We use the DMRG to study the correlated electron states favored by the Coulomb interaction projected onto the narrow bands of twisted bilayer graphene within a spinless one-valley model. The Hilbert space of the narrow bands is constructed from a pair of hybrid Wannier states with opposite Chern numbers. Depending on the parameters in the BM model, the DMRG in this basis determines the ground state at one particle per unit cell to be either QAH state or a state with no Hall effect which is nearly a product state. Based on this form, we then apply the variational method to study their competition, thus identifying three states: the QAH, a gapless $C_2T$ symmetric nematic, and a gapped $C_2T$ symmetric stripe. All three states are nearly degenerate at the realistic parameters of the BM model. The single particle spectrum of the nematic contains either a quadratic node or two close Dirac nodes near $Γ$. Motivated by the Landau level degeneracy found in this state, we propose it to be the state observed at the charge neutrality point once spin and valley degeneracies are restored. The optimal period for the $C_2T$ stripe state is found to be $2$ unit cells. In addition, using the fact that the topological charge of the nodes in the $C_2T$ nematic phase is no longer described simply by their winding numbers once the translation symmetry is broken, but rather by certain elements of a non-Abelian group that was recently pointed out, we identify the mechanism of the gap opening within the $C_2T$ stripe state. Although the nodes at the Fermi energy are locally stable, they can be annihilated after braiding with other nodes connecting them to adjacent (folded) bands. Therefore, if the translation symmetry is broken, the gap at one particle per unit cell can open even if the system preserves the $C_2T$ and valley $U(1)$ symmetries, and the gap to remote bands remains open.
The degeneracy of Landau levels flanking charge neutrality in twisted bilayer graphene is known to change from eight-fold to four-fold when the twist angle is reduced to values near the magic angle of $\approx 1.05^\circ$. This degeneracy lifting has been reproduced in experiments by multiple groups, and is known to occur even in devices which do not harbor the correlated insulators and superconductors. We propose $C_3$ symmetry breaking as an explanation of such robust degeneracy lifting, and support our proposal by numerical results on the Landau level spectrum in near-magic-angle twisted bi…
Using scanning probe microscopy and spectroscopy, we explore the spatial symmetry of the electronic wavefunctions of twisted bilayer graphene at the "magic angle" of 1.1 degrees. This small twist angle leads to a long wavelength moiré unit cell on the order of 13 nm and the appearance of two flat bands. As the twist angle is decreased, correlation effects increase until they are maximized at the magic angle. At this angle, the wavefunctions at the charge neutrality point show only C2 symmetry due to the emergence of a charge ordered state. As the system is doped, the symmetry of the wavefuncti…
Magic-angle twisted bilayer graphene (MATBG) exhibits a rich variety of electronic states, including correlated insulators, superconductors, and topological phases. Understanding the microscopic mechanisms responsible for these phases requires determining the interplay between electron-electron interactions and quantum degeneracy due to spin and valley degrees of freedom. Signatures of strong electron-electron correlations have been observed at partial fillings of the flat electronic bands in recent spectroscopic measurements. Transport experiments have shown changes in the Landau level degene…
We extend the analysis of the Vafa $\mathcal{N}=4$ SUSY model of FQHE and discuss other observables which characterize the FQHE topological order. We consider in particular the braiding properties of quasi-holes with generic charge. As one naturally expects, any quasi-hole is associated with an irreducible representation of $SU(2)$ and the statistics of a bunch of $d$ punctures is described by the Knizhnik-Zamolodchikov connection specialized to the corresponding representation. We also discuss the higher genus generalization of the Vafa model of FQHE and the corresponding $tt^{*}$ geometry.
We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with four-fold degeneracy. After calculating the maximally localized superlattice Wannier wavefunctions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well defined experimental signatures.
Universal Gates from Braiding and Fusing Anyons on Quantum Hardware
Harvard University（哈佛大学）；Quantinuum；Stony Brook University（石溪大学）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学杨振宁理论物理研究所）；University of Chicago（芝加哥大学）
In magic angle twisted bilayer graphene (MATBG), the moiré superlattice potential gives rise to narrow electronic bands1 which support a multitude of many-body quantum phases. Further richness arises in the presence of a perpendicular magnetic field, where the interplay between moiré and magnetic length scales leads to fractal Hofstadter subbands. In this strongly correlated Hofstadter platform, multiple experiments have identified gapped topological and correlated states, but little is known about the phase transitions between them in the intervening compressible regimes. Here, using a scanni…
We study a two-dimensional spin model obtained by "Higgsing" the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on the L_x * L_y square lattice under the periodic boundary condition (PBC). There are p degrees of freedom per orbital and three orbitals per unit cell in the spin model. The resulting spin model is a stabilizer code consisting of three mutually commuting projectors that are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying LGT. The spin model thus obtained is exactly solvable, with th…
In the vicinity of the magic angle in twisted bilayer graphene (TBG), the two low-energy van Hove singularities (VHSs) become exceedingly narrow1-10 and many exotic correlated states, such as superconductivity, ferromagnetism, and topological phases, are observed11-16. Heterostrain, which is almost unavoidable in the TBG, can modify its single-particle band structure and lead to novel properties of the TBG that have never been considered so far. Here, we show that heterostrain in a TBG near the magic angle generates a new zero-energy flat band between the two VHSs. Doping the TBG to partially …
We investigate correlated insulating states in magic-angle twisted bilayer graphene (TBG) by the exact diagonalization method applied to the extended Hubbard model with interaction parameters recently evaluated in the realistic effective model. Our model can handle the competing interactions among Wannier orbitals owing to their significant overlap, which is a crucial but overlooked aspect of the magic-angle TBG. We propose two candidates for the correlated insulating states: spin- and valley-ferromagnetic band insulator and the Dirac semimetallic state for two flavors with peculiar renormaliz…
We introduce both an exactly solvable model and a coupled-layer construction for an exotic, three-dimensional phase of matter with immobile topological excitations that carry a protected internal degeneracy. Unitary transformations on this degenerate Hilbert space may be implemented by braiding certain point-like excitations. This provides a new way of extending non-Abelian statistics to three-dimensions.
We investigate the correlated electrons in the magic-angle twisted bilayer graphene by using the slave-rotor mean-field theory. Owing to the extended figure of Wannier orbitals, we study the two-orbital cluster Hubbard model with spin-valley fourfold degeneracy, focusing around half filling of valence bands below the neutrality point. The theory predicts multiple Mott insulator phases at fractional fillings not only for integer charges per moir{é} site, and it demonstrates that long-range electron hopping is highly suppressed because multiple-charge excitations are induced. Furthermore, the Ke…
I propose that non-Abelian topological order can emerge from the organization of quantum particles into identical indistinguishable copies of the same quantum many-body state. Quantum indistinguishability (symmetrization) of the collectivities leads to topological degeneracy in the subspace of elementary excitations, giving rise to non-Abelian braiding statistics. The non-Abelian hidden order of a symmetrized structure is manifested in its entanglement properties, and the corresponding non-Abelian fusion and braiding rules can be derived by analyzing the set of symmetrized states on a surface …
We present a framework for understanding the recently observed cascade transitions and the Landau level degeneracies at every integer filling of twisted bilayer graphene. The Coulomb interaction projected onto narrow bands causes the charged excitations at an integer filling to disperse, forming new bands. If the excitation moves the filling away from the charge neutrality point, then it has a band minimum at the moire Brillouin zone center with a small mass that compares well with the experiment; if towards the charge neutrality point, then it has a much larger mass and a higher degeneracy. A…
We present a tight-binding calculation of a twisted bilayer graphene at magic angle $θ\sim 1.08^\circ$, allowing for full, in- and out-of-plane, relaxation of the atomic positions. The resulting band structure displays as usual four narrow mini bands around the neutrality point, well separated from all other bands after the lattice relaxation. A thorough analysis of the mini-bands Bloch functions reveals an emergent $D_6$ symmetry, despite the lack of any manifest point group symmetry in the relaxed lattice. The Bloch functions at the $Γ$ point are degenerate in pairs, reflecting the so-called…
Twisted bilayer graphene has been argued theoretically to host exceptionally flat bands when the angle between the two layers falls within a magic range near 1.1$^\circ$. This is now strongly supported by experiment, which furthermore reveals dramatic correlation effects in the magic range due to the relative dominance of interactions when the bandwidth is suppressed. Experimentally, quantum oscillations exhibit different Landau level degeneracies when the angles fall in or outside the magic range; these observations can contain crucial information about the low energy physics. In this paper, …
Charge carriers in magic angle graphene come in eight flavors described by a combination of their spin, valley, and sublattice polarizations. When the inversion and time reversal symmetries are broken by the substrate or by strong interactions, the degeneracy of the flavors can be lifted and their corresponding bands can be filled sequentially. Due to their non-trivial band topology and Berry curvature, each of the bands is classified by a topological Chern number, leading to the quantum anomalous Hall and Chern insulator states at integer fillings $ν$ of the bands. It has been recently predic…
We study theoretically many-body properties of magic-angle twisted bilayer graphene for different doping levels. Our investigation is focused on the emergence, stability, and manifestations of nematicity of the ordered low-temperature electronic state. It is known that, at vanishing interactions, the low-energy spectrum of the system studied consists of four almost-flat almost-degenerate bands. Electron-electron repulsion lifts this degeneracy. To account for such an interaction effect, a numerical mean-field theory is used. Assuming that the ground state has spin-density-wave-like order, we i…
Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding q…
We provide a self-consistent mean-field framework to study the effect of strong interactions in a quantum spin Hall insulator on the honeycomb lattice. We identify an exotic phase for large spin-orbit coupling and intermediate Hubbard interaction. This phase is gapped in the bulk, has gapless edge states and does not break any symmetry. Instead, we find a four-fold topological degeneracy of the ground state on the torus and fractionalized excitations with semionic mutual braiding statistics. Our work highlights the important theme that interesting phases arise in the regime of strong spin-orbi…
Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit pi/8-phase gate…
Twisted bilayer graphene (TBG) has taken the spotlight in the condensed matter community since the discovery of correlated phases at the so-called magic angle. Interestingly, the role of a substrate on the electronic properties of TBG has not been completely elucidated. Up to now, most of the theoretical works carried out in order to understand this effect have been done using continuum models. In this work, we have gone one step ahead and have studied heterostructures of TBG and hBN using an atomistic tight-binding model together with semi-classical molecular dynamics to take into account rel…
Three-Dimensional Non-Foliated Fractional Quantum Hall Phases with Irrational Anyons in Twisted van der Waals Multilayers
Pohang University of Science and Technology（浦项科技大学）；Daegu Gyeongbuk Institute of Science and Technology (DGIST)（大邱庆北科学技术院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Institute for Basic Science（基础科学研究院）
When layers of graphene are rotationally misaligned by the magic angle, the moiré superlattice features extremely flat bands. Due to the enhanced density of states, the Coulomb interaction induces a variety of instabilities. The most prominent occur at integer filling and are therefore commonly attributed to spontaneous polarization of the moiré unit cell's `flavor' degrees of freedom -- spin, valley, and the flat-band degeneracy. As the dominant member of the hierarchy, these correlated states are thought to crucially determine further instabilities at lower energy scales, such as superconduc…
Motivated by experimental progress in the growth of heavy transition metal oxides, we theoretically study a class of lattice models of interacting fermions with strong spin-orbit coupling. Focusing on interactions of intermediate strength, we derive a low-energy effective field theory for a fully gapped, topologically ordered, fractionalized state with an eight-fold ground-state degeneracy. This state is a fermionic symmetry-enriched topological phase with particle-number conservation and time-reversal symmetry. The topological terms in the effective field theory describe a quantized magnetoel…
We construct generalized Hofstadter models that possess "color-entangled" flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously propose…
Condensation of quantum loops naturally leads to topological phases with Abelian excitations. Here, I propose that non-Abelian topological phases can arise from merging two (or several) identical Abelian quantum loop condensates. I define merging through a symmetrization operation, which makes the two loop condensates indistinguishable, inducing the possibility of a topological degeneracy in the space of quasiparticles. To illustrate the construction, the case of two identical toric-code quantum loop gases is considered. A spin-1 model for the two dimensional square lattice is presented for wh…
By applying a self-consistent Hartree-Fock approximation, we show that the mechanism of dynamical symmetry breaking can account for the insulating phase that develops about the charge neutrality point of twisted bilayer graphene around the magic angle. ($i$) If the Coulomb interaction is screened by metallic gates, the opening of a gap between the lowest-energy valence and conduction bands proceeds through the breakdown of chiral symmetry at strong coupling. Increasing the dielectric screening, however, we find a critical coupling at which chiral symmetry breaking is suppressed, triggering a v…
We study the electronic structure of a double-twisted few layer graphite (DTFLG), which consists of three few layer graphite (FLG), i.e. ABA-stacked graphene multilayer, stacked with two twist angles. We consider two categories of DTFLG, alternately twisted case and chirally twisted one, according to the rotation direction of the two twist angles. We show that, once the middle FLG of DTFLG is not thinner than trilayer, both kinds of DTFLG can remarkably host two pairs of degenerate moiré flat bands (MFBs) at $E_f$, twice that of the magic angle twisted bilayer graphene (TBG). The doubled MFBs …
Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green's observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from …
In this work, we determine states of electronic order of small-angle twisted bilayer graphene. Ground states are determined for weak and strong couplings which are representatives for varying distances of the twist-angle from its magic value. In the weak-coupling regime, charge density waves emerge which break translational and $C_{3}$-rotational symmetry. In the strong coupling-regime, we find rotational and translational symmetry breaking Mott insulating states for all commensurate moiré band fillings. Depending on the local occupation of superlattice sites hosting up to four electrons, glob…
We study the zero-temperature many-body properties of twisted bilayer graphene with a twist angle equal to the so-called `first magic angle'. The system low-energy single-electron spectrum consists of four (eight, if spin label is accounted) weakly-dispersing partially degenerate bands, each band accommodating one electron per Moir{é} cell per spin projection. This weak dispersion makes electrons particularly susceptible to the effects of interactions. Introducing several excitonic order parameters with spin-density-wave-like structure, we demonstrate that (i)~the band degeneracy is partially …
We analyze a description of twisted graphene bilayers, that incorporates deformation of the layers due to the nature modern interlayer potentials, and a modification of the hopping parameters between layers in the light of the classic Slonczewski-Weiss-McClure parametrisation. We shall show that flat bands result in all cases, but that their nature can be rather different. We will show how to construct a more general reduction to a continuum model, and show that even though such a model can be constructed, its complexity increases, requiring more coupling parameters to be included, and the ful…
