plane mirror symmetry. The interplay between the magnetic anisotropy and electronic band-topology changes the Chern number and hence the topological phases. Our DFT study is corroborated with comprehensive analysis of relevant symmetries as well as a detailed explanation of topological phase transitions using a generic tight binding model.

Bandgap formation in topological metamaterials with spatially modulated resonators

Within the field of elastic metamaterials, topological metamaterials have recently received much attention due to their ability to host topologically robust edge states. Introducing local resonators to these metamaterials also opens the door for many applications such as energy harvesting and reconfigurable metamaterials. However, the interactions between phenomena from local resonance and modulation patterning are currently unknown. This work fills that gap by studying multiple cases of spatially modulated metamaterials with local resonators to reveal the mechanisms behind bandgap formation. Their dispersion relations are determined analytically for infinite chains and validated numerically using eigenvalue analysis. The inverse method is used to determine the imaginary wavenumber components from which each bandgap is characterized by its formation mechanism. The topological nature of the bandgaps is also explored through calculating the Chern number and integrated density of states. The band structures are obtained for various sources of modulation as well as multiple resonator parameters to illustrate how both local resonance and modulation patterning interact together to influence the band structure. Other unique features of these metamaterials are further demonstrated through the mode shapes obtained using the eigenvectors. The results reveal a complex band structure that is highly tunable, and the observations given here can be used to guide designers in choosing resonator parameters and patterning to fit a variety of applications.

High-performance quantum anomalous Hall effect in monolayer Ti2Sb2KRb and Ti2Bi2NaK

Quantum anomalous Hall (QAH) insulators are an ideal platform for developing topological electronic devices, but their low observation temperature limits the applications. In this study, based on first-principles calculations, monolayer Ti2Sb2KRb and Ti2Bi2NaK are demonstrated to be QAH insulators with topological gaps 43 and 57 meV, respectively. Their Chern numbers are calculated to be C=−2. The study of electronic structures indicates that the ferromagnetic topological property is induced by the energy band inversion of dxy and dx2−y2 orbitals for Ti atoms near the Dirac cone. Both monolayer Ti2Sb2KRb and Ti2Bi2NaK exhibit a perpendicular magnetic anisotropy, and their Curie temperatures are estimated to be 480 and 478 K, respectively. The ferromagnetic coupling is induced by the small crystal-field splitting energy caused by Sb and Bi atom's large radius. Our study suggests that monolayer Ti2Sb2KRb and Ti2Bi2NaK are promising candidates for room temperature QAH insulators.