We develop an effective extended Hubbard model to describe the low-energy electronic properties of the twisted bilayer graphene. By using the Bloch states in the effective continuum model and with the aid of the maximally localized algorithm, we construct the Wannier orbitals and obtain an effective tight-binding model on the emergent honeycomb lattice. We found the Wannier state takes a peculiar three-peak form in which the amplitude maxima are located at the triangle corners surrounding the center. We estimate the direct Coulomb interaction and the exchange interaction between the Wannier states. At the filling of two electrons per super cell, in particular, we find an unexpected coincidence in the direct Coulomb energy between a charge-ordered state and a homogeneous state, which would possibly lead to an unconventional many-body state.

Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronicstructure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectivelylocalised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with arXiv:1907.09788v1 [cond-mat.mtrl-sci]

1Recent discovery of both gapped and gapless topological phases in weakly correlated electron systems has introduced various relativistic particles and a number of exotic phenomena in condensed matter physics [1][2][3][4][5] . The Weyl fermion 6-8 is a prominent example of three dimensional (3D), gapless topological excitation, which has been experimentally identified in inversion symmetry breaking semimetals 4,5 . However, their realization in spontaneously time reversal symmetry (TRS) breaking magnetically ordered states of correlated materials has so far remained hypothetical 7, 9, 10 . Here, we report a set of experimental evidence for elusive magnetic Weyl fermions in Mn 3 Sn, a non-collinear antiferromagnet that exhibits a large anomalous Hall effect even at room temperature 11 . Detailed comparison between our angle resolved photoemission spectroscopy (ARPES) measurements and density functional theory (DFT) calculations reveals significant bandwidth renormalization and damping effects due to the strong correlation among Mn 3d electrons. Moreover, our transport measurements have unveiled strong evidence for the chiral anomaly of Weyl fermions, namely, the emergence of positive magnetoconductance only in the presence of parallel electric and magnetic fields. The magnetic Weyl fermions of Mn 3 Sn have a significant technological potential, since a weak field (∼ 10 mT) is adequate for controlling the distribution of Weyl points and the large fictitious field (∼ a few 100 T) in the momentum space. Our discovery thus lays the foundation for a new field of science and technology involving the magnetic Weyl excitations of strongly correlated electron systems.Traditionally, topological properties have been considered for the systems supporting gapped bulk excitations 1 . However, over the past few years three dimensional gapless systems such asWeyl and Dirac semimetals have been discovered, which combine two seemingly disjoint notions 2 of gapless bulk excitations and band topology [2][3][4][5] . In 3D inversion or TRS breaking systems, two nondegenerate energy bands can linearly touch at pairs of isolated points in the momentum (k) space, giving rise to the Weyl quasiparticles. The touching points or Weyl nodes act as the unit strength (anti) monopoles of underlying Berry curvature [4][5][6][7] , leading to the protected zero energy surface states also known as the Fermi-arcs 4,5,7 , and many exotic bulk properties such as large anomalous Hall effect (AHE) 12 , optical gyrotropy 13 , and chiral anomaly 6,[14][15][16][17][18][19] . Interestingly, the Weyl fermions can describe low energy excitations of both weakly and strongly correlated electron systems. In weakly correlated, inversion symmetry breaking materials, where the symmetry breaking is entirely caused by the crystal structure rather than the collective properties of electrons, the ARPES has provided evidence for long-lived bulk Weyl fermions and the surface Fermi arcs 4, 5 .On the other hand, the magnetic Weyl fermions have been predicted for several...

A temperature gradient in a ferromagnetic conductor can generate a transverse voltage drop perpendicular to both the magnetization and heat current. This anomalous Nernst e ect has been considered to be proportional to the magnetization 1-7 , and thus observed only in ferromagnets. Theoretically, however, the anomalous Nernst e ect provides a measure of the Berry curvature at the Fermi energy 8,9 , and so may be seen in magnets with no net magnetization. Here, we report the observation of a large anomalous Nernst e ect in the chiral antiferromagnet Mn 3 Sn (ref. 10). Despite a very small magnetization ∼0.002 µ B per Mn, the transverse Seebeck coe cient at zero magnetic field is ∼0.35 µV K −1 at room temperature and reaches ∼0.6 µV K −1 at 200 K, which is comparable to the maximum value known for a ferromagnetic metal. Our first-principles calculations reveal that this arises from a significantly enhanced Berry curvature associated with Weyl points near the Fermi energy 11 . As this e ect is geometrically convenient for thermoelectric power generation-it enables a lateral configuration of modules to cover a heat source 6 -these observations suggest that a new class of thermoelectric materials could be developed that exploit topological magnets to fabricate e cient, densely integrated thermopiles.Current intensive studies on thermally induced electron transport in ferromagnetic materials have opened various avenues for research on thermoelectricity and its application [12][13][14][15] . This trend has also triggered renewed interest in the anomalous Nernst effect (ANE) in ferromagnetic metals [3][4][5][6][7]15 , which is the spontaneous transverse voltage drop induced by heat current and is known to be proportional to magnetization (Fig. 1a). On the other hand, the recent Berry phase formulation of the transport properties has led to the discovery that a large anomalous Hall effect (AHE) may arise not only in ferromagnets, but in antiferromagnets and spin liquids, in which the magnetization is vanishingly small 10, [16][17][18][19][20][21][22] . As the first case in antiferromagnets, Mn 3 Sn has been experimentally found to exhibit a large AHE 10 . While the AHE is obtained by an integration of the Berry curvature for all of the occupied bands, the ANE is determined by the Berry curvature at E F (refs 8,9). Thus, the observation of a large AHE does not guarantee the observation of a large ANE. Furthermore, the ANE measurement should be highly useful to clarify the Berry curvature spectra near E F and to verify the possibility of the Weyl metal recently proposed for Mn 3 Sn (ref. 11).Mn 3 Sn has a hexagonal crystal structure with a space group of P6 3 /mmc (ref. 23). Mn atoms form a breathing type of kagome lattice in the ab-plane (Fig. 1b), and the Mn triangles constituting the kagome lattice are stacked on top along the c axis forming a tube of face-sharing octahedra. On cooling below the Néel temperature of 430 K, Mn magnetic moments of ∼3µ B lying in the ab-plane form a coplanar, chiral magnetic structure chara...