Helical edge states and fractional quantum Hall effect in a graphene electron–hole bilayer

Sanchez-Yamagishi, Javier; Luo, Jason; Young, Andrea; Hunt, Benjamin; Watanabe, Kenji; Taniguchi, Takashi; Ashoori, R. C.; Jarillo‐Herrero, Pablo

doi:10.1038/nnano.2016.214

Cited by 88 publications

(108 citation statements)

References 38 publications

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“…This type of cuts were measured in Ref. 25 as a function of B. We find good quantitative agreement with the experiment using U 2 = 0.28U , and conversely can rule out the U 2 = 0 model.…”

Section: Non-local Interactionssupporting

confidence: 89%

“…A similar phase diagram has been experimentally measured in Ref. 25. Deviations occur mostly in the shape of each region.…”

Section: Pseudo-qsh Under An Interlayer Biassupporting

confidence: 82%

“…We mention also that in Ref. 25 the two FI regions corresponding to opposite filling factors ( (1) and (3) in Fig. 2d) display conductances ranging between 0.8 and 1.3 e 2 /h, that is around half the value that is expected according to our model.…”

Section: Pseudo-qsh Under An Interlayer Biassupporting

confidence: 59%

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Quantum spin Hall effect in twisted bilayer graphene

Finocchiaro¹,

Guinea²,

San-José³

2017

2D Mater.

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Motivated by a recent experiment (Sanchez-Yamagishi et.al, arXiv:1602.06815) reporting evidence of helical spin-polarized edge states in layer-biased twisted bilayer graphene under a magnetic flux, we study the possibility of stabilising a Quantum Spin Hall (QSH) phase in such a system, without Zeeman or spin-orbit couplings, and with a QSH gap induced instead by electronic interactions. We analyse how magnetic flux, electric field, interlayer rotation angle, and interactions (treated at a mean field level) combine to produce a pseudo-QSH with broken time-reversal symmetry, and spin-polarized helical edge states. The effect is a consequence of a robust interaction-induced ferrimagnetic ordering of the Quantum Hall ground state under an interlayer bias, provided the two rotated layers are effectively decoupled at low energies. We discuss in detail the electronic structure, and the constraints on system parameters, such as the angle, interactions and magnetic flux, required to reach the pseudo-QSH phase. We find, in particular, that purely local electronic interactions are not sufficient to account for the experimental observations, which demand at least nearest-neighbour interactions to be included.

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“…This type of cuts were measured in Ref. 25 as a function of B. We find good quantitative agreement with the experiment using U 2 = 0.28U , and conversely can rule out the U 2 = 0 model.…”

Section: Non-local Interactionssupporting

confidence: 89%

“…A similar phase diagram has been experimentally measured in Ref. 25. Deviations occur mostly in the shape of each region.…”

Section: Pseudo-qsh Under An Interlayer Biassupporting

confidence: 82%

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Quantum spin Hall effect in twisted bilayer graphene

Finocchiaro¹,

Guinea²,

San-José³

2017

2D Mater.

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“…This situation might lead to protected modes between different regions of the system, dramatically changing the low-energy properties of the whole material. This is the natural scenario in van der Waals heterostructures, where Moire patterns [9][10][11] could coexist with any topological state [12,13]. A more controlled situation is the proposals for topological superconductivity involving nanowires, where the topological state is controlled locally by electric gates [14,15].…”

Section: Introductionmentioning

confidence: 99%

Real-space mapping of topological invariants using artificial neural networks

et al. 2018

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Topological invariants allow one to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wave functions under twisted boundary conditions. However, those procedures do not allow one to calculate a topological invariant by evaluating the system locally, and thus require information about the wave functions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a one-dimensional topological superconductor and a two-dimensional quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the kernel polynomial method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.

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“…The spin structure of the graphene Landau levels [41] can then potentially support a FQSH state at ν = −1/3 + 1/3 [42]. Twisted bilayer graphene is also a possible candidate, as it has been shown to realize an integer QSH state [43,44]. Recent experiments have demonstrated advances in coupling superconductors to semiconductor 2DEGS [45,46], e.g.…”

mentioning

confidence: 99%

Charge2e/3Superconductivity and Topological Degeneracies without Localized Zero Modes in Bilayer Fractional Quantum Hall States

Barkeshli

2016

Phys. Rev. Lett.

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It has been recently shown that non-Abelian defects with localized parafermion zero modes can arise in conventional Abelian fractional quantum Hall (FQH) states. Here we propose an alternate route to creating, manipulating, and measuring topologically protected degeneracies in bilayer FQH states coupled to superconductors, without the creation of localized parafermion zero modes. We focus mainly on electron-hole bilayers, with a ±1/3 Laughlin FQH state in each layer, with boundaries that are proximity-coupled to a superconductor. We show that the superconductor induces charge 2e/3 quasiparticle-pair condensation at each boundary of the FQH state, and that this leads to (1) topologically protected degeneracies that can be measured through charge sensing experiments and (2) a fractional charge 2e/3 AC Josephson effect. We demonstrate that an analog of non-Abelian braiding is possible, despite the absence of a localized zero mode. We discuss several practical advantages of this proposal over previous work, and also extensions to electron-electron bilayers at ν = 2/3 + 1/3 coupled to superconductivity, and to electron-hole bilayers with only interlayer tunneling.A profound feature of topologically ordered states of matter is the existence of topologically protected degeneracies [1]. These are degeneracies in the ground state spectrum of a given Hamiltonian for which no local operator can distinguish the topologically distinct states, thus rendering them robust to local perturbations. Topological degeneracies are predicted to occur when a topological phase is defined on a space with nontrivial topology, such as a torus [2,3], when the system hosts non-Abelian quasiparticle excitations [4][5][6][7], and in topological superconductors when Majorana zero modes are bound to cores of vortices or the ends of one-dimensional wires [8][9][10][11][12][13][14][15].It has recently been shown theoretically that a wide class of non-Abelian defects can be realized in conventional Abelian fractional quantum Hall (FQH) states, such as the Laughlin FQH states, by suitably controlling the tunneling processes along the edge modes. In particular, it was shown that in bilayer FQH states, domain walls between regions of interlayer and intra-layer tunneling can give rise to a novel type of non-Abelian defect that generalizes Majorana zero modes by yielding m states per pair of defects, where the integer m depends on the filling fraction ν of the FQH state [16]. These non-Abelian defects were subsequently shown to also arise as lattice defects in certain lattice spin models [17], extending [18]. They were then also shown to arise in FQH systems coupled to superconductors, at domain walls between regions of normal and Andreev backscattering at the edge [19][20][21], where they were referred to as parafermion [22], or fractionalized Majorana, zero modes. The general theory of these defects [23][24][25][26][27], a number of experimental proposals in FQH systems [28][29][30], other exotic phenomena at the boundary of FQH states and superconduct...

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Helical edge states and fractional quantum Hall effect in a graphene electron–hole bilayer

Cited by 88 publications

References 38 publications

Quantum spin Hall effect in twisted bilayer graphene

Quantum spin Hall effect in twisted bilayer graphene

Real-space mapping of topological invariants using artificial neural networks

Charge2e/3Superconductivity and Topological Degeneracies without Localized Zero Modes in Bilayer Fractional Quantum Hall States

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