Double and Quadruple Flat Bands Tuned by Alternative Magnetic Fluxes in Twisted Bilayer Graphene.

Twisted bilayer graphene (TBG) can host the moiré energy flat bands with twofold degeneracy serving as a fruitful playground for strong correlations and topological phases. However, the number of degeneracy is not limited to two. Introducing a spatially alternative magnetic field, we report that the induced magnetic phase becomes an additional controllable parameter and leads to an undiscovered generation of fourfold degenerate flat bands. This emergence stems from the band inversion at the Γ point near the Fermi level with a variation of both twisted angle and magnetic phase. We present the conditions for the emergence of multifold degenerate flat bands, which are associated with the eigenvalue degeneracy of a Birman-Schwinger operator. Using holomorphic functions, which explain the origin of the double flat bands in the conventional TBG, we can generate analytical wave functions in the magnetic TBG to show absolute flatness with fourfold degeneracy. Moreover, we identify an orbital-related intervalley coherent state as the many-body ground state at charge neutrality. In contrast, the conventional TBG has only two moiré energy flat bands, and the highly degenerate flat bands with additional orbital channels in this magnetic platform might bring richer correlation physics.

Topological quadratic-node semimetal in a photonic microring lattice.

Graphene, with its two linearly dispersing Dirac points with opposite windings, is the minimal topological nodal configuration in the hexagonal Brillouin zone. Topological semimetals with higher-order nodes beyond the Dirac points have recently attracted considerable interest due to their rich chiral physics and their potential for the design of next-generation integrated devices. Here we report the experimental realization of the topological semimetal with quadratic nodes in a photonic microring lattice. Our structure hosts a robust second-order node at the center of the Brillouin zone and two Dirac points at the Brillouin zone boundary-the second minimal configuration, next to graphene, that satisfies the Nielsen-Ninomiya theorem. The symmetry-protected quadratic nodal point, together with the Dirac points, leads to the coexistence of massive and massless components in a hybrid chiral particle. This gives rise to unique transport properties, which we demonstrate by directly imaging simultaneous Klein and anti-Klein tunnelling in the microring lattice.

Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs.

Majorana zero modes (MZMs) obey non-Abelian statistics and are considered building blocks for constructing topological qubits1,2. Iron-based superconductors with topological bandstructures have emerged as promising hosting materials, because isolated candidate MZMs in the quantum limit have been observed inside the topological vortex cores3-9. However, these materials suffer from issues related to alloying induced disorder, uncontrolled vortex lattices10-13 and a low yield of topological vortices5-8. Here we report the formation of an ordered and tunable MZM lattice in naturally strained stoichiometric LiFeAs by scanning tunnelling microscopy/spectroscopy. We observe biaxial charge density wave (CDW) stripes along the Fe-Fe and As-As directions in the strained regions. The vortices are pinned on the CDW stripes in the As-As direction and form an ordered lattice. We detect that more than 90 per cent of the vortices are topological and possess the characteristics of isolated MZMs at the vortex centre, forming an ordered MZM lattice with the density and the geometry tunable by an external magnetic field. Notably, with decreasing the spacing of neighbouring vortices, the MZMs start to couple with each other. Our findings provide a pathway towards tunable and ordered MZM lattices as a platform for future topological quantum computation.

Yu-Shiba-Rusinov States in a Superconductor with Topological Z_{2} Bands.

A Yu-Shiba-Rusinov (YSR) state is a localized in-gap state induced by a magnetic impurity in a superconductor. Recent experiments used an STM tip to manipulate the exchange coupling between an Fe adatom and the FeTe_{0.55}Se_{0.45} superconductor possessing a Z_{2} nontrivial band structure with topological surface states. As the tip moves close to the single Fe adatom, the energy of the in-gap state modulates and exhibits a zero-energy crossing followed by an unusual return to zero energy, which cannot be understood by coupling the magnetic impurity to the superconducting topological surface Dirac cone. Here, we numerically and analytically study the YSR states in superconductors with nontrivial Z_{2} bands and show the emergence of the two zero-energy crossings as a function of the exchange coupling between the magnetic impurity and the bulk states. We analyze the role of the topological surface states and compare in-gap states to systems with trivial Z_{2} bands. The spin polarization of the YSR states is further studied for future experimental measurement.

Dirac Fermion Cloning, Moiré Flat Bands, and Magic Lattice Constants in Epitaxial Monolayer Graphene.

Tuning interactions between Dirac states in graphene has attracted enormous interest because it can modify the electronic spectrum of the 2D material, enhance electron correlations, and give rise to novel condensed-matter phases such as superconductors, Mott insulators, Wigner crystals, and quantum anomalous Hall insulators. Previous works predominantly focus on the flat band dispersion of coupled Dirac states from different twisted graphene layers. In this work, a new route to realizing flat band physics in monolayer graphene under a periodic modulation from substrates is proposed. Graphene/SiC heterostructure is taken as a prototypical example and it is demonstrated experimentally that the substrate modulation leads to Dirac fermion cloning and, consequently, the proximity of the two Dirac cones of monolayer graphene in momentum space. Theoretical modeling captures the cloning mechanism of the Dirac states and indicates that moiré flat bands can emerge at certain magic lattice constants of the substrate, specifically when the period of modulation becomes nearly commensurate with the ( 3 × 3 ) R 30 o \[(\sqrt 3 \; \times \;\sqrt 3 )R{30^o}\] supercell of graphene. The results show that epitaxial single monolayer graphene on suitable substrates is a promising platform for exploring exotic many-body quantum phases arising from interactions between Dirac electrons.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points.

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.

Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals.

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this Letter, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the oriented nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf link. We extend our theory to 3D non-Hermitian multiband exceptional line semimetals. Our work provides a recipe to understand the transition of the knot topology for protected nodal lines.

Scalable Majorana vortex modes in iron-based superconductors.

The iron-based superconductor FeTe x Se1-x is one of the material candidates hosting Majorana vortex modes residing in the vortex cores. It has been observed by recent scanning tunneling spectroscopy measurement that the fraction of vortex cores having zero-bias peaks decreases with increasing magnetic field on the surface of FeTe x Se1-x . The hybridization of two Majorana vortex modes cannot simply explain this phenomenon. We construct a three-dimensional tight-binding model simulating the physics of over a hundred Majorana vortex modes in FeTe x Se1-x . Our simulation shows that the Majorana hybridization and disordered vortex distribution can explain the decreasing fraction of the zero-bias peaks observed in the experiment; the statistics of the energy peaks off zero energy in our Majorana simulation are in agreement with the experiment. These agreements lead to an important indication of scalable Majorana vortex modes in FeTe x Se1-x . Thus, FeTe x Se1-x can be one promising platform having scalable Majorana qubits for quantum computing.

Realization of Symmetry-Enforced Two-Dimensional Dirac Fermions in Nonsymmorphic α-Bismuthene.

Two-dimensional (2D) Dirac-like electron gases have attracted tremendous research interest ever since the discovery of free-standing graphene. The linear energy dispersion and nontrivial Berry phase play a pivotal role in the electronic, optical, mechanical, and chemical properties of 2D Dirac materials. The known 2D Dirac materials are gapless only within certain approximations, for example, in the absence of spin-orbit coupling (SOC). Here, we report a route to establishing robust Dirac cones in 2D materials with nonsymmorphic crystal lattice. The nonsymmorphic symmetry enforces Dirac-like band dispersions around certain high-symmetry momenta in the presence of SOC. Through µ-ARPES measurements, we observe Dirac-like band dispersions in α-bismuthene. The nonsymmorphic lattice symmetry is confirmed by µ-low-energy electron diffraction and scanning tunneling microscopy. Our first-principles simulations and theoretical topological analysis demonstrate the correspondence between nonsymmorphic symmetry and Dirac states. This mechanism can be straightforwardly generalized to other nonsymmorphic materials. The results enlighten the search of symmetry-enforced Dirac fermions in the vast uncharted world of nonsymmorphic 2D materials.

Engineering of many-body Majorana states in a topological insulator/s-wave superconductor heterostructure.

We study a vortex chain in a thin film of a topological insulator with proximity-induced superconductivity-a promising platform to realize Majorana zero modes (MZMs)-by modeling it as a two-leg Majorana ladder. While each pair of MZMs hybridizes through vortex tunneling, we hereby show that MZMs can be stabilized on the ends of the ladder with the presence of tilted external magnetic field and four-Majorana interaction. Furthermore, a fruitful phase diagram is obtained by controlling the direction of magnetic field and the thickness of the sample. We reveal many-body Majorana states and interaction-induced topological phase transitions and also identify trivial-superconducting and commensurate/incommensurate charge-density-wave states in the phase diagram.

Topological nodal-line fermions in spin-orbit metal PbTaSe2.

Topological semimetals can support one-dimensional Fermi lines or zero-dimensional Weyl points in momentum space, where the valence and conduction bands touch. While the degeneracy points in Weyl semimetals are robust against any perturbation that preserves translational symmetry, nodal lines require protection by additional crystalline symmetries such as mirror reflection. Here we report, based on a systematic theoretical study and a detailed experimental characterization, the existence of topological nodal-line states in the non-centrosymmetric compound PbTaSe2 with strong spin-orbit coupling. Remarkably, the spin-orbit nodal lines in PbTaSe2 are not only protected by the reflection symmetry but also characterized by an integer topological invariant. Our detailed angle-resolved photoemission measurements, first-principles simulations and theoretical topological analysis illustrate the physical mechanism underlying the formation of the topological nodal-line states and associated surface states for the first time, thus paving the way towards exploring the exotic properties of the topological nodal-line fermions in condensed matter systems.

Stabilization of Majorana modes in magnetic vortices in the superconducting phase of topological insulators using topologically trivial bands.

It has been shown that doped topological insulators, up to a certain level of doping, still preserve some topological signatures of the insulating phase such as axionic electromagnetic response and the presence of a Majorana mode in the vortices of a superconducting phase. Multiple topological insulators such as HgTe, ScPtBi, and other ternary Heusler compounds have been identified and generically feature the presence of a topologically trivial band between the two topological bands. In this Letter we show that the presence of such a trivial band can stabilize the topological signature over a much wider range of doping. Specifically, we calculate the structure of vortex modes in the superconducting phase of doped topological insulators, a model that captures the features of HgTe and the ternary Heusler compounds. We show that, due to the hybridization with the trivial band, Majorana modes are preserved over a large, extended doping range for p doping. In addition to presenting a viable system where much less fine-tuning is required to observe the Majorana modes, our analysis opens a route to study other topological features of doped compounds that cannot be modeled using the simple Bi(2)Se(3) Dirac model.