Topological Chern Number Research Articles

Variational MonteCarlo Study of the 1/9-Magnetization Plateau in Kagome Antiferromagnets.

Motivated by very recent experimental observations of the 1/9-magnetization plateaus in YCu_{3}(OH)_{6+x}Br_{3-x} and YCu_{3}(OD)_{6+x}Br_{3-x}, our study delves into the magnetic-field-induced phase transitions in the nearest-neighbor antiferromagnetic Heisenberg model on the kagome lattice using the variational MonteCarlo technique. We uncover a phase transition from a zero-field Dirac spin liquid to a field-induced magnetically disordered phase that exhibits the 1/9-magnetization plateau. Through a comprehensive analysis encompassing the magnetization distribution, spin correlations, chiral order parameter, topological entanglement entropy, ground-state degeneracy, Chern number, and excitation spectrum, we pinpoint the phase associated with this magnetization plateau as a chiral Z_{3} topological quantum spin liquid and elucidate its diverse physical properties.

Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Superconductivity in monolayer and few-layer graphene. II. Topological edge states and Chern numbers

Superconductivity in monolayer and few-layer graphene. II. Topological edge states and Chern numbers

Quantized Berry winding from an emergent $\mathcal{PT}$ symmetry

Linear crossings of energy bands occur in a wide variety of materials. In this paper we address the question of the quantization of the Berry winding characterizing the topology of these crossings in dimension D=2D=2. Based on the historical example of 22-bands crossing occuring in graphene, we propose to relate these Berry windings to the topological Chern number within a D=3D=3 dimensional extension of these crossings. This dimensional embedding is obtained through a choice of a gap-opening potential. We show that the presence of an (emergent) \mathcal{PT}𝒫𝒯 symmetry, local in momentum and antiunitary, allows the quantization of the Berry windings as multiples of \piπ. We illustrate this quantization mechanism on a variety of three-band crossings.

Entanglement spectrum and entropy in Floquet topological matter

Entanglement is one of the most fundamental features of quantum systems. In this work, we obtain the entanglement spectrum and entropy of Floquet noninteracting fermionic lattice models and build their connections with Floquet topological phases. Topological winding and Chern numbers are introduced to characterize the entanglement spectrum and eigenmodes. Correspondences between the spectrum and topology of entanglement Hamiltonians under periodic boundary conditions and topological edge states under open boundary conditions are further established. The theory is applied to Floquet topological insulators in different symmetry classes and spatial dimensions. Our work thus provides a useful framework for the study of rich entanglement patterns in Floquet topological matter.

Observation of bulk-edge correspondence in topological pumping based on a tunable electric circuit

In topological phases, localized edge states protected by the bulk topological number appear. This phenomenon is known as the bulk-edge correspondence, which has been theoretically established in various topological systems including topological pumping. There have been experimental works on topological edge states. However, direct observation of bulk topological numbers remains challenging. Here, we experimentally observe both topological edge states and bulk Chern numbers near the topological phase transition in a tunable electric circuit system. The admittance matrix of the system is accurately engineered by using variable capacitors, leading to the implementation of classical analogue of the quantum Hall phase and the topological transition. The topological numbers of the edge states are determined experimentally, where the locations of the edge states are flipped in the topological transition. The corresponding bulk Chern numbers are determined by the Středa’s formula to the impedance spectrum. Our electric circuit scheme is a promising platform to provide a crucial understanding of topological phenomena, and opens up the possibility toward functional energy transfer mechanisms.

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.

Integer quantum Hall effect of two-component hard-core bosons in a topological triangular lattice

We study the many-body ground states of two-component hardcore bosons in topological triangular lattice models. Utilizing exact diagonalization and density-matrix renormalization group calculations, we demonstrate that at commensurate two-thirds filling per lattice site, two-component bosonic integer quantum hall (BIQH) effect emerges with the associated $\mathbf{K}=\begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix}$ matrix under strong intercomponent Hubbard repulsion. The topological nature is further elucidated by (i) a unique ground state degeneracy with a robust spectrum gap, (ii) a quantized topological Chern number matrix $\mathbf{C}=\mathbf{K}^{-1}$, and (iii) two counterpropagating edge branches. Moreover, with increasing nearest-neighbor repulsions, the ground state undergoes a first-order transition from a BIQH liquid to a commensurate solid order.

Higher Chern number states in curved periodic nanowires

The coupling between the spin and momentum degrees of freedom due to spin–orbit interactions (SOI) suggests that the strength of the latter can be modified by controlling the motion of the charge carriers. In this paper, we investigate how the effective SOI can be modulated by constraining the motion of charge carriers to curved waveguides thereby introducing real-space geometric curvature in their motion. The change in the SOI can in turn induce topological phase transitions in the system. Specifically, we study how the introduction of periodic sinusoidal curvature in nanowires with intrinsic SOC can induce the onset of mid-gap topologically protected edge states, which can be characterized by a topological invariant or Chern number. The Chern number corresponds to the number of discrete charges that would be pumped across the length of the nanowire when the phase of a sliding gate potential relative to that of the sinusoidal curvature is varied adiabatically over a complete period. In addition, coupling to an external magnetization can be utilized as an experimental knob to modify the Chern number by displacing the energies of the curvature-induced bands relative to one another. The magnetization can be tuned to achieve large discrete jumps in the number of pump charges per phase period.

Topological Quantum Phase Transitions of Anisotropic Antiferromagnetic Kitaev Model Driven by Magnetic Field

AbstractThe evolution of quantum spin liquid states (QSL) of the anisotropic antiferromagnetic (AFM) Kitaev model with the [001] magnetic field by utilizing the finite‐temperature Lanczos method (FTLM) is investigated. In this anisotropic Kitaev model with and (K is the energy unit), due to the competition between anisotropy and magnetic field, the system emerges four exotic quantum phase transitions (QPTs) when and , while only two QPTs when . At these magnetic‐field tuning quantum phase transition points, the low‐energy excitation spectrums appear level crossover, and the specific heat, magnetic susceptibility and Wilson ratio display anomalies; accordingly, the topological Chern number may also change. These results demonstrate that the anisotropic interacting Kitaev model with modulating magnetic field displays more rich phase diagrams, in comparison with the isotropic Kitaev model.

Response of quantum spin Hall insulators to Zeeman fields, and device design based on stanene

The nontrivial topology of materials has been one of the keys for breaking through to a new stage of electronics involving phenomena such as topologically protected quantum spin Hall (QSH) states and quantum anomalous Hall (QAH) states. In this work, we study the Kane-Mele model with next-nearest-neighbor Rashba spin-orbit coupling (SOC), which accurately describes the QSH effect of two-dimensional group IV A materials. By investigating electronic structures with different SOCs and Zeeman fields, we obtain various phase transitions and plot rich phase diagrams where a time-reversal-symmetry-broken QSH insulator and three types of QAH insulators are identified based on calculations of topological invariants, including total and spin Chern numbers and Wannier function centers. In addition, the edge modes and spin texture within the bulk gap are also nontrivial. Based on the results of the model, we also study a real QSH material, stanene, with a large bulk gap and propose a scheme based on its transport properties to design a spin-resolved filter.

Topological Imbert–Fedorov shifts in silicene

In this paper, we study the topological Imbert–Fedorov (IF) shifts of a light beam reflected from a silicene–substrate system in the presence of an externally applied electric field and circularly polarized beam. Silicene is a topological material and due to the strong spin–orbit interaction has a rich phase diagram. Using the famous Kubo formalism, we derive analytical expressions for the longitudinal and transverse Hall conductivities of silicene. By employing angular spectrum analysis, the analytical expression of the spatial IF shift is obtained. We show that electric field/circularly polarized light beam modulated positive and negative spatial beam shifts can be obtained by tuning the angle of incidence near the Brewster angle and the incident photonic energy in the terahertz regime. Furthermore, we also investigate the IF shifts in different topological phases of the silicene by driving the system through distinct topological quantum phase transitions (TQPTs), i.e., from topologically nontrivial to a semi-metallic state and further to a band insulating state by manipulating the interplay between the electric field and circularly polarized light beam. We believe these results are helpful for developing novel optoelectronic, spintronic, and valleytronic devices and may provide a possible technique for probing Berry curvature, topological Chern numbers, and TQPTs by direct optical measurement.

Bosonic Halperin (441) Fractional Quantum Hall Effect at Filling Factor ν = 2/5

Quantum Hall effects with multicomponent internal degrees of freedom facilitate the playground of novel emergent topological orders. Here, we explore the correlated topological phases of two-component hardcore bosons at a total filling factor ν = 2/5 in topological lattice models under the interplay of intracomponent and intercomponent repulsions. We give the numerical demonstration of the emergence of Halperin (441) fractional quantum Hall effect based on exact diagonalization and density-matrix renormalization group methods. We elucidate its topological features including the degeneracy of the ground state, fractionally quantized topological Chern number matrix and chiral edge modes.

Topological fluctuating electron-hole Cooper pairs in graphene-GaAs heterostructures

Fluctuating Cooper pairs formed by spatially separated electrons and holes are precursors of their equilibrium condensation. Their presence strongly impacts transport phenomena and interlayer tunneling in double-layer systems above the transition temperature. Here, we consider a hybrid graphene/quantum-well double-layer system and focus on the dynamics of fluctuating Cooper pairs formed by conventional electrons and Dirac holes. We demonstrate that the chiral nature of Dirac holes is manifested in the presence of two (almost) degenerate competing pairing channels, which are intertwined by effective pseudospin-orbit interactions. We argue that the spectrum of the Ginzburg-Landau Hamiltonian describing the energetics of fluctuating Cooper pairs is geometrically nontrivial and can be characterized by the half-integer topological Chern number. We derive a kinetic equation for fluctuating Cooper pairs and demonstrate that their nontrivial geometries generate two anomalous velocities of distinct geometrical origins. These velocities are intricately connected with the Berry curvature and the quantum metric for the Ginzburg-Landau Hamiltonian, respectively. The resulting anomalous contributions to conductivity are singular at the transition temperature, and we discuss possible setups for their experimental observation.

Electron transport probing the electrically tunable topological phase transition in a Dirac semimetal

Electron transport in a Dirac semimetal (DSM) quantum wire without/with an external electric field is investigated by using the Green-function-based Landauer-B\"uttiker formula. Normal-topological phase transition is certificated by the different characteristics between the DSM quantum wire without and with the electric field, including the orbital decomposition for the energy bands, transversally local charge and spin polarization density distributions, topological Chern number, transport behaviors, and disorder effects. Topologically trivial surface states are found in the former system and their transport can be localized by disorders. However, topologically nontrivial surface states are generated in the latter one and their transport is robust against strong disorders when the electric field strength is larger than a critical value. In addition, switching effect of the electron transport in the DSM quantum wire can be achieved by varying the electric field strength. Further studies indicate that the switching effect shows a strong robustness against disorder, displaying the feasibility of designing a topological field effect transistor based on the considered system.

Topological charge pumping in quasiperiodic systems characterized by the Bott index

We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, regardless of the detail of quasiperiodic models. We apply this formalism to the Fibonacci-Rice-Mele model which we made from Fibonacci lattice, a well-known quasiperiodic system, and Rice-Mele model. We find that these quasiperiodic systems show topological charge pumping with a multi-level behavior due to the fractal nature of the Fibonacci lattice. Such multi-level pumping behaviors can be understood by a real space renormalization group analysis.

Spin-momentum locked interface modes based on transverse resonance and Zak phase in finite thickness dielectric slabs

The field of topological photonics has made great strides in the past decade with many new designs based on bandgap and band inversion structures that provide robust, unidirectional, and reflection-free propagation of energy. The topological invariant or Chern number of a metamaterial guarantees the existence of topologically protected edge modes. However, its mathematical application to real systems is not always straightforward and can be greatly simplified by reducing the dimensional complexity of the problem by the calculation of a Zak phase that determines the topological phase in just one dimension. This work explores two methods of creating interface modes with finite height dielectric slabs: (1) transverse resonance through variable edge truncation of a photonic crystal (PhC) or cell sliding and (2) a uni-axial topological phase by means of scaling the internal features of a unit cell. The proposed metamaterial devices use the same C4v symmetric unit cell structure on both sides of the interface and are finite in all three dimensions, allowing for easy fabrication, excitation, and implementation in real-world applications. The all-dielectric design also enables an easy transition to and from conventional PhC waveguides and lends itself well to operation in frequencies spanning across the microwave and optical spectrum without concerns of additional metallic losses in the THz region.

Fractional quantum Hall effect of Bose-Fermi mixtures

Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the $\mathbf{K}=\begin{pmatrix} m & 1\\ 1 & n\\ \end{pmatrix}$ matrix (even $m$ for boson and odd $n$ for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological $(mn-1)$-fold ground-state degeneracy equivalent to the determinant of the $\mathbf{K}$ matrix, (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the $\mathbf{K}$ matrix, and (iii) two parallel-propagating chiral edge branches with level counting $1,2,5,10$ consistent with the conformal field theory description.

Moiré magnons in twisted bilayer magnets with collinear order

We explore the moir\'e magnon bands in twisted bilayer magnets with next-nearest neighboring Dzyaloshinskii-Moriya interactions, assuming that the out-of-plane collinear magnetic order is preserved under weak interlayer coupling. By calculating the magnonic band structures and the topological Chern numbers for four representative cases, we find that (i) the valley moir\'e bands are extremely flat over a wide range of continuous twist angles; (ii) the topological Chern numbers of the lowest few flat bands vary significantly with the twist angle; and (iii) the lowest few topological flat bands in bilayer antiferromagnets entail nontrivial thermal spin transport in the transverse direction; These properties make twisted bilayer magnets an ideal platform to study the magnonic counterparts of moir\'e electrons, where the statistical distinction between magnons and electrons leads to fundamentally new physical behavior.

Dynamical quantum phase transition from a critical quantum quench

We study the dynamical quantum phase transition of the critical quantum quench, in which the prequenched Hamiltonian, or the postquenched Hamiltonian, or both of them are set to be the critical points of equilibrium quantum phase transitions, we find half-quantized or unquantized dynamical topological order parameter and dynamical Chern number; these results and also the existence of dynamical quantum phase transition are all closely related to the singularity of the Bogoliubov angle at the gap-closing momentum. The effects of the singularity may also be canceled out if both the prequenched and postquenched Hamiltonians are critical, then the dynamical topological order parameter and dynamical Chern number restore to integer ones. Our findings show that the widely accepted definitions of dynamical topological order parameter and dynamical Chern number are problematic for the critical quenches in the perspective of topology, which call for new definitions of them.