Chern Number Research Articles

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.

Evidence for π-Shifted Cooper Quartets and Few-Mode Transport in PbTe Nanowire Three-Terminal Josephson Junctions.

Josephson junctions are typically characterized by a single phase difference across two superconductors. This conventional two-terminal Josephson junction can be generalized to a multiterminal device where the Josephson energy contains terms with contributions from multiple independent phase variables. Such multiterminal Josephson junctions (MTJJs) are being considered as platforms for engineering effective Hamiltonians with nontrivial topologies, such as Weyl crossings and higher-order Chern numbers. These prospects rely on the ability to create MTJJs with nonclassical multiterminal couplings in which only a few quantum modes are populated. Here, we demonstrate these requirements in a three-terminal Josephson junction fabricated on selective-area-grown (SAG) PbTe nanowires. We observe signatures of a π-shifted Josephson effect, consistent with interterminal couplings mediated by four-particle quantum states called Cooper quartets. We further observe a supercurrent coexistent with a non-monotonic evolution of the conductance with gate voltage, indicating transport mediated by a few quantum modes in both two- and three-terminal devices.

Topological hidden phase transition in honeycomb bilayers with a high Chern number

Topological hidden phase transition in honeycomb bilayers with a high Chern number

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.

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.

Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang.

Coexistence of Chiral Edge and Localized States in a High Chern Number Topological Photonic Alloy

Coexistence of Chiral Edge and Localized States in a High Chern Number Topological Photonic Alloy

Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.

Spinon Pairing Induced by Chiral In-Plane Exchange and the Stabilization of Odd-Spin Chern Number Spin Liquid in Twisted MoTe_{2}.

The unusual structure and symmetry of low-energy states in twisted transition metal dichalcogenides leads to large in-plane spin-exchange interactions between spin-valley locked holes. We demonstrate that this exchange interaction can stabilize a gapped spin-liquid phase with a quantized spin-Chern number of 3 when the twist angle is sufficiently small and the system lies in a Mott insulating phase. The gapped spin liquid may be understood as arising from spinon pairing in the DIII Altland-Zirnbauer symmetry class. Applying an out-of-plane electric field or increasing the twist angle is shown to drive a transition, respectively, to an anomalous Hall insulator or an in-plane antiferromagnet. Recent experiments indicate that a spin-Chern number 3 phase occurs in twisted MoTe_{2} at small twist angles with a transition to a quantum anomalous Hall phase as the twist angle is increased above a critical value of about 2.5° in the absence of an applied electric field.

Integrated terahertz topological valley-locked power divider with arbitrary power ratios

Integrated power dividers (PDs) are essential in terahertz (THz) communication and radar systems, but miniaturization often leads to performance degradation due to fabrication inaccuracies and sharp bends. Topological photonics offers a solution to these issues, yet creating THz power dividers with arbitrary splitting ratios remains challenging. We present a design methodology for on-chip topological THz power dividers with customizable splitting ratios using valley-locked photonic crystals. These crystals feature a tri-layered structure with two distinct valley Chern number layers and an intermediate semimetal layer. Utilizing the Jackiw–Rebbi model, we show that the characteristic impedance of the valley-locked photonic crystals, and thus the power division ratio, can be tuned by adjusting the semimetal layer width. Our approach is validated through simulations and experiments for both equal (1:1) and unequal (4:9) power ratios. This method enables efficient navigation around sharp bends and robust THz on-chip connectivity.

Structure of the Kuranishi spaces of pairs of Kähler manifolds and polystable Higgs bundles

AbstractLet be a compact Kähler manifold and be a Higgs bundle over it. We study the structure of the Kuranishi space for the pair when the Higgs bundle admits a harmonic metric or equivalently when the Higgs bundle is polystable and the Chern classes are 0. Under such assumptions, we show that the Kuranishi space of the pair is isomorphic to the direct product of the Kuranishi space of and the Kuranishi space of . Moreover, when is a Riemann surface and is stable and the degree is 0, we show that the deformation of the pair is unobstructed and calculate the dimension of the Kuranishi space.

Phononic Weyl nodal lines and Weyl pairs in van der Waals heavy fermion material CeSiI

Abstract Topological phonon is a new frontier in the field of topological materials. Different from electronic structures, phonons are bosons and most topological phonons are metallic form. Previous studies about topological phonon states were focused on threedimensional (3D) materials. Owing to the lack of material candidates, two-dimensional (2D) and van der Waals f-electron-related topological phonons were rarely reported. Based on first-principles calculations, we investigated the topological phononic state in the heavy fermion material CeSiI with a layered structure. Both three-dimensional (3D) bulk and two-dimensional (2D) monolayers have topological nontrivial states in the rarely seen f-electron dominated van der Waals metal. Owing to the PT and C 3z symmetries, Weyl nodal lines with non-zero Chern numbers exist on the hinge of the Brillouin zone (BZ). Protected by C 3z rotation symmetry, three pairs of Weyl points with ±π Berry phase exist at point K near the Frequency of 8 and 10 THz. In addition to the bulk topological charges, corresponding surface/edge states were also systematically analyzed, which gives a consistent understanding. Our results proposed another interesting point in the newly discovered rear earth heavy fermion material CeSiI and are helpful for future experimental study of its topological phonons.

Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions

In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev’s holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) — i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields — as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.

Supersymmetry dictated topology in periodic gauge fields and realization in strained and twisted 2D materials

Supersymmetry (SUSY) of a Hamiltonian dictates double degeneracy between a pair of superpartners (SPs) transformed by supercharge, except at zero energy where modes remain unpaired in many cases. Here we explore a SUSY of complete isospectrum between SPs-with paired zero modes-realized by 2D electrons in zero-flux periodic gauge fields, which can describe twisted or periodically strained 2D materials. We find their low-energy sector containing zero (or threshold) modes must be topologically non-trivial, by proving that Chern numbers of the two SPs have a finite difference dictated by the number of zero modes and energy dispersion in their vicinity. In 30° twisted bilayer (double bilayer) transition metal dichalcogenides subject to periodic strain, we find one SP is topologically trivial in its lowest miniband, while the twin SP of identical dispersion has a Chern number of 1 (2), in stark contrast to time-reversal partners that have to be simultaneously trivial or nontrivial. For systems whose physical Hamiltonian corresponds to the square root of a SUSY Hamiltonian, such as twisted or strained bilayer graphene, we reveal that topological properties of the two SUSY SPs are transferred respectively to the conduction and valence bands, including the contrasted topology in the low-energy sector and identical topology in the high-energy sector. This offers a unified perspective for understanding topological properties in many flat-band systems described by such square-root models. Both types of SUSY systems provide unique opportunities for exploring correlated and topological phases of matter.

Uniqueness of Landau levels and their analogs with higher Chern numbers

Landau levels are the eigenstates of a charged particle in two dimensions under a magnetic field and are at the heart of the integer and fractional quantum Hall effects, which are two prototypical phenomena showing topological features. Following recent discoveries of fractional quantum Hall phases in van der Waals materials, there is a rapid progress in understanding of the precise condition under which the fractional quantum Hall phases can be stabilized. It is now understood that the key to obtaining the fractional quantum Hall phases is the energy band whose eigenstates are holomorphic functions in both real and momentum space coordinates. Landau levels are indeed examples of such energy bands with an additional special property of having flat geometrical features. In this paper, we prove that, in fact, the only energy eigenstates having holomorphic wave functions with a flat geometry are the Landau levels and their higher Chern number analogs. Since it has been known that any holomorphic eigenstates can be constructed from the ones with a flat geometry such as the Landau levels, our uniqueness proof of the Landau levels allows one to construct any possible holomorphic eigenstate with which the fractional quantum Hall phases can be stabilized. Published by the American Physical Society 2024

On the intersection form of fillings

AbstractWe prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [Lazarev, Geom. Funct. Anal. 30 (2020), no. 1, 188–254]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.

Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension

Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension

Valley selections and control of topological phases by bicircular laser field in transition-metal dichalcogenides

The energy spectra of transition metal dichalcogenides are influenced by three orbitals, forming a three-band model with strong spin-orbit interaction. Analyzing these bands through a two-band projection suffices for exploring electronic and optical properties. We examine topological phases and the spin-valley Hall effect, derived from spin-valley Chern numbers. Spin-orbit interaction breaks inversion symmetry but preserves time-reversal symmetry, resulting in non-zero spin-valley-dependent Hall conductivities, crucial for both spin and valley Hall effects. Interestingly, direct optical manipulation of the valley degree of freedom can be achieved using a trefoil field generated by a bicircular field. Selectable valley conductivity occurs when the field orientation matches the lattice symmetry, leading to a quantum anomalous Hall effect without magnetization. This approach suggests the potential for valley selection via field orientation, with applications in valleytronics and spintronics devices, enabling qubit encoding.

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.

Magnetic Exchange Mechanism and Quantized Anomalous Hall Effect in Bi2Se3 Film with a CrWI6 Monolayer.

Magnetizing the surface states of topological insulators without damaging their topological features is a crucial step for realizing the quantum anomalous Hall (QAH) effect and remains a challenging task. The TI-ferromagnetic material interface system was constructed and studied by the density functional theory (DFT). A two-dimensional magnetic semiconductor CrWI6 has been proven to effectively magnetize topological surface states (TSSs) via the magnetic proximity effect. The non-trivial phase was identified in the Bi2Se3 (BS) films with six quantum layers (QL) within the CrWI6/BS/CrWI6 heterostructure. BS thin films exhibit the generation of spin splitting near the TSSs, and a band gap of approximately 2.9 meV is observed at the Γ in the Brillouin zone; by adjusting the interface distance of the heterostructure, we increased the non-trivial band gap to 7.9 meV, indicating that applying external pressure is conducive to realizing the QAH effect. Furthermore, the topological non-triviality of CrWI6/6QL-BS/CrWI6 is confirmed by the nonzero Chern number. This study furnishes a valuable guideline for the implementation of the QAH effect at elevated temperatures within heterostructures comprising two-dimensional (2D) magnetic monolayers (MLs) and topological insulators.