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Recently, a quantum anomalous Hall (QAH) state was observed in AB-stacked moir & eacute; MoTe2/WSe2 heterobilayers at half filling. More recent layer-resolved magnetic circular dichroism measurements revealed that spin-polarized moir & eacute; bands from both the MoTe2 and the WSe2 layers are involved at the formation of the QAH state. This scenario is not expected by existing theories. In this work, we suggest that the observed QAH state is a new state of matter, namely, a topological px + ipy intervalley coherent state. We point out that the massive Dirac spectrum of the MoTe2 moir & eacute; bands, together with the Hund's interaction and the Coulomb interactions, gives rise to this novel QAH state. Through a self-consistent Hartree-Fock analysis, we study how this px + ipy-pairing phase could appear. Besides explaining several key features of the experiments, our theory predicts that the order parameter would involve the pairing of electrons and holes with a definite momentum mismatch such that the pairing would generate a new unit cell which is three times the size of the original moir & eacute; unit cell, due to the order parameter modulations.

The ability to host Majorana modes, which are of great interest for more fault-tolerant quantum computation, keeps topological superconductors in the focus of research. Here, we report experimental data revealing 100-nm-wide quantum anomalous Hall insulator (QAHI) nanoribbons as a promising platform for the realization of zero-energy Majorana modes. One part of the nanoribbon is covered with superconducting niobium, while the other part is connected to a gold lead via two-dimensional QAHI regions. Andreev reflection spectroscopy reveals multiple in-gap conductance peaks in different devices. In the presence of an increasing magnetic field perpendicular to the film, the multiple-peak structure evolves into a single zero-bias conductance peak (ZBCP). Theoretical simulations suggest that the measurements are consistent with the scenario that the increasing field drives the nanoribbons from a multi-channel occupied regime to a single-channel regime, which is the necessary condition for the observation of zero-energy Majorana modes. © 2023 The Author(s)

Diode effects are of great interest for both fundamental physics and modern technologies. Electrical diode effects (nonreciprocal transport) have been observed in Weyl systems. Optical diode effects arising from the Weyl fermions have been theoretically considered but not probed experimentally. Here, we report the observation of a nonlinear optical diode effect (NODE) in the magnetic Weyl semimetal CeAlSi, where the magnetization introduces a pronounced directionality in the nonlinear optical second-harmonic generation (SHG). We demonstrate a six-fold change of the measured SHG intensity between opposite propagation directions over a bandwidth exceeding 250 meV. Supported by density-functional theory, we establish the linearly dispersive bands emerging from Weyl nodes as the origin of this broadband effect. We further demonstrate current-induced magnetization switching and thus electrical control of the NODE. Our results advance ongoing research to identify novel nonlinear optical/transport phenomena in magnetic topological materials and further opens new pathways for the unidirectional manipulation of light. © The Author(s) 2024.

Recent experiments on Bernal bilayer graphene (BLG) deposited on monolayer WSe2 revealed robust, ultraclean superconductivity coexisting with sizable induced spin-orbit coupling. Here, we propose BLG/WSe2 as a platform to engineer gate-defined planar topological Josephson junctions, where the normal and superconducting regions descend from a common material. More precisely, we show that if superconductivity in BLG/WSe2 is gapped and emerges from a parent state with intervalley coherence, then Majorana zero-energy modes can form in the barrier region upon applying weak in-plane magnetic fields. Our results spotlight a potential pathway for "internally engineered"topological superconductivity that minimizes detrimental disorder and orbital-magnetic-field effects. © 2023 American Physical Society.

Recently, the Josephson diode effect (JDE), in which the superconducting critical current magnitudes differ when the currents flow in opposite directions, has attracted great interest. In particular, it was demonstrated that gate-defined Josephson junctions based on magic-angle twisted bilayer graphene showed a strong nonreciprocal effect when the weak-link region is gated to a correlated insulating state at half filling (two holes per moiré cell). However, the mechanism behind such a phenomenon is not yet understood. In this Letter, we show that the interaction-driven valley polarization, together with the trigonal warping of the Fermi surface, induce the JDE. The valley polarization, which lifts the degeneracy of the states in the two valleys, induces a relative phase difference between the first and the second harmonics of the supercurrent and results in the JDE. We further show that the nontrivial current phase relation, which is responsible for the JDE, also generates the asymmetric Shapiro steps.

In this Letter, we study superconducting moiré homobilayer transition metal dichalcogenides where the Ising spin-orbit coupling (SOC) is much larger than the moiré bandwidth. We call such noncentrosymmetric superconductors, moiré Ising superconductors. Because of the large Ising SOC, the depairing effect caused by the Zeeman field is negligible and the in-plane upper critical field (Bc2) is determined by the orbital effects. This allows us to study the effect of large orbital fields. Interestingly, when the applied in-plane field is larger than the conventional orbital Bc2, a finite-momentum pairing phase would appear which we call the orbital Fulde-Ferrell (FF) state. In this state, the Cooper pairs acquire a net momentum of 2qB, where 2qB=eBd is the momentum shift caused by the magnetic field B and d denotes the layer separation. This orbital field-driven FF state is different from the conventional FF state driven by Zeeman effects in Rashba superconductors. Remarkably, we predict that the FF pairing would result in a giant superconducting diode effect under electric gating when layer asymmetry is induced. An upturn of the Bc2 as the temperature is lowered, coupled with the giant superconducting diode effect, would allow the detection of the orbital FF state.

Recently, topological flat bands and the spin Hall effect have been experimentally observed in AB-stacked MoTe2/WSe2 heterostructures. In this paper, we systematically study the Berry curvature effects in moire transition metal dichalcogenide (TMD) heterobilayers. We point out that the moire potential of the remote conduction bands would induce a sizable periodic pseudomagnetic field (PMF) on the valence band. This periodic PMF creates net Berry curvature flux in each valley of the moire Brillouin zone. The combination of the effect of the Berry curvature and the spin-valley locking can induce the spin Hall effect being observed in the experiment. Interestingly, the valley-contrasting Berry curvature distribution generated by the PMF can be probed through shift currents, which are dc currents induced by linearly polarized light through nonlinear responses. Our work sheds light on the novel quantum phenomena induced by Berry curvature in moire TMD heterobilayers.

In recent years, it has been shown that Berry curvature monopoles and dipoles play essential roles in the anomalous Hall and the nonlinear Hall effects, respectively. In this work, we demonstrate that Berry curvature multipoles (the higher moments of Berry curvatures at the Fermi energy) can induce higher-order nonlinear anomalous Hall (NLAH) effects. Specifically, an AC Hall voltage perpendicular to the current direction emerges, where the frequency is an integer multiple of the frequency of the applied current. Importantly, by analyzing the symmetry properties of all the 3D and 2D magnetic point groups, we note that the quadrupole, hexapole, and even higher Berry curvature moments can cause the leading-order frequency multiplication in certain materials. To provide concrete examples, we point out that the third-order NLAH voltage can be the leading-order Hall response in certain antiferromagnets due to Berry curvature quadrupoles, and the fourth-order NLAH voltage can be the leading response in the surface states of topological insulators induced by Berry curvature hexapoles. Our results are established by symmetry analysis, effective Hamiltonian, and first-principles calculations. Other materials which support the higher-order NLAH effect are further proposed, including 2D antiferromagnets and ferromagnets, Weyl semimetals, and twisted bilayer graphene near the quantum anomalous Hall phase. © 2023 American Physical Society.

The investigation of two-dimensional atomically thin superconductors— especially those hosting topological states—attracts growing interest in condensed-matter physics. Here we report the observation of spin–orbit– parity coupled superconducting state in centrosymmetric atomically thin 2M-WS2, a material that has been predicted to exhibit topological band inversions. Our magnetotransport measurements show that the in-plane upper critical feld not only exceeds the Pauli paramagnetic limit but also exhibits a strongly anisotropic two-fold symmetry in response to the in-plane magnetic feld direction. Furthermore, tunnelling spectroscopy measurements conducted under high in-plane magnetic felds reveal that the superconducting gap possesses an anisotropic magnetic response along diferent in-plane magnetic feld directions, and it persists much above the Pauli limit. Self-consistent mean-feld calculations show that this unusual behaviour originates from the strong spin–orbit–parity coupling arising from the topological band inversion in 2M-WS2, which efectively pins the spin of states near the topological band crossing and gives rise to an anisotropic renormalization of the efect of external Zeeman felds. Our results identify the unconventional superconductivity in atomically thin 2M-WS2, which serves as a promising platform for exploring the interplay between superconductivity, topology and strong spin–orbit–parity coupling.

Recently, it has been pointed out that the twisting of bilayer WSe2 would generate topologically non-trivial flat bands near the Fermi energy. In this work, we show that twisted bilayer WSe2 (tWSe2) with uniaxial strain exhibits a large nonlinear Hall (NLH) response due to the non-trivial Berry curvatures of the flat bands. Moreover, the NLH effect is greatly enhanced near the topological phase transition point which can be tuned by a vertical displacement field. Importantly, the nonlinear Hall signal changes sign across the topological phase transition point and provides a way to identify the topological phase transition and probe the topological properties of the flat bands. The strong enhancement and high tunability of the NLH effect near the topological phase transition point renders tWSe2 and related moire materials available platforms for rectification and second harmonic generations. © 2022, The Author(s).

Recent studies have shown that moiré flat bands in twisted bilayer graphene (TBG) can acquire nontrivial Berry curvatures when aligned with a hexagonal boron nitride substrate, which can be manifested as a correlated Chern insulator near the 3/4 filling. In this Letter, we show that the large Berry curvatures in the moiré bands lead to a strong nonlinear Hall (NLH) effect in strained TBG with general filling factors. Under a weak uniaxial strain ∼0.1%, the Berry curvature dipole which characterizes the nonlinear Hall response can be as large as ∼200Å, exceeding the values of previously known nonlinear Hall materials by two orders of magnitude. The dependence of the giant NLH effect as a function of electric gating, strain, and twist angle is further investigated systematically. Importantly, we point out that the giant NLH effect appears generically for a twist angle near the magic angle due to the strong susceptibility of nearly flat moiré bands to symmetry breaking induced by strains, which can even induce a topological band inversion. Our results establish TBG as a promising platform for investigating nonlinear effects such as the NLH effect, the nonlinear Nernst effect, and the nonlinear thermal Hall effect due to its giant Berry curvature dipole.

Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs. © 2021, The Author(s).

In recent years, it has been shown that Berry curvature monopoles and dipoles play essential roles in the anomalous Hall effect and the nonlinear Hall effect respectively. In this work, we demonstrate that Berry curvature multipoles (the higher moments of Berry curvatures at the Fermi energy) can induce higher-order nonlinear anomalous Hall (NLAH) effect. Specifically, an AC Hall voltage perpendicular to the current direction emerges, where the frequency is an integer multiple of the frequency of the applied current. Importantly, by analyzing the symmetry properties of all the 3D and 2D magnetic point groups, we note that the quadrupole, hexapole and even higher Berry curvature moments can cause the leading-order frequency multiplication in certain materials. To provide concrete examples, we point out that the third-order NLAH voltage can be the leading-order Hall response in certain antiferromagnets due to Berry curvature quadrupoles, and the fourth-order NLAH voltage can be the leading response in the surface states of topological insulators induced by Berry curvature hexapoles. Our results are established by symmetry analysis, effective Hamiltonian and first-principles calculations. Other materials which support the higher-order NLAH effect are further proposed, including 2D antiferromagnets and ferromagnets, Weyl semimetals and twisted bilayer graphene near the quantum anomalous Hall phase.

In a recent work [S. Manna, P. Wei, Y. Xie, K. T. Law, P. A. Lee, and J. S. Moodera, Proc. Natl. Acad. Sci. 117, 8775 (2020)10.1073/pnas.1919753117], signatures of a pair of Majorana bound states (MBSs) were found in an experimental platform formed by EuS islands deposited on top of a gold surface which was made superconducting through proximity coupling to a superconductor. In this paper, we provide a theoretical understanding for how MBSs can be formed in EuS/Au/superconductor heterostructures. We focus on the strip geometry where a narrow ferromagnetic strip is deposited on a planar structure. We first explicitly map out the topological phase diagram of the EuS/Au/superconductor heterostructure using the lattice Green's function method. Importantly, we find that the chemical potential step between the region with and without EuS covering is a crucial ingredient for the creation of MBS of this setup. Next, we focus on the Bogoliugov quasiparticles that are bound to the region under the EuS by Andreev reflections from the surrounding superconductors. Moreover, we obtain the topological regimes analytically using the scattering matrix method. Notably, we confirm that the normal reflections induced by the chemical potential step are essential for creating finite topological regimes. Furthermore, the area of the topological regimes shows periodic oscillation as a function of chemical potential as well as the sample width. We conclude by showing that the feromagnetic strip geometry holds a number of advantages over other quasi-one-dimensional schemes that have been proposed. © 2021 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Quantum materials with strong spin-orbit coupling are partially interesting, especially when there further exhibits topology or superconductivity. In this thesis, we present our studies on several systems: noncentrosymmetric achiral materials, quantum anomalous hall and superconductor heterostructures, and the topological monolayer WTe2, where the interplay of spin-orbit coupling, topology, and superconductivity gives rise to novel physics. Specifically, in chapter II, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there always are doubly degenerate lines connecting time-reversal invariant momenta with achiral little groups. In chapter III, we demonstrate that making a cut (a narrow vacuum regime) in the bulk of a quantum anomalous Hall insulator (QAHI) creates a topologically protected single helical channel with counter-propagating electron modes, and inducing superconductivity on the helical channel through proximity effects will create Majorana zero-energy modes (MZMs) at the ends of the cut. In chapter IV, we unveil some distinctive superconducting properties of centrosymmetric 1T′-WTe2 which arise from the coupling of spin, momentum, and band parity degrees of freedom. As a result of this spin-orbit-parity coupling (SOPC), the spin susceptibility is anisotropic with respect to in-plane directions and can result in possible anisotropic and strong gate dependence Bc2 even in this centrosymmetric system. Our studies identify several interesting physical properties in both noncenstrosymmetric and centrosymmetric quantum materials with the interplay of spin-orbit coupling, topology, and superconductivity.

Recent experiments reported gate-induced superconductivity in the monolayer 1T'-WTe2 which is a two-dimensional topological insulator in its normal state. The in-plane upper critical field B-c(2) is found to exceed the conventional Pauli paramagnetic limit B-p by one to three times. The enhancement cannot be explained by conventional spin-orbit coupling which vanishes due to inversion symmetry. In this Letter, we unveil some distinctive superconducting properties of centrosymmetric 1T'-WTe2 which arise from the coupling of spin, momentum and band parity degrees of freedom. As a result of this spin-orbit-parity coupling (SOPC): (i) there is a first-order superconductor-metal transition at B-c2 that is much higher than the Pauli paramagnetic limit B-p, (ii) spin-susceptibility is anisotropic with respect to in-plane directions and can result in possible anisotropic B-c2, and (iii) the B-c2 exhibits a strong gate dependence as the spin-orbit-parity coupling is significant only near the topological band crossing points. The importance of SOPC on the topologically nontrivial inter-orbital pairing phase is also discussed. Our theory generally applies to centrosymmetric materials with topological band inversions.

Td-WTe2 (non-centrosymmetric and orthorhombic), a type-II Weyl semimetal, is expected to have higher-order topological phases with topologically protected, helical one-dimensional hinge states when its Weyl points are annihilated. However, the detection of these hinge states is difficult due to the semimetallic behaviour of the bulk. In this study, we have spatially resolved the hinge states by analysing the magnetic field interference of the supercurrent in Nb–WTe2–Nb proximity Josephson junctions. The Josephson current along the a axis of the WTe2 crystal, but not along the b axis, showed a sharp enhancement at the edges of the junction, and the amount of enhanced Josephson current was comparable to the upper limits of a single one-dimensional helical channel. Our experimental observations suggest a higher-order topological phase in WTe2 and its corresponding anisotropic topological hinge states, in agreement with theoretical calculations. Our work paves the way for the study of hinge states in topological transition-metal dichalcogenides and analogous phases. © 2020, The Author(s), under exclusive licence to Springer Nature Limited.

With the recent discovery of the quantum anomalous Hall insulator (QAHI), which exhibits the conductive quantum Hall edge states without external magnetic field, it becomes possible to create a topological superconductor (SC) by introducing superconductivity into these edge states. In this case, 2 distinct topological superconducting phases with 1 or 2 chiral Majorana edge modes were theoretically predicted, characterized by Chern numbers (N) of 1 and 2, respectively. We present spectroscopic evidence from Andreev reflection experiments for the presence of chiral Majorana modes in an Nb/(Cr0.12Bi0.26Sb0.62)2Te3 heterostructure with distinct signatures attributed to 2 different topological superconducting phases. The results are in qualitatively good agreement with the theoretical predictions. © 2020 National Academy of Sciences. All rights reserved.

An external magnetic field is needed to drive a nanowire in proximity to an s-wave superconductor into a topological regime which supports Majorana end states. However, magnetic field generally suppresses the proximity superconducting gap induced on the nanowire. In recent experiments using InSb nanowires coupled to conventional superconductors such as Al and NbTiN, the induced proximity gap vanishes at magnetic fields B ∼ 1 T. This results in a small superconducting gap on the wire and a narrow topological regime which is proportional to the strength of the magnetic field. In this work, we show that by placing nanowires in proximity to recently discovered Ising superconductors such as the atomically thin transition-metal dichalcogenide NbSe2, the topological superconducting gap on the wire can be maintained at a large magnetic field as strong as B ∼ 10 T. This robust topological superconducting gap is induced by the unique equal-spin triplet Cooper pairs of the parent Ising superconductor. The strong magnetic field allows a topological regime ten times larger than those in InSb wires coupled to conventional superconductors. Our work establishes a realistic platform for building robust Majorana-based qubits.