Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles θ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of α ∼ 1/θ, they recur with the simple periodicity of ∆α ≃ 3/2, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to α 8 we capture important features of the first magic angle θ ≈ 1.09 • (α ≈ 0.586), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.

We consider conditions for the existence of boundary modes in non-Hermitian systems with edges of arbitrary co-dimension. Through a universal formulation of formation criteria for boundary modes in terms of local Green functions, we outline a generic perspective on the appearance of such modes and generate corresponding dispersion relations. In the process, we explain the skin effect in both topological and non-topological systems, exhaustively generalizing bulk-boundary correspondence in the presence of non-Hermiticity. This is accomplished via a doubled Green's function, inspired by doubled Hamiltonian methods used to classify Floquet and, more recently, non-Hermitian topological phases. Our work constitutes a general tool, as well as, a unifying perspective for this rapidly evolving field. Indeed, as a concrete application we find that our method can expose novel non-Hermitian topological regimes beyond the reach of previous methods.arXiv :1902.07217v2 [cond-mat.mes-hall]

When two monolayers of graphene are stacked with a small relative twist angle, the resulting band structure exhibits a remarkably flat pair of bands at a sequence of 'magic angles' where correlation effects can induce a host of exotic phases. Here, we study a class of related models of n-layered graphene with alternating relative twist angle ±θ which exhibit magic angle flat bands coexisting with several Dirac dispersing bands at the Moiré K point. Remarkably, we find that the Hamiltonian for the multilayer system can be mapped exactly to a set of decoupled bilayers at different angles, revealing a remarkable hierarchy mathematically relating all these magic angles to the TBG case. For the trilayer case (n = 3), we show that the sequence of magic angle is obtained by multiplying the bilayer magic angles by √ 2, whereas the quadrilayer case (n = 4) has two sequences of magic angles obtained by multiplying the bilayer magic angles by the golden ratio ϕ = ( √ 5 + 1)/2 ≈ 1.62 and its inverse. We also show that for larger n, we can tune the angle to obtain several narrow (almost flat) bands simultaneously and that for n → ∞, there is a continuum of magic angles for θ 2 o . Furthermore, we show that tuning several perfectly flat bands for a small number of layers is possible if the coupling between different layers is different. The setup proposed here can be readily achieved by repeatedly applying the "tear and stack" method without the need of any extra tuning of the twist angle and has the advantage that the first magic angle is always larger than the bilayer case. :1901.10485v2 [cond-mat.str-el] arXiv

Electrons in quantum materials exhibiting coexistence of dispersionless (flat) bands piercing dispersive (steep) bands can give rise to strongly correlated phenomena, and are associated with unconventional superconductivity. It is known that in twisted trilayer graphene steep Dirac cones can coexist with band flattening, but the phenomenon is not stable under layer misalignments. Here we show that such a twisted sandwiched graphene (TSWG) -a threelayer van der Waals heterostructure with a twisted middle layer -can have very stable flat bands coexisting with Dirac cones near the Fermi energy when twisted to 1.5 • . These flat bands require a specific high-symmetry stacking order, and our atomistic calculations predict that TSWG always relaxes to it. Additionally, with external fields, we can control the relative energy offset between the Dirac cone vertex and the flat bands. Our work establishes twisted sandwiched graphene as a new platform for research into strongly interacting phases, and topological transport beyond Dirac and Weyl semimetals.Graphene, an atomically thin crystal of carbon, provides an experimentally favorable platform for two dimensional (2D) Dirac physics as it exhibits ultrarelativistic Dirac cones in its band structure, described with massless quasiparticles when weak spin-orbit coupling is neglected. 1 1 arXiv:1907.00952v1 [cond-mat.str-el] 1 Jul 2019Bilayers of graphene in the energetically favorable Bernal (AB) stacking have quadratic dispersion and quasiparticles with well-defined effective mass. 1 Twisted bilayer graphene (TBG) -two rotationally mismatched graphene layers -can be fabricated at the so-called magic angle near 1.1 • , where it hosts ultraheavy fermions with remarkably flat, almost dispersionless electronic bands 2-5 of a topological origin. [5][6][7][8] The twist angle serves as a precise control of the interlayer coupling between the graphene monolayers, revealing the band flattening phenomena as an ultimate manifestation of hybridization of Dirac cones. Flat bands and the corresponding large density of electronic states can lead to novel strongly correlated phenomena. Indeed, since the recent discovery of correlated insulators and unconventional superconductivity in TBG, [9][10][11][12] van der Waals multilayer stacks have been further explored as a platform of exotic correlated physics. In particular, effectively 2D heterostructures consisting of flat sheets of graphene, transition metal dichalcogenides, and hexagonal boron nitride have been successful candidates for the moiré-induced correlated phenomena. [13][14][15][16][17][18][19][20][21][22][23] Recent experimental progress in studying correlations in multilayer heterostructures with more than two twisted graphene layers 13, 24-26 has led to a search for novel multilayer platforms with a particular focus on the trilayer geometry. 14,[27][28][29][30] In this work, we provide a detailed ab initio study of a unique extension of the TBG system: the twisted graphene sandwich (Fig. 1b), which is a promising construct of a...

We investigate twisted double bilayer graphene (TDBG), a four-layer system composed of two AB-stacked graphene bilayers rotated with respect to each other by a small angle. Our ab initio band structure calculations reveal a considerable energy gap at the charge point neutrality that we assign to the intrinsic symmetric polarization (ISP). We then introduce the ISP effect into the tight-binding parameterization and perform calculations on TDBG models that include lattice relaxation effects down to very small twist angles. We identify a narrow region around the magic angle θ • = 1.3 • characterized by a manifold of remarkably flat bands gapped out from other states even without external electric fields. To understand the fundamental origin of the magic angle in TDBG, we construct a continuum model that points to a hidden mathematical link to the twisted bilayer graphene (TBG) model, thus indicating that the band flattening is a fundamental feature of TDBG, and is not a result of external fields.