arXiv subjects

Eslam Khalaf

Publications and source records attributed to Eslam Khalaf.

At least 55 records · Page 3Linked to original sources

arXiv2021

Field-tuned and zero-field fractional Chern insulators in magic angle graphene

In contrast to the fractional quantum Hall (FQH) effect, where electron density fixes the applied magnetic field, fractional Chern insulators (FCIs) can realize FQH states in comparatively weak or even zero magnetic fields. Previous theoretical work highlighted magic angle graphene as a promising FCI platform, satisfying the twin requirements of flat bands and lowest-Landau-level-like quantum geometry. Indeed, recent experiments have demonstrated FCIs in magic angle graphene with weak magnetic fields. Here we conduct a detailed theoretical study of the most prominent FCI state observed, and clarify the role of the magnetic field in stabilizing this state. We introduce two new technical tools: first, we generalize the notion of ideal quantum geometry to Hofstadter minibands and, second, we extend the Hartree-Fock theory of magic-angle graphene to finite field, to account for the interaction generated bandwidth. We show that magnetic field both dramatically reduces the effective bandwidth and improves the quantum geometry for hosting FCIs. Using density matrix renormalization group (DMRG) simulations of a microscopic model of magic angle graphene, we establish the regime of bandwidth and quantum geometry indicators where FCIs are stabilized. Further characterizing the finite-field bands by the same quantities we show how a zero-field charge density wave state gives way to an FCI state at a magnetic flux consistent with experiment. We also speculate on the other FCIs seen in the same experiments, including anomalous incompressible states and even-denominator fractions which may host non-Abelian states. Finally, when bandwidth is the limiting factor, we propose a range of experimental parameters where FCIs should appear at zero magnetic field.

cond-mat.str-el↗

arXiv2021

Strong Coupling Theory of Magic-Angle Graphene: A Pedagogical Introduction

We give a self contained review of a recently developed strong coupling theory of magic-angle graphene. An advantage of this approach is that a single formulation can capture both the insulating and superconducting states, and with a few simplifying assumptions, can be treated analytically. We begin by reviewing the electronic structure of magic angle graphene's flat bands, in a limit that exposes their peculiar band topology and geometry. We highlight how similarities between the flat bands and the lowest Landau level give insight into the effect of interactions. For example, at certain fractional fillings, we note the promise for realizing fractional Chern states. At integer fillings, this approach points to flavor ordered insulators, which can be captured by a sigma-model in its ordered phase. Unexpectedly, topological textures of the sigma model carry electric charge which allows us to extend the same theory to describe the doped phases away from integer filling. We show how this approach can lead to superconductivity on disordering the sigma model, and estimate the T$_c$ for the superconductor. We highlight the important role played by an effective super-exchange coupling both in pairing and in setting the effective mass of Cooper pairs. Seeking to enhance this coupling helps predict new superconducting platforms, including the recently discovered alternating twist trilayer platform. We also contrast our proposal from strong coupling theories for other superconductors.

cond-mat.str-el↗

arXiv2021

From electrons to baby skyrmions in Chern ferromagnets: A topological mechanism for spin-polaron formation in twisted bilayer graphene

The advent of Moiré materials has galvanized interest in the nature of charge carriers in topological bands. In contrast to conventional materials where charge carriers are electron-like quasiparticles, topological bands allow for more exotic possibilities where charge is carried by nontrivial topological textures, such as skyrmions. However, the real space description of skyrmions is ill-suited to address the limit of small or `baby' skyrmions which consist of an electron and a few spin flips. Here, we study the formation of the smallest skyrmions -- spin polarons, formed as bound states of an electron and a spin flip -- in Chern ferromagnets. We show that, quite generally, there is an attraction between an electron and a spin flip that is purely topological in origin and of $p$-wave symmetry, which promotes the formation of spin polarons. Applying our results to the topological bands of twisted bilayer graphene, we identify a range of parameters where spin polarons are formed and are lower in energy than electrons. In particular, spin polarons are found to be energetically cheaper on doping correlated insulators at integer fillings towards charge neutrality, consistent with the absence of quantum oscillations and the rapid onset of flavor polarization (cascade) transition in this regime. Our study sets the stage for pairing of spin polarons, helping bridge skyrmion pairing scenarios and momentum space approaches to superconductivity.

cond-mat.mes-hall↗

arXiv2021

TB or not TB? Contrasting properties of twisted bilayer graphene and the alternating twist $n$-layer structures ($n=3, 4, 5, \dots$)

The emergence of alternating twist multilayer graphene (ATMG) as a generalization of twisted bilayer graphene (TBG) raises the question - in what important ways do these systems differ? Here, we utilize ab-initio relaxation and single-particle theory, analytical strong coupling analysis, and Hartree-Fock to contrast ATMG with $n=3,4,5,\ldots$ layers and TBG. $\textbf{I}$: We show how external fields enter in the decomposition of ATMG into TBG and graphene subsystems. The parallel magnetic field has little effect for $n$ odd due to mirror symmetry, but surprisingly also for any $n > 2$ if we are are the largest magic angle. $\textbf{II}$: We compute the relaxation of the multilayers leading to effective parameters for each TBG subsystem. We find that the second magic angle for $n=5$, $θ= 1.14$ is closest to the "chiral" model and thus may be an optimal host for fractional Chern insulators. $\textbf{III}$: We integrate out non-magic subsystems and reduce ATMG to its magic angle TBG subsystem with a screened interaction. $\textbf{IV}$: We perform an analytic strong coupling analysis of the external fields and corroborate it with Hartree-Fock. An in-plane magnetic field in TBG drives a transition to a valley Hall state or a gapless "magnetic semimetal." A displacement field $V$ has little effect on TBG, but induces a gapped phase in ATMG for small $V$ for $n = 4$ and above a critical $V$ for $n = 3$. For $n\geq 3$, we find the superexchange coupling - believed to set the scale of superconductivity in the skyrmion mechanism - increases with $V$ at angles near and below the magic angle. $\textbf{V}$: We complement our strong coupling approach with a weak coupling theory of ATMG pair-breaking. While for $n=2$ orbital effects of the in-plane magnetic field can give a critical field of the same order as the Pauli field, for $n>2$ we expect the critical field to exceed the Pauli limit.

cond-mat.str-el↗

arXiv2021

Nematic topological semimetal and insulator in magic angle bilayer graphene at charge neutrality

We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the Moiré flat bands of twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structures and interactions and focus on the charge neutrality point, where experiments have variously reported either insulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remains unbroken, we find three types of self-consistent solutions with competitive ground state energy (i) insulators that break $C_2 {\mathcal T}$ symmetry, including valley Chern insulators (ii) spin or valley polarized insulators and (iii) rotation $C_3$ symmetry breaking semimetals whose gaplessness is protected by the topology of the Moiré flat bands. We find that the relative stability of these states can be tuned by weak strains that break $C_3$ rotation. The nematic semimetal and also, somewhat unexpectedly, the $C_2 {\mathcal T}$ breaking insulators, are stabilized by weak strain. These ground states may be related to the semi-metallic and insulating behaviors seen at charge neutrality, and the sample variability of their observation. We also compare with the results of STM measurements near charge neutrality.

cond-mat.str-el↗

arXiv2021

Fermionic Monte Carlo study of a realistic model of twisted bilayer graphene

The rich phenomenology of twisted bilayer graphene (TBG) near the magic angle is believed to arise from electron correlations in topological flat bands. An unbiased approach to this problem is highly desirable, but also particularly challenging, given the multiple electron flavors, the topological obstruction to defining tight binding models and the long-ranged Coulomb interactions. While numerical simulations of realistic models have thus far been confined to zero temperature, typically excluding some spin or valley species, analytic progress has relied on fixed point models away from the realistic limit. Here we present for the first time unbiased Monte Carlo simulations of realistic models of magic angle TBG at charge-neutrality. We establish the absence of a sign problem for this model in a momentum space approach, and describe a computationally tractable formulation that applies even on breaking chiral symmetry and including band dispersion. Our results include (i) the emergence of an insulating Kramers inter-valley coherent ground state in competition with a correlated semi-metal phase, (ii) detailed temperature evolution of order parameters and electronic spectral functions which reveal a `pseudogap' regime, in which gap features are established at a higher temperature than the onset of order and (iii) predictions for electronic tunneling spectra and their evolution with temperature. Our results pave the way towards uncovering the physics of magic angle graphene through exact simulations of over a hundred electrons across a wide temperature range.

cond-mat.str-el↗

arXiv2021

Fractional Chern insulators in magic-angle twisted bilayer graphene

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue toward manipulating non-abelian excitations. Early theoretical studies have predicted their existence in systems with energetically flat Chern bands and highlighted the critical role of a particular quantum band geometry. Thus far, however, FCI states have only been observed in Bernal-stacked bilayer graphene aligned with hexagonal boron nitride (BLG/hBN), in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field and limiting its potential for applications. By contrast, magic angle twisted bilayer graphene (MATBG) supports flat Chern bands at zero magnetic field, and therefore offers a promising route toward stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in MATBG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically-trivial charge density wave states. Unlike the BLG/hBN platform, we demonstrate that the principal role of the weak magnetic field here is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum band geometry favorable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in moiré systems with native flat Chern bands.

cond-mat.mes-hall↗

arXiv2021

Boundary-obstructed topological phases

Symmetry protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap. Here, we introduce a new notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions. More specifically, we say two bulk Hamiltonians belong to distinct boundary obstructed topological `phases' (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other for symmetric open boundaries without closing the gap at at least one high symmetry region on the surface. BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner or hinge charges. We show that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model. In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner or hinge charges. We also provide a general framework to study boundary obstructions in free-fermion systems in terms of Wannier band representations (WBR), an extension of the recently-developed band representation formalism to Wannier bands. WBRs capture the notion of topological obstructions in the Wannier bands which can then be used to study topological obstructions in the boundary spectrum by means of the correspondence between the Wannier and boundary spectra. This establishes a form of bulk-boundary correspondence for BOTPs by relating the bulk band representation to the boundary topology.

cond-mat.mes-hall↗

arXiv2021

Charged Skyrmions and Topological Origin of Superconductivity in Magic Angle Graphene

Spontaneous symmetry breaking plays a pivotal role in many areas of physics, engendering a variety of excitations from sound modes in solids to pions in nuclear physics. Equally important excitations are solitons, nonlinear configurations of the symmetry breaking field, which can enjoy exceptional stability as in the Skyrme model of nuclear forces. Here we argue that similar models may describe magic angle graphene, a remarkable new material . When the angle between two sheets of graphene is near the magic angle of $\sim 1^\circ$, insulating behavior is observed, which gives way to superconductivity on changing the electron density. We propose a unifying description of both the order underlying the insulator as well as the superconductor. While the symmetry breaking condensate leads to the ordered phase, topological solitons in the condensate - skyrmions - are shown to be bosons that carry an electric charge of 2e. Condensation of skyrmions leads to a superconductor whose pairing strength, symmetry and other properties are inferred. More generally, we show how topological textures can mitigate Coulomb repulsion to pair electrons and provide a new route to superconductivity. Our mechanism potentially applies to much wider class of systems but crucially invokes certain key ingredient such as inversion symmetry present in magic angle graphene. We discuss how these insights not only clarify why certain correlated moire materials do not superconduct, they also point to promising new platforms where robust superconductivity is anticipated.

cond-mat.str-el↗

arXiv2021

Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene

The interplay between strong electron-electron interactions and band topology can lead to novel electronic states that spontaneously break symmetries. The discovery of flat bands in magic-angle twisted bilayer graphene (MATBG) with nontrivial topology has provided a unique platform in which to search for new symmetry-broken phases. Recent scanning tunneling microscopy and transport experiments have revealed a sequence of topological insulating phases in MATBG with Chern numbers $C=\pm 3, \, \pm 2, \, \pm 1$ near moiré band filling factors $ν= \pm 1, \, \pm 2, \, \pm 3$, corresponding to a simple pattern of flavor-symmetry-breaking Chern insulators. Here, we report high-resolution local compressibility measurements of MATBG with a scanning single electron transistor that reveal a new sequence of incompressible states with unexpected Chern numbers observed down to zero magnetic field. We find that the Chern numbers for eight of the observed incompressible states are incompatible with the simple picture in which the $C= \pm 1$ bands are sequentially filled. We show that the emergence of these unusual incompressible phases can be understood as a consequence of broken translation symmetry that doubles the moiré unit cell and splits each $C=\pm 1$ band into a $C=\pm 1$ band and a $C=0$ band. Our findings significantly expand the known phase diagram of MATBG, and shed light onto the origin of the close competition between different correlated phases in the system.

cond-mat.mes-hall↗

arXiv2020

Symmetry constraints on superconductivity in twisted bilayer graphene: Fractional vortices, $4e$ condensates or non-unitary pairing

When two graphene sheets are twisted relative to each other by a small angle, enhanced correlations lead to superconductivity whose origin remains under debate. Here, we derive some general constraints on superconductivity in twisted bilayer graphene (TBG), independent of its underlying mechanism. Neglecting weak coupling between valleys, the global symmetry group of TBG consists of independent spin rotations in each valley in addition to valley charge rotations, $ {\rm SU}(2) \times {\rm SU}(2) \times {\rm U}_V(1) $. This symmetry is further enhanced to a full ${\rm SU}(4)$ in the idealized chiral limit. In both cases, we show that any charge $2e$ pairing must break the global symmetry. Additionally, if the pairing is unitary the resulting superconductor admits fractional vortices. This leads to the following general statement: Any superconducting condensate in either symmetry class has to satisfy one of three possibilities: (i) the superconducting pairing is non-unitary, (ii) the superconducting condensate has charge $2e$ but admits at least half quantum vortices which carry a flux of $h/4e$, or (iii) the superconducting condensate has charge $2me$, $m>1$, with vortices carrying $h/2me$ flux. The latter possibility can be realized by a symmetric charge $4e$ superconductor ($m=2$). Non-unitary pairing (i) is expected in TBG for superconductors observed in the vicinity of flavor polarized states. On the other hand, in the absence of flavor polarization, e.g. in the vicinity of charge neutrality, one of the two exotic possibilities (ii) and (iii) is expected. We sketch how all three scenarios can be realized in different limits within a strong coupling theory of superconductivity based on skyrmions. Finally we discuss the effect of symmetry lowering anisotropies and experimental implications of these scenarios.

cond-mat.supr-con↗

arXiv2020

Electric field tunable unconventional superconductivity in alternating twist magic-angle trilayer graphene

We construct a van der Waals heterostructure consisting of three graphene layers stacked with alternating twisting angles $\pmθ$. At the average twist angle $θ\sim 1.56^{\circ}$, a theoretically predicted magic angle for the formation of flat electron bands, narrow conduction and valence moiré bands appear together with a linearly dispersing Dirac band. Upon doping the half-filled moiré valence band with holes, or the half-filled moiré conduction band with electrons, displacement field tunable superconductivity emerges, reaching a maximum critical temperature of 2.1 K at optimal doping and displacement field. By tuning the doping level and displacement fields, we find that superconducting regimes occur in conjunction with flavour polarization of moiré bands bounded by a van Hove singularity (vHS) at high displacement fields. This experimental observation is found to be inconsistent with a weak coupling description, suggesting that the observed moiré superconductivity has an unconventional nature.

cond-mat.supr-con↗

arXiv2020

Soft modes in magic angle twisted bilayer graphene

We present a systematic study of the low-energy collective modes for different insulating states at integer fillings in twisted bilayer graphene. In particular, we provide a simple counting rule for the total number of soft modes, and analyze their energies and symmetry quantum numbers in detail. To study the soft mode spectra, we employ time dependent Hartree-Fock whose results are reproduced analytically via an effective sigma model description. We find two different types of low-energy modes - (i) approximate Goldstone modes associated with breaking an enlarged U(4)$\times$U(4) symmetry and, surprisingly, a set of (ii) nematic modes with non-zero angular momentum under three-fold rotation. The modes of type (i) include true gapless Goldstone modes associated with exact symmetries in addition to gapped "pseudo-Goldstone" modes associated with approximate symmetries. While the modes of type (ii) are always gapped, we show that their gap decreases as the Berry curvature grows more concentrated. For realistic parameter values, the gapped soft modes of both types have comparable gaps of only a few meV, and lie completely inside the mean-field bandgap. The entire set of soft modes emerge as Goldstone modes of a different idealized model in which Berry flux is limited to a solenoid, which enjoys an enlarged U(8) symmetry. Furthermore, we discuss the number of Goldstone modes for each symmetry-broken state, distinguishing the linearly vs quadratically dispersing modes. Finally, we present a general symmetry analysis of the soft modes for all possible insulating Slater determinant states at integer fillings that preserve translation symmetry, independent of the energetic details. The resulting soft mode degeneracies and symmetry quantum numbers provide a fingerprint of the different insulting states enabling their experimental identification from a measurement of their soft modes.

cond-mat.str-el↗

arXiv2020

Ground State and Hidden Symmetry of Magic Angle Graphene at Even Integer Filling

In magic angle twisted bilayer graphene, electron-electron interactions play a central role resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question and potentially linked to the relatively high temperature superconductivity observed in the same devices. Here we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC exhibits a pattern of alternating circulating currents which triples the graphene unit cell leading to an "orbital magnetization density wave". Although translation and time reversal symmetry are broken, a combined `Kramers' time reversal symmetry is preserved. Our analytic arguments are built on first identifying an approximate ${\rm U}(4) \times {\rm U}(4)$ symmetry, resulting from the remarkable properties of the tBG band structure, which helps select a low energy manifold of states, which are further split to favor the K-IVC. This low energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry lowering perturbations can stabilize other insulators and the semi-metallic state, and discuss the ground state at half filling and a comparison with experiments.

cond-mat.str-el↗

arXiv2020

Fractional Chern Insulator States in Twisted Bilayer Graphene: An Analytical Approach

Recent experiments on bilayer graphene twisted near the magic angle have observed spontaneous integer quantum Hall states in the presence of an aligned hexagonal boron nitride (hBN) substrate. These states arise from valley ferromagnetism, and the complete filling of Chern bands. A natural question is whether fractional filling of the same bands would lead to fractional quantum Hall states, i.e. to fractional Chern insulators (FCIs). Here, we argue that the magic angle graphene bands have favorable quantum geometry for realizing FCI phases. We show that in the tractable `chiral' limit, the flat bands wavefunctions are an analytic function of the crystal momentum. This remarkable property fixes the quantum metric up to an overall momentum dependent scale factor, the local Berry curvature, whose variation is itself small. Thus the three conditions associated with FCI stability (i) narrow bands (ii) quantum metric satisfying the isotropic ideal droplet condition and (iii) relatively uniform Berry curvature are all satisfied. Our work emphasizes continuum real space approaches to FCIs in contrast to earlier works which mostly focused on tight binding models. This enables us to construct a Laughlin wavefunction on a real-space torus that is a zero energy ground state of pseudopotential-like interactions. The latter can be realized for Coulomb interaction in the limit of very short screening length. We also map the problem onto an equivalent one of Dirac particles in an inhomogeneous but periodic magnetic field and no crystal potential. Finally we discuss evolution of the band geometry on tuning away from the chiral limit and show numerically that some of the desirable properties continue to hold at a quantitative level for realistic parameter values.

cond-mat.str-el↗

arXiv2019

A search for correlation-induced adiabatic paths between distinct topological insulators

Correlations in topological states of matter provide a rich phenomenology, including a reduction in the topological classification of the interacting system compared to its non-interacting counterpart. This happens when two phases that are topologically distinct on the non-interacting level become adiabatically connected once interactions are included. We use a quantum Monte Carlo method to study such a reduction. We consider a 2D charge-conserving analog of the Levin-Gu superconductor whose classification is reduced from $\mathbb{Z}$ to $\mathbb{Z}_4$. We may expect any symmetry-preserving interaction that leads to a symmetric gapped ground state at strong coupling, and consequently a gapped symmetric surface, to be sufficient for such reduction. Here, we provide a counter example by considering an interaction which (i) leads to a symmetric gapped ground state at sufficient strength and (ii) does not allow for any adiabatic path connecting the trivial phase to the topological phase with $w=4$. The latter is established by numerically mapping the phase diagram as a function of the interaction strength and a parameter tuning the topological invariant. Instead of the adiabatic connection, the system exhibits an extended region of spontaneous symmetry breaking separating the topological sectors. Frustration reduces the size of this long-range ordered region until it gives way to a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that an interaction which trivializes the surface of a gapped topological phase is necessary but not sufficient to establish an adiabatic path within the reduced classification. In other words, the class of interactions which trivializes the surface is different from the class which establishes an adiabatic connection in the bulk.

cond-mat.str-el↗

arXiv2019

Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene

Two monolayers of graphene twisted by a small `magic' angle exhibit nearly flat bands leading to correlated electronic states and superconductivity, whose precise nature including possible broken symmetries, remain under debate. Here we theoretically study a related but different system with reduced symmetry - twisted {\em double} bilayer graphene (TDBLG), consisting of {\em two} Bernal stacked bilayer graphene sheets, twisted with respect to one another. Unlike the monolayer case, we show that isolated flat bands only appear on application of a vertical displacement field $D$. We construct a phase diagram as a function of twist angle and $D$, incorporating interactions via a Hartree-Fock approximation. At half filling, ferromagnetic insulators are stabilized, typically with valley Chern number $C_v=2$. Ferromagnetic fluctuations in the metallic state are argued to lead to spin triplet superconductivity from pairing between electrons in opposite valleys. Response of these states to a magnetic field applied either perpendicular or parallel to the graphene sheets is obtained, and found to compare favorably with a recent experiment. We highlight a novel orbital effect arising from in-plane fields that can exceed the Zeeman effect and plays an important role in interpreting experiments.

cond-mat.str-el↗

arXiv2019

Magic Angle Hierarchy in Twisted Graphene Multilayers

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 $\pm θ$ 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 {\it 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 $\sqrt{2}$, whereas the quadrilayer case ($n = 4$) has two sequences of magic angles obtained by multiplying the bilayer magic angles by the golden ratio $φ= (\sqrt{5} + 1)/2 \approx 1.62$ 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 $n \rightarrow \infty$, there is a continuum of magic angles for $θ\lesssim 2^o$. 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.

cond-mat.str-el↗