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.

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...

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We introduce a cluster extension of multipole moments to discuss the anomalous Hall effect (AHE) in both ferromagnetic (FM) and antiferromagnetic (AFM) states in a unified framework. We first derive general symmetry requirements for the AHE in the presence or absence of the spin-orbit coupling, by considering the symmetry of the Berry curvature in k space. The cluster multipole (CMP) moments are then defined to quantify the macroscopic magnetization in non-collinear AFM states, as a natural generalization of the magnetization in FM states. We identify the macroscopic CMP order which induces the AHE. The theoretical framework is applied to the non-collinear AFM states of Mn3Ir, for which an AHE was predicted in a first-principles calculation, and Mn3Z (Z=Sn, Ge), for which a large AHE was recently discovered experimentally. We further compare the AHE in Mn3Z and bcc Fe in terms of the CMP. We show that the AHE in Mn3Z is characterized with the magnetization of a cluster octupole moment in the same manner as that in bcc Fe characterized with the magnetization of the dipole moment.

We use ultra-high resolution, tunable, VUV laser-based, angle-resolved photoemission spectroscopy (ARPES) and temperature and field dependent resistivity and thermoelectric power (TEP) measurements to study the electronic properties of WTe2, a compound that manifests exceptionally large, temperature dependent magnetoresistance. The temperature dependence of the TEP shows a change of slope at T=175 K and the Kohler rule breaks down above 70-140 K range. The Fermi surface consists of two electron pockets and two pairs of hole pockets along the X-Γ-X direction. Upon increase of temperature from 40K, the hole pockets gradually sink below the chemical potential. Like BaFe2As2, WTe2 has clear and substantial changes in its Fermi surface driven by modest changes in temperature. In WTe2, this leads to a rare example of temperature induced Lifshitz transition, associated with the complete disappearance of the hole pockets. These dramatic changes of the electronic structure naturally explain unusual features of the transport data.The discovery of giant magnetoresistance (GMR) in Fe/Cr superlattice [1,2], opened a new era of applications in magnetic field sensors, read heads in high density hard disks, random access memories, and galvanic isolators [3]. In the quest of achieving large MR in crystalline materials, a class of manganese oxides colossal magnetoresistance (CMR) materials was discovered that exhibits a large change in resistance with applied magnetic fields but only at low temperatures [4][5][6] The temperature dependent electrical resistivity, Hall coefficient and thermoelectric power of tungstenditelluride have been known for several decades [13] and a three-carrier semi-metal band model [13,14] was proposed to explain the electrical resistivity. Later on, density-functional based augmented spherical wave (ASW) electronic structure calculations and relatively low resolution ARPES [15] were used to study the electronic properties of WTe 2 and further supported the semimetallic nature of this material. However due to the low resolution of these early experiments, no details about the Fermi surface or band dispersion were obtained. Recent ARPES [11] and quantum oscillation [16] results revealed presence of small electron and hole pockets of roughly similar size. These findings were consistent with carrier compensation mechanism as the primary source of the MR effect [10,12]. Furthermore, this study also reported a change of the size of the Fermi pockets between 20K and 100K. More recently, Jiang et al.[17] claimed observation of strong spin-orbital coupling effect and proposed that the backscattering protection mechanism also may play a role in the large nonsaturating MR of WTe 2 .In this letter, we present the results from temperature dependent transport and ultra-high resolution, tunable VUV laser based ARPES[18] measurements. Our data show that there are two electron pockets and four hole pockets along the X-Γ-X direction and a fully occupied light "hole" band at the center of the Brillouin Zone (BZ) located j...