Strong coupling theory of magic-angle graphene: A pedagogical introduction

Ledwith, Patrick J.; Khalaf, Eslam; Vishwanath, Ashvin

doi:10.1016/j.aop.2021.168646

Cited by 44 publications

(30 citation statements)

References 54 publications

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“…The discovery of superconductivity and correlated insulators in twisted bilayer graphene (TBG) [1,2] close to the theoretically predicted magic angle θ TBG ≈ 1.1 o [3][4][5][6] has stimulated a huge amount of experimental [7][8][9][10][11][12][13][14][15][16][17] and theoretical work [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][33][34][35][36][37][38][39][40][41][42][43] on the system. In addition to the correlated insulators and superconductivity observed in the early works, more recent discoveries include the quantum anomalous Hall states with [44][45][46][47] and without…”

mentioning

confidence: 94%

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

Ledwith¹,

Khalaf²,

Zhu³

et al. 2021

Preprint

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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 a combination of techniques including ab-initio relaxation and single-particle theory, analytical strong coupling analysis, and Hartree-Fock to contrast ATMG with n = 3, 4, 5, . . . layers and TBG. I: We show how external fields enter in the decomposition of ATMG into twisted bilayer graphene and graphene subsystems. The parallel magnetic field is expected to have a much smaller effect when n is odd due to mirror symmetry, but surprisingly also for any n > 2 if we are are the largest magic angle. II: We compute the lattice relaxation of the multilayers leading to the effective parameters for each TBG subsystem as well as small mixing between the subsystems. We find that the second magic angle for n = 5, θ ≈ 1.14, provides the closest realization of the "chiral" model and is protected from mixing by mirror symmetry. It may be an optimal host for fractional Chern insulators. III: When there are no external fields, we integrate out the non-magic subsystems and reduce ATMG to the magic angle TBG subsystem with a screened interaction. IV: We perform an analytic strong coupling analysis of the effect of external fields and corroborate our results with numerical Hartree Fock simulations. For TBG itself, we find that an in-plane magnetic field can drive a phase transition to a valley Hall state or a gapless "magnetic semimetal" while having a weaker effect on n ≥ 3 ATMG at the first magic angle. In contrast, displacement field (V ) has very little effect on TBG, but induces a gapped phase in ATMG for small V for n = 4 and above a finite critical V for n = 3. For n ≥ 3, we extract the superexchange coupling -believed to set the scale of superconductivity in the skyrmion mechanism -and show that it increases with V at angles near and below the magic angle. V: We complement our strong coupling approach with a phenomenological 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. Contents A. n = 2 B. n = 3 C. General n VII. Conclusions VIII. Acknowledgements References SI. Review of the mapping twisted multilayer graphene to TBG A. Trilayer n = 3 B. Tetralayer n = 4 C. n = 5 SII. Lattice relaxation A. Summary of relaxation calculation B. Effect of lattice relaxation on magic angles and the multilayer mapping SIII. Intra-TBG effects of external fields A. External fields in n = 2 MATBG 1. Electric field 2. Magnetic field B. Even n¿2

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confidence: 94%

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

Ledwith¹,

Khalaf²,

Zhu³

et al. 2021

Preprint

Self Cite

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“…Previous works showed some peculiarities about frustration properties [9,76,77], yet it was not clear why such states were at the middle of the band. The same happens with the analytical form of zero-modes, which were identified as reminiscent of a Hall effect ground state without a clear explanation for this phenomena [63,70].…”

Section: Introductionmentioning

confidence: 67%

“…The mathematical properties and structure of the wavefunction have been rigorously studied in several works [14,[64][65][66][67]. As one can imagine the graphene layer as two triangular sublattices each one with an equal magnetic flux but with opposite sign, therefore, TBG graphene consists of coupled magnetic fluxes with opposite sign between layers [68][69][70]. This produces a strong skyrmion behavior in which electrons form vortexes, reflected in the presence of strong electron-electron correlation on specific locations across the Moire superlattice [71].…”

Section: Introductionmentioning

confidence: 99%

Why the first magic-angle is different from others in twisted graphene bilayers: interlayer currents, kinetic and confinement energy and wavefunction localization

Navarro-Labastida,

Espinosa-Champo,

Aguilar-Mendez

et al. 2021

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The chiral Hamiltonian for twisted graphene bilayers is analyzed in terms of its squared Hamiltonian which removes the particle-hole symmetry and thus one bipartite lattice, allowing to write the Hamiltonian in terms of a 2 × 2 matrix. This brings to the front the three main physical actors of twisted systems: kinetic energy, confinement potential and an interlayer interaction operator which is divided in two parts: a non-Abelian interlayer operator and an operator which contains the energy associated with the shift of momentum between the layers. Here, each of these components is analyzed as a function of the angle of rotation, as well as in terms of the wave-function localization properties. In particular, it is proved that the non-Abelian operator represents interlayer currents on a given bipartite sublattice, i.e., a second-neighbor interlayer current. Such contribution dominates the momentum shift term at magic angles different from the first. Therefore, a crossover is seen between such contributions and thus the first magic angle is different from other higher order magic angles. Such behavior is confirmed numerically as well as the relationships between the expected values of the kinetic, confinement and interlayer interaction energy.

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“…Based on the Dirac Hamiltonian obtained in Sec. II B, we derive the BM Hamiltonian for the twisted bilayer system in this section [4,7,36]. To this end, let us introduce the Dirac momentum k M explicitly in the Hamiltonian by replacing −2i∂ → −2i∂ − kM and…”

Section: Twisted Bilayer Systemmentioning

confidence: 99%

“…The flat bands of the TBG occurs even in generic multilayer systems: The relationship between magic angles and the number of layers has been conjectured in [29]. Detailed discussions including above can be found in the series of papers [30][31][32][33][34][35] and a review [36].…”

Section: Introductionmentioning

confidence: 98%

Moiré Landau levels of a $C_4$-symmetric twisted bilayer system in the absence of a magnetic field

Soeda,

Asaga,

Fukui

2022

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It is widely known that the twisted bilayer graphene (TBG) shows flat bands at magic angles, which can be well described by the effective continuum model derived by Bistritzer and MacDonald (BM). We propose in this paper a similar twisted bilayer system but defined on the square lattice with π-flux per plaquette, and study its spectrum using the BM Hamiltonian with a mass term which is originated from the staggered potential. The basic difference between the TBG and the present model is simply rotational symmetry, C3 vs C4, as well as a mass term. Nevertheless, the feature of the flat bands is quite different: Those of the TBG appear at magic angles only, while the present model shows many flat bands, which are reminiscent of Landau levels, quite stably at any angles even in the absence of a magnetic field other than π-flux which keeps time reversal (TR) symmetry. Moreover, flat bands emerge in the mass gap of the Dirac spectrum, and each state composing these flat bands is well-localized at the position forming the moiré lattice. It turns out that the moiré potential serves as a periodic magnetic field, which can give energies smaller that the gap around moiré lattice positions. We derive a local Hamiltonian valid around the moiré lattice sites and show that it indeed reproduces the energies of the flat bands within the mass gap. Since these mid-gap states are localized at the moiré lattice, they form degenerate levels, which may be referred to as moiré Landau levels, although the mechanism of degeneracies are different from the conventional Landau levels. Interestingly, doubled fermions of the BH Hamiltonian associated with two layers have opposite charges when they couple with the effective moiré magnetic filed, which concern TR symmetry. This is a generic feature of the C4 symmetric moiré system described by the BM Hamiltonian.

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Strong coupling theory of magic-angle graphene: A pedagogical introduction

Cited by 44 publications

References 54 publications

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

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

Why the first magic-angle is different from others in twisted graphene bilayers: interlayer currents, kinetic and confinement energy and wavefunction localization

Moiré Landau levels of a $C_4$-symmetric twisted bilayer system in the absence of a magnetic field

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