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We propose to implement tunable higher-order topological states in a heterojunction consisting of a two-dimensional (2D) topological insulator and the recently discovered altermagnets, whose unique spin-polarization in both real and reciprocal space and null magnetization are in contrast to conventional ferromagnets and antiferromagnets. Based on symmetry analysis and effective edge theory, we show that the special spin splitting in altermagnets with different symmetries, such as $d$-wave, can introduce Dirac mass terms with opposite signs on the adjacent boundaries of the topological insulator, resulting in the higher-order topological state with mass-domain bound corner states. Moreover, by adjusting the direction of the N\'{e}el vector, we can manipulate such topological corner states by moving their positions. By first-principles calculations, taking a 2D topological insulator bismuthene with a square lattice on an altermagnet MnF$_2$ as an example, we demonstrate the feasibility of creating and manipulating the higher-order topological states through altermagnets. Finally, we discuss the experimental implementation and detection of the tunable topological corner states, as well as the potential non-Abelian braiding of the Dirac corner fermions., Comment: Submitted to Physical Review Letters on 25 July 2023; Accepted by PRB as a Letter on 22 April 2024

The mutual competition and synergy of magnetism and superconductivity provide us with a very valuable opportunity to access topological superconductivity and Majorana Fermions. Here, we devise a heterostructure consisting of an $s$-wave superconductor, a 2D topological insulator and an altermagnet, which is classified as the third magnet and featured by zero magnetization but spin polarization in both real and reciprocal spaces. We find that the altermagnet can induce mass terms at the edges that compete with electron pairing, and mass domains are formed at the corners of sample, resulting in zero-energy Majorana corner modes (MCMs). The presence or absence of MCMs can be engineered by only changing the direction of the N\'{e}el vector. Moreover, uniaxial strain can effectively manipulate the patterns of the MCMs, such as moving and interchanging MCMs. Experimental realization, remarkable advantages of our proposal and possible braiding are discussed., Comment: Accepted by PRB, 5 pages main text + 13 pages SM

The realization of Majorana corner modes generally requires unconventional superconducting pairing or $s$-wave pairing. However, the bulk nodes in unconventional superconductors and the low $T_c$ of $s$-wave superconductors are not conducive to the experimental observation of Majorana corner modes. Here we show the emergence of a Majorana corner mode at each corner of a two-dimensional topological insulator in proximity to a $d+id'$ pairing superconductor, such as heavily doped graphene or especially a twisted bilayer of a cuprate superconductor, e.g., Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, which has recently been proposed as a fully gapped chiral $d_{x^2-y^2}+id_{xy}$ superconductor with $T_c$ close to its native 90 K, and an in-plane magnetic field. By numerical calculation and intuitive edge theory, we find that the interplay of the proximity-induced pairing and Zeeman field can introduce opposite Dirac masses on adjacent edges of the topological insulator, which creates one zero-energy Majorana mode at each corner. Our scheme offers a feasible route to achieve and explore Majorana corner modes in a high-temperature platform without bulk superconductor nodes., Comment: 6 pages, 3 figures

We introduce a two-dimensional Chern insulator in proximity to a $d$-wave pseudogap state of the high-T$_c$ superconducting material as an effective platform to realize the higher order topological system. The proximity-induced $d$-density-wave (DDW) order in the Chern insulator layer serves as an effective mass. The edge states will be fully gapped by this DDW order. In the real space, the sign of the DDW order parameter changes at the system corners due to the $d$-wave factor, leading to the gapless corner states, indicating that this system may be in a higher order topological state. The higher order topology in this coupled system is confirmed based on the calculation of the edge polarization and the quadrupole moment. In the superconducting state where the superconducting order and the DDW order coexist, the Majorana corner states emerge., Comment: 10 pages, 6 figures, including the supplemental material

Applying a microscopic model, we study theoretically the quasiparticle excitation of a twodimensional topological insulator (TI) being in proximity to a high-Tc superconductor. In the momentum space, the proximity induced pairing term in the TI layer includes both the singlet channel and triplet channel, leading to the C4 rotational symmetry breaking of the energy bands and the quasiparticle spectra. For a cylinder geometry, the zero energy edge states may appear but they are localized at the upper boundary. For the finite-size system with open boundaries, the zero energy states at the upper boundary disappear and the Majorana bound states emerge at the two lower corners. All of the results can be understood well through exploring the pairing order parameter and the anomalous Green's function., Comment: 8 pages,5 figures

Recurrent neural networks have been widely used in sequence learning tasks. In previous studies, the performance of the model has always been improved by either wider or deeper structures. However, the former becomes more prone to overfitting, while the latter is difficult to optimize. In this paper, we propose a simple new model named Thick-Net, by expanding the network from another dimension: thickness. Multiple parallel values are obtained via more sets of parameters in each hidden state, and the maximum value is selected as the final output among parallel intermediate outputs. Notably, Thick-Net can efficiently avoid overfitting, and is easier to optimize than the vanilla structures due to the large dropout affiliated with it. Our model is evaluated on four sequential tasks including adding problem, permuted sequential MNIST, text classification and language modeling. The results of these tasks demonstrate that our model can not only improve accuracy with faster convergence but also facilitate a better generalization ability., Comment: 11 pages, 4 figures