We theoretically report that, with in-plane magnetization, the quantum anomalous Hall effect (QAHE) can be realized in two-dimensional atomic crystal layers with preserved inversion symmetry but broken out-of-plane mirror reflection symmetry. We take the honeycomb lattice as an example, where we find that the low-buckled structure, which makes the system satisfy the symmetric criteria, is crucial to induce QAHE. The topologically nontrivial bulk gap carrying a Chern number of C = ±1 opens in the vicinity of the saddle points M , where the band dispersion exhibits strong anisotropy. We further show that the QAHE with electrically tunable Chern number can be achieved in Bernalstacked multilayer systems, and the applied interlayer potential differences can dramatically decrease the critical magnetization to make the QAHE experimentally feasible.

At the interface between two-dimensional materials with different topologies, topologically protected one-dimensional states (also named as zero-line modes) arise. Here, we focus on the quantum anomalous Hall effect based zero-line modes formed at the interface between regimes with different Chern numbers. We find that, these zero-line modes are chiral and unilaterally conductive due to the breaking of time-reversal invariance. For a beam splitter consisting of two intersecting zero lines, the chirality ensures that current can only be injected from two of the four terminals. Our numerical results further show that, in the absence of contact resistance, the (anti-)clockwise partitions of currents from these two terminals are the same owing to the current conservation, which effectively simplifies the partition laws. We find that the partition is robust against relative shift of Fermi energy, but can be effectively adjusted by tuning the relative magnetization strengths at different regimes or relative angles between zero lines. PACS numbers:Introduction-. The presence of edge states that are topologically protected from backscattering is one of the striking hall-marks of topologically nontrivial insulators [1][2][3][4][5]. According to the rigorous bulk-edge correspondence rule [6,7], edge states appear at the boundary of two-dimensional topological systems, like quantum Hall effect [8], quantum anomalous Hall effect [9,10], and quantum spin-Hall effect (or two-dimensional topological insulators) [11,12]. These edge states are localized at the boundaries that are interfaces between topological materials and topologically trivial vacuum. Thus, these boundaries can be generalized to interfaces between two topologically distinct materials, like the interface between quantum anomalous Hall effect and quantum valley Hall effect [13], quantum spin-Hall effect and quantum valley Hall effect [14], or the graphene nanoroad between two structurally different boron-nitride sheets [15]. One widely explored system is the zero-line modes (ZLMs) occurred at the interface, across which the valley Chern numbers varies [1,16]. These ZLMs are protected from long-range scattering potential by large momentum separation and exhibit zero bend resistance in the absence of atomic defects [15,17]. Such modes are experimentally feasible [19,20] in Bernal stacked multilayer graphenes with out-of-plane electric field [21][22][23] and have attracted much attention from both theoreticians [1,17,18,[24][25][26][27][28][29][30][31][32][33][34] and experimentalists [19,20,35].

We demonstrate that zero-energy Majorana bound state, in the ferromagnetic insulator (FI)-superconductor (SC) junction formed on the edge of two-dimensional topological insulator, exhibits three types of spin-triplet pairing correlations and its spin polarization direction is position independent in ferromagnetic insulator. When an electron is injected with a spin (anti-)parallel to this direction, equal-spin Andreev reflection exhibits the widest (narrowest) resonance peak. Similar behaviour is found when the coupling between two Majorana bound states in a FI-SC-FI junction is invoked, though an additional weak spin-singlet pairing correlation is generated. These signatures can readily facilitate the experimental detection of spin-triplet correlations and spin polarization of Majorana bound states. PACS numbers: 74.45.+c, 74.78.Na, 74.20.RP, 74.25.F-Introduction-. Majorana fermions are exotic particles that are their own antiparticles [1], and have been suggested to exist as Majorana bound states (MBSs) in condensed matter systems [2]. Two spatially separated MBSs can define a qubit that stores information non-locally and is robust against local sources of decoherence [5], which together with its non-Abelian statistics [3, 4] make it exhibit potential applications in quantum information and quantum computation [6]. Several theoretical proposals were raised to realize such states, like topological insulators proximity-coupled with superconductors [7-10], semiconductor-superconductor heterostructures [11][12][13][14], and magnetic-atomic chains on superconductors [15]. Recently, intensive theoretical and experimental efforts have been made to verify the existence of MBSs in these systems by employing charge transport properties [16][17][18][19][20][21][22][23][24][25][26][27][28]. However, few attention has been paid to the spin related phenomena of MBSs [29][30][31][32]. Furthermore, the in-depth classification of spin-triplet correlations and spin polarization of MBSs are yet unclear [33,34], especially those in twodimensional topological insulator systems. And these characteristics are closely related to the resulting unusual spinrelated transport, like the intriguing selective equal-spin Andreev reflection [29,31].

In van der Waals multilayers of triangular lattice, trigonal warping occurs universally due to the interlayer hopping. We theoretically investigate the effect of trigonal warping upon distinctive topological phases, like the quantum anomalous Hall effect (QAHE) and the quantum valley Hall effect (QVHE). Taking Bernal-stacked bilayer graphene as an example, we find that the trigonal warping plays a crucial role in the formation of QAHE in large exchange field and/or interlayer potential difference by inducing extra band inversion points at momentum further away from high symmetric point. The presence of trigonal warping shrinks the phase space of QAHE and QVHE, leading to the emergence of valley-polarized QAHE with high Chern numbers ranging from C = −7 to 7. These results suggest that the universal trigonal warping may play important role when the Bloch states at momentum away from high-symmetric points are involved.

Here, we propose that Floquet engineering offers a strategy to realize the nonequilibrium quantum anomalous Hall effect (QAHE) with tunable Chern number. Using first-principles calculations and Floquet theorem, we unveil that QAHE related to valley polarization (VP-QAHE) is formed from the hybridization of Floquet sidebands in the two-dimensional family MSi 2 Z 4 (M = Mo, W, V; Z = N, P, As) by irradiating circularly polarized light (CPL). Through the tuning of frequency, intensity, and handedness of CPL, the Chern number of VP-QAHE is highly tunable and up to C = ±4, which attributes to light-induced trigonal warping and multiple-band inversion at different valleys. The chiral edge states and quantized plateau of Hall conductance are visible inside the global band gap, thereby facilitating the experimental measurement. Our work not only establishes Floquet engineering of nonequilibrium VP-QAHE with tunable Chern number in realistic materials but also provides an avenue to explore emergent topological phases under light irradiation.

Due to their nonlocality, Majorana bound states have been proposed to induce current-current correlations (CCCs) that are completely different from those induced by low-energy fermionic Andreev bound states. Such characteristics can be used as a signature to detect Majorana bound states. Herein, we studied the Majorana and fermionic Andreev bound states in a two-dimensional topological insulator system. We found that nonlocality occurs for both types of bound states and that their coupling strengths depend on system parameters in the same pattern. Majorana and fermionic Andreev bound states show the same differential CCCs characteristics, thereby indicating a universal behavior for both types of bound states. The maximal cross differential CCCs are robust to the structural asymmetry of the system.

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We investigate spin dependence of the nonlocal transport induced by Majorana fermions in a one-dimensional ferromagnet-ferromagnetic-insulator-superconductor-ferromagneticinsulator-ferromagnet junction on the edge of a two-dimensional topological insulator. The results show that coupled Majorana fermions lead to the nonlocal transport processes including electron tunneling and crossed Andreev reflection, which can be tuned by adjusting the spin polarizations of the Majorana fermions. By manipulating the bands in the two ferromagnets, the nonlocal transport can be selected as either pure electron tunneling or pure crossed Andreev reflection, the transmission probability of which could be 100%. Furthermore, the pure electron tunneling and the pure crossed Andreev reflection are well controlled by the spin directions of the electron states in the two ferromagnets.