On the origin of magnetic anisotropy in two dimensional CrI3_3

The observation of ferromagnetic order in a monolayer of CrI$_3$ has been recently reported, with a Curie temperature of 45 Kelvin and off-plane easy axis. Here we study the origin of magnetic anisotropy, a necessary ingredient to have magnetic order in two dimensions, combining two levels of modeling, density functional calculations and spin model Hamiltonians. We find two different contributions to the magnetic anisotropy of the material, both favoring off-plane magnetization and contributing to open a gap in the spin wave spectrum. First, ferromagnetic super-exchange across the $\simeq$ 90 degree Cr-I-Cr bonds, are anisotropic, due to the spin orbit interaction of the ligand I atoms. Second, a much smaller contribution that comes from the single ion anisotropy of the $S=3/2$ Cr atom. Our results permit to establish the XXZ Hamiltonian, with a very small single ion anisotropy, as the adequate spin model for this system. Using spin wave theory we estimate the Curie temperature and we highlight the essential role played by the gap that magnetic anisotropy induces on the magnon spectrum.Comment: 8 pages, 5 figure

Noncollinear magnetic phases and edge states in graphene quantum Hall bars

Application of a perpendicular magnetic field to charge neutral graphene is expected to result in a variety of broken symmetry phases, including antiferromagnetic, canted and ferromagnetic. All these phases open a gap in bulk but have very different edge states and non-collinear spin order, recently confirmed experimentally. Here we provide an integrated description of both edge and bulk for the various magnetic phases of graphene Hall bars making use of a non-collinear mean field Hubbard model. Our calculations show that, at the edges, the three types of magnetic order are either enhanced (zigzag) or suppressed (armchair). Interestingly, we find that preformed local moments in zigzag edges interact with the quantum Spin Hall like edge states of the ferromagnetic phase and can induce back-scattering.Comment: 5 pages, 4 figure

Quantum spin Hall phase in multilayer graphene

The so called quantum spin Hall phase is a topologically non trivial insulating phase that is predicted to appear in graphene and graphene-like systems. In this work we address the question of whether this topological property persists in multilayered systems. We consider two situations: purely multilayer graphene and heterostructures where graphene is encapsulated by trivial insulators with a strong spin-orbit coupling. We use a four orbital tight-binding model that includes the full atomic spin-orbit coupling and we calculate the $Z_{2}$ topological invariant of the bulk states as well as the edge states of semi-infinite crystals with armchair termination. For homogeneous multilayers we find that even when the spin-orbit interaction opens a gap for all the possible stackings, only those with odd number of layers host gapless edge states while those with even number of layers are trivial insulators. For the heterostructures where graphene is encapsulated by trivial insulators, it turns out that the interlayer coupling is able to induce a topological gap whose size is controlled by the spin-orbit coupling of the encapsulating materials, indicating that the quantum spin Hall phase can be induced by proximity to trivial insulators.Comment: 7 pages, 6 figure

Topological features of hydrogenated graphene

Hydrogen adatoms are one of the most the promising proposals for the functionalization of graphene. Hydrogen induces narrow resonances near the Dirac energy, which lead to the formation of magnetic moments. Furthermore, they also create local lattice distortions which enhance the spin-orbit coupling. The combination of magnetism and spin-orbit coupling allows for a rich variety of phases, some of which have non trivial topological features. We analyze the interplay between magnetism and spin-orbit coupling in ordered arrays of hydrogen on graphene monolayers, and classify the different phases that may arise. We extend our model to consider arrays of adsorbates in graphene-like crystals with stronger intrinsic spin-orbit couplings.Comment: 6 pages, 4 figure

Real space mapping of topological invariants using artificial neural networks

Topological invariants allow to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wavefunctions under twisted boundary conditions. However, those procedures do not allow to calculate a topological invariant by evaluating the system locally, and thus require information about the wavefunctions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a 1-D topological superconductor and a 2-D quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the Kernel Polynomial Method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.Comment: 9 pages, 6 figure

Impurity-induced triple point fermions in twisted bilayer graphene

Triple point fermions are elusive electronic excitations that generalize Dirac and Weyl modes beyond the conventional high energy paradigm. Yet, finding real materials naturally hosting these excitations at the Fermi energy has remained challenging. Here we show that twisted bilayer graphene is a versatile platform to realize robust triple point fermions in two dimensions. In particular, we establish that the introduction of localized impurities lifts one of the two degenerate Dirac cones, yielding triple point fermions at charge neutrality. Furthermore, we show that the valley polarization is preserved for certain impurity locations in the moire supercell for both weak and strong impurity potentials. We finally show that in the presence of interactions, a symmetry broken state with local magnetization can develop out of the triple point bands, which can be selectively controlled by electrostatic gating. Our results put forward twisted bilayer graphene as a simple solid-state platform to realize triple point fermions at charge neutrality, and demonstrate the non-trivial role of impurities in moire systems.Comment: 8 pages, 7 figure