Chern Insulator Research Articles

Hyperbolic fractional Chern insulators

Hyperbolic fractional Chern insulators

Topological linear response of hyperbolic Chern insulators

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to -e^2C_{ij}/h−e2Cij/h, where C_{ij}Cij is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.

Topological phase transitions via attosecond x-ray absorption spectroscopy

We present a numerical experiment that demonstrates the possibility to capture topological phase transitions via an x-ray absorption spectroscopy scheme. We consider a Chern insulator whose topological phase is tuned via a second-order hopping. We perform time-dynamics simulations of the out-of-equilibrium laser-driven electron motion that enables us to model a realistic attosecond spectroscopy scheme. In particular, we use an ultrafast scheme with a circularly polarized IR pump pulse and an attosecond x-ray probe pulse. A laser-induced dichroism-type spectrum shows a clear signature of the topological phase transition. We are able to connect these signatures with the Berry structure of the system. This work extend the applications of attosecond absorption spectroscopy to systems presenting a non-trivial topological phase.

Quantized Integrated Shift Effect in Multigap Topological Phases.

We show that certain three-dimensional multigap topological insulators can host quantized integrated shift photoconductivities due to bulk invariants that are defined under reality conditions imposed by additional symmetries. We recast the quantization in terms of the integrated torsion tensor and the non-Abelian Berry connection constituting Chern-Simons forms. Physically, we recognize that the topological quantization emerges purely from virtual transitions contributing to the optical response. Our findings provide another quantized electromagnetic dc response due to the nontrivial band topology, beyond the quantum anomalous Hall effect of Chern insulators and quantized circular photogalvanic effect found in Weyl semimetals.

Topological traveling-wave radiation based on chiral edge states in photonic Chern insulators.

Topological chiral edge states in photonic Chern insulators, which exhibit the one-way propagation property with reflection-free behavior and excellent robustness against perturbations, have become an important frontier in topological photonics. Currently, there have been many studies on its robust propagation behavior, but few studies have focused on its radiation properties. In developing functional photonic devices (e.g., topological antennas) for real-world applications, studying traveling-wave radiation characteristics based on chiral edge states deserves more attention. In this Letter, we propose a topological traveling-wave radiation system based on chiral edge states in photonic Chern insulators that show a steerable beam and reflection-free property. By introducing the perturbation controlled by magnetic fields, we demonstrate that exploiting a certain strength of perturbation can flexibly manipulate the radiation beam. In particular, the proposed dual-channel topological traveling-wave radiation system can not only improve the gain but also overcome the issue of impedance matching. The proposed configurations provide a novel way to manipulate topological-wave radiation and have potential applications for designing topological traveling-wave antennas with a reflection-free property.

Fractional Chern Insulators in Twisted Bilayer MoTe_{2}: A Composite Fermion Perspective.

The discovery of fractional Chern insulators (FCIs) in twisted bilayer MoTe_{2} has sparked significant interest in fractional topological matter without external magnetic fields. Unlike the flat dispersion of Landau levels, moiré electronic states are influenced by lattice effects within a nanometer-scale superlattice. This Letter examines the impact of these lattice effects on the topological phases in twisted bilayer MoTe_{2}, uncovering a family of FCIs with Abelian anyonic quasiparticles. Using a composite fermion approach, we identify a sequence of FCIs with fractional Hall conductivities σ_{xy}=[C/(2C+1)](e^{2}/h) linked to partial filling ν_{h} of holes of the topmost moiré valence band. These states emerge from incompressible composite fermion bands of Chern number C within a complex Hofstadter spectrum. This approach explains FCIs with Hall conductivities σ_{xy}=(2/3)e^{2}/h and σ_{xy}=(3/5)e^{2}/h at fractional fillings ν_{h}=2/3 and ν_{h}=3/5 observed in experiments, and uncovers other fractal FCI states. The Hofstadter spectrum reveals new phenomena, distinct from Landau levels, including a higher-order Van Hove singularity (HOVHS) at half-filling, leading to novel quantum phase transitions. This Letter offers a comprehensive framework for understanding FCIs in transition metal dichalcogenide moiré systems and highlights mechanisms for topological quantum criticality.

Proof of bulk-edge correspondence for band topology by Toeplitz algebra

Abstract We rigorously yet concisely prove the bulk-edge correspondence for general d-dimensional (dD) topological insulators in complex Altland-Zirnbauer classes, which states that the bulk topological number equals to the edge-mode index. Specifically, an essential formula is discovered that links the quantity expressed by Toeplitz algebra, i.e., hopping terms on the lattice with an edge, to the Fourier series on the bulk Brillouin zone. We then apply it to chiral models and utilize exterior differential calculations, instead of the sophisticated K -theory, to show that the winding number of bulk system equals to the Fredholm index of 1D edge Hamiltonian, or to the sum of edge winding numbers for higher odd dimensions. Moreover, this result is inherited to the even dimensional Chern insulators as each of them can be mapped to an odd dimensional chiral model. It is revealed that the Chern number of bulk system is identical to the spectral flow of 2D edge Hamiltonian, or to the negative sum of edge Chern numbers for higher even dimensions. Our methods and conclusions are friendly to physicists and could be easily extended to other physical scenarios.

Thermal crossover from a Chern insulator to a fractional Chern insulator in pentalayer graphene

Thermal crossover from a Chern insulator to a fractional Chern insulator in pentalayer graphene

Fractional Chern insulator candidate on a twisted bilayer checkerboard lattice

Fractional Chern insulator candidate on a twisted bilayer checkerboard lattice

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.

Boundary-dominated photonic Chern insulators with hyperbolic lattice geometry

Boundary-dominated photonic Chern insulators with hyperbolic lattice geometry

Strain-modulated antiferromagnetic Chern insulator in NiOsCl$_6$ monolayer

Abstract Recently, Chern insulators in an antiferromagnetic (AFM) phase have been suggested theoretically and predicted in a few materials. However, the experimental observation of two-dimensional AFM quantum anomalous Hall effect is still a challenge to date. In this work, we propose that an AFM Chern insulator can be realized in a two-dimensional monolayer of NiOsCl$_6$ modulated by a compressive strain. Strain modulation is accessible experimentally and used widely in predicting and tuning topological nontrivial phases. With first-principles calculations, we have investigated the structural, magnetic and electronic properties of NiOsCl$_6$. Its stability has been confirmed through molecular dynamical simulations, elasticity constant and phonon spectrum. It has a collinear AFM order, with opposite magnetic moments of 1.3 $\mu_B$ on each Ni/Os atom, respectively, and the Néel temperature is estimated to be 93 $K$. In the absence of strain, it functions as an AFM insulator with a direct gap with spin-orbital coupling included. Compressive strain will induce a transition from a normal insulator to a Chern insulator characterized by a Chern number $C = 1$, with a band gap of about 30 meV. This transition is accompanied by a structural distortion. Remarkably, the Chern insulator phase persists within the 3$\%$-10$\%$ compressive strain range, offering an alternative platform for the utilization of AFM materials in spintronic devices.

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.

Meandering conduction channels and the tunable nature of quantized charge transport

The discovery of the quantum Hall effect has established the foundation of the field of topological condensed matter physics. An amazingly accurate quantization of the Hall conductance, now enshrined in quantum metrology, is stable against any reasonable perturbation due to its topological protection. Conversely, the latter implies a form of censorship by concealing any local information from the observer. The spatial distribution of the current in a quantum Hall system is such a piece of information, which, thanks to spectacular recent advances, has now become accessible to experimental probes. It is an old question whether the original and intuitively compelling theoretical picture of the current, flowing in a narrow channel along the sample edge, is the physically correct one. Motivated by recent experiments locally imaging quantized current in a Chern insulator (Bi, Sb)[Formula: see text]Te[Formula: see text] heterostructure [Rosen et al., Phys. Rev. Lett. 129, 246602 (2022); Ferguson et al., Nat. Mater. 22, 1100-1105 (2023)], we theoretically demonstrate the possibility of a broad "edge state" generically meandering away from the sample boundary deep into the bulk. Further, we show that by varying experimental parameters one can continuously tune between the regimes with narrow edge states and meandering channels, all the way to the charge transport occurring primarily within the bulk. This accounts for various features observed in, and differing between, experiments. Overall, our findings underscore the robustness of topological condensed matter physics, but also unveil the phenomenological richness, hidden until recently by the topological censorship-most of which, we believe, remains to be discovered.

Quantum geometry in condensed matter

Abstract One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of electrons in condensed matter can be characterized by the geometry of the Hilbert space of their wave functions. Such quantum geometry, comprising of Berry curvature and quantum metric, can thus exert profound influences on various properties of materials. The dipoles of both Berry curvature and quantum metric produce nonlinear transport. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. The uniformly distributed momentum-space quantum geometry stabilizes the fractional Chern insulators and results in the fractional quantum anomalous Hall effect. We here review in detail quantum geometry in condensed matter, paying close attention to its effects on nonlinear transport, superconductivity, and topological properties. Possible future research directions in this field are also envisaged.

Rich magnon topology in triangular lattice magnets

The two-dimensional magnet has been an emerging and rapidly growing field. The nontrivial topological phenomenon in these materials is an attracting subject. Yet, the realization of such magnets exhibiting topological magnons remains a challenge. Here, employing the linear spin-wave theory and the first-principles calculations, we propose that variety of topological phases exist in the triangular ferromagnet. These include magnon Chern insulators and high-order topological insulators. Interestingly, these topological states can coexist within a certain parameter space, leading to a hybrid topological state. We propose that these topological phases can be realized via atomic substitutions inMnSe2orMnTe2single-layers. The following detailed analysis suggests that non-uniform Dzyaloshinsky-Moriya interactions are crucial in achieving topological magnons. Our work unveil a unique approach to obtaining non-trivial topological magnons in two-dimensional materials.

Quantum-metric-induced quantum Hall conductance inversion and reentrant transition in fractional Chern insulators

The quantum metric of single-particle wave functions in topological flat bands plays a crucial role in determining the stability of fractional Chern insulating (FCI) states. Here, we unravel that the quantum metric causes the many-body Chern number of the FCI states to deviate sharply from the expected value associated with partial filling of the single-particle topological flat band. Furthermore, the variation of the quantum metric in momentum space induces band dispersion through interactions, affecting the stability of the FCI states. This causes a reentrant transition into the Fermi liquid from the FCI phase as the interaction strength increases. Published by the American Physical Society 2024

Simulating a Chern Insulator with C = ±2 on Synthetic Floquet Lattice

Abstract The synthetic Floquet lattice, generated by multiple strong drives with mutually incommensurate frequencies, provides a powerful platform for quantum simulation of topological phenomena. In this study, we propose a 4-band tight-binding model of the Chern insulator with a Chern number C = ±2 by coupling two layers of the half Bernevig–Hughes–Zhang lattice and subsequently mapping it onto the Floquet lattice to simulate its topological properties. To determine the Chern number of our Floquet-version model, we extend the energy pumping method proposed by Martin et al. [2017 Phys. Rev. X 7 041008] and the topological oscillation method introduced by Boyers et al. [2020 Phys. Rev. Lett. 125 160505], followed by numerical simulations for both methodologies. The simulation results demonstrate the successful extraction of the Chern number using either of these methods, providing an excellent prediction of the phase diagram that closely aligns with the theoretical one derived from the original bilayer half Bernevig–Hughes–Zhang model. Finally, we briefly discuss a potential experimental implementation for our model. Our work demonstrates significant potential for simulating complex topological matter using quantum computing platforms, thereby paving the way for constructing a more universal simulator for non-interacting topological quantum states and advancing our understanding of these intriguing phenomena.

Theory of topological exciton insulators and condensates in flat Chern bands

Excitons are the neutral quasiparticles that form when Coulomb interactions create bound states between electrons and holes. Due to their bosonic nature, excitons are expected to condense and exhibit superfluidity at sufficiently low temperatures. In interacting Chern insulators, excitons may inherit the nontrivial topology and quantum geometry from the underlying electron wavefunctions. We theoretically investigate the excitonic bound states and superfluidity in flat-band insulators pumped with light. We find that the exciton wavefunctions exhibit vortex structures in momentum space, with the total vorticity being equal to the difference of Chern numbers between the conduction and valence bands. Moreover, both the exciton binding energy and the exciton superfluid density are proportional to the Brillouin-zone average of the quantum metric and the Coulomb potential energy per unit cell. Spontaneous emission of circularly polarized light from radiative decay is a detectable signature of the exciton vorticity. We propose that the vorticity can also be experimentally measured via the nonlinear anomalous Hall effect, whereas the exciton superfluidity can be detected by voltage-drop quantization through a combination of quantum geometry and Aharonov-Casher effect. Topological excitons and their superfluid phase could be realized in flat bands of twisted Van der Waals heterostructures.

Tutorial 2.0: computing topological invariants in 3D photonic crystals

The field of topological photonics has been on the rise due to its versatility in manufacturing and its applications as topological lasers or unidirectional waveguides. Contrary to 1D or 2D photonic crystals, the transversal and vectorial nature of light in 3D precludes using standard methods for diagnosing topology. This tutorial describes the problems that emerge in computing topological invariants in 3D photonic crystals and the diverse strategies for overcoming them. Firstly, we introduce the fundamentals of light propagation in 3D periodic media and expose the complications of directly implementing the usual topological diagnosis tools. Secondly, we describe the properties of electromagnetic Wilson loops and how they can be used to diagnose topology and compute topological invariants in 3D photonic crystals. Finally, we apply the previously described methods to several examples of 3D photonic crystals showing different topological phases, such as Weyl nodes and walls, 3D photonic Chern insulators, and photonic axion insulators.