Engineering topics

Lian, Biao

Publications and source records attributed to Lian, Biao.

At least 19 records

DOE OSTI2024

Correlated topological flat bands in rhombohedral graphite

Flat bands and nontrivial topological physics are two important topics of condensed matter physics. With a unique stacking configuration analogous to the Su–Schrieffer–Heeger model, rhombohedral graphite (RG) is a potential candidate for realizing both flat bands and nontrivial topological physics. Here, in this study, we report experimental evidence of topological flat bands (TFBs) on the surface of bulk RG, which are topologically protected by bulk helical Dirac nodal lines via the bulk-boundary correspondence. Moreover, upon in situ electron doping, the surface TFBs show a splitting with exotic doping evolution, with an order-of-magnitude increase in the bandwidth of the lower split band, and pinning of the upper band near the Fermi level. These experimental observations together with Hartree–Fock calculations suggest that correlation effects are important in this system. Our results demonstrate RG as a platform for investigating the rich interplay between nontrivial band topology, correlation effects, and interaction-driven symmetry-broken states.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Quantum fragmentation in the extended quantum breakdown model

We introduce a one-dimensional (1D) extended quantum breakdown model comprising a fermionic and a spin degree of freedom per site, and featuring a spatially asymmetric breakdown-type interaction between the fermions and spins. Furthermore, our model resembles the breakdown process of particles incident into a cloud chamber with nonzero quantum amplitudes of both exciting and not exciting the local vapor atoms. We analytically show that, in the absence of any magnetic field for the spins, the model exhibits Hilbert space fragmentation within each symmetry sector into exponentially many Krylov subspaces and hence displays nonthermal dynamics. Here, we demonstrate that the fragmentation naturally occurs in an entangled basis and thus provides an example of “quantum fragmentation.” Besides establishing the nature of fragmentation analytically, we also study the long-time behavior of the entanglement entropy and its deviation from the expected Page value as a probe of ergodicity in the system. Upon introducing a magnetic field for the spins, most of the Krylov subspaces merge and the model becomes chaotic. Finally, we study the effects of strong randomness on the system and observe behavior similar to that of many-body localized systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Imaging real-space flat band localization in kagome magnet FeSn

Kagome lattices host flat bands due to their frustrated lattice geometry, which leads to destructive quantum interference of electron wave functions. Here, we report imaging of the kagome flat band localization in real-space using scanning tunneling microscopy. We identify both the Fe 3 Sn kagome lattice layer and the Sn 2 honeycomb layer with atomic resolution in kagome antiferromagnet FeSn. On the Fe 3 Sn lattice, at the flat band energy determined by the angle resolved photoemission spectroscopy, tunneling spectroscopy detects an unusual state localized uniquely at the Fe kagome lattice network. We further show that the vectorial in-plane magnetic field manipulates the spatial anisotropy of the localization state within each kagome unit cell. Our results are consistent with the real-space flat band localization in the magnetic kagome lattice. We further discuss the magnetic tuning of flat band localization under the spin–orbit coupled magnetic kagome lattice model.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Topological kagome magnets and superconductors

A kagome lattice naturally features Dirac fermions, flat bands and van Hove singularities in its electronic structure. The Dirac fermions encode topology, flat bands favour correlated phenomena such as magnetism, and van Hove singularities can lead to instabilities towards long-range many-body orders, altogether allowing for the realization and discovery of a series of topological kagome magnets and superconductors with exotic properties. Recent progress in exploring kagome materials has revealed rich emergent phenomena resulting from the quantum interactions between geometry, topology, spin and correlation. Here we review these key developments in this field, starting from the fundamental concepts of a kagome lattice, to the realizations of Chern and Weyl topological magnetism, to various flat-band many-body correlations, and then to the puzzles of unconventional charge-density waves and superconductivity. We highlight the connection between theoretical ideas and experimental observations, and the bond between quantum interactions within kagome magnets and kagome superconductors, as well as their relation to the concepts in topological insulators, topological superconductors, Weyl semimetals and high-temperature superconductors. In conclusion, these developments broadly bridge topological quantum physics and correlated many-body physics in a wide range of bulk materials and substantially advance the frontier of topological quantum matter.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Visualization of Tunable Weyl Line in A–A Stacking Kagome Magnets

Abstract Kagome magnets provide a fascinating platform for a plethora of topological quantum phenomena, in which the delicate interplay between frustrated crystal structure, magnetization, and spin–orbit coupling (SOC) can engender highly tunable topological states. Here, utilizing angle‐resolved photoemission spectroscopy, the Weyl lines are directly visualized with strong out‐of‐plane dispersion in the A–A stacked kagome magnet GdMn 6 Sn 6 . Remarkably, the Weyl lines exhibit a strong magnetization‐direction‐tunable SOC gap and binding energy tunability after substituting Gd with Tb and Li, respectively. These results not only illustrate the magnetization direction and valence counting as efficient tuning knobs for realizing and controlling distinct 3D topological phases, but also demonstrate AMn 6 Sn 6 (A = rare earth, or Li, Mg, or Ca) as a versatile material family for exploring diverse emergent topological quantum responses.

Cheng, Zi‐Jia↗

DOE OSTI2021

Twisted symmetric trilayer graphene. II. Projected Hartree-Fock study

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a twisted-bilayer-graphene- (TBG-) like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in Calugaru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings ν = -3, -2, -1, and 0 measured from charge neutrality. We study the phase diagram with w 0 /w 1 , the ratio of AA and AB interlayer hoppings, and the displacement field, which introduces an interlayer potential U and hybridizes the TBG-like bands with the Dirac bands. At small U, we find the ground states at all fillings ν are in the same phases as the tensor products of a Dirac semimetal with the filling ν TBG insulator ground states, which are spin-valley polarized at ν = -3, and fully (partially) intervalley coherent at ν = -2, 0(ν = -1) in the flat bands. An exception is ν = -3 with w 0 /w 1 ≳ 0.7 , which possibly becomes a metal with competing orders at small U due to charge transfers between the Dirac and flat bands. At strong U where the bandwidths exceed interactions, all the fillings ν enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate U, semimetal or insulator phases with zero intervalley coherence may arise for ν = -2, -1, 0. Finally, our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.

2-dimensional systems↗

DOE OSTI2021

Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle

Significance Multiple flat bands in high-quality twisted bilayer graphene close to the theoretically predicted second magic angle are observed. These well-isolated flat moiré bands host a nontrivial topology which is evidenced by a connecting multiband Hofstadter butterfly spectrum. This work provides a perspective for understanding the emergent quantum phases (i.e., strong correlation and band topology) in twisted bilayer graphene and the fractal Hofstadter spectra of multiple topological bands.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Delocalization Transition of a Disordered Axion Insulator

The axion insulator is a higher-order topological insulator protected by inversion symmetry. We show that, under quenched disorder respecting inversion symmetry on average, the topology of the axion insulator stays robust, and an intermediate metallic phase in which states are delocalized is unavoidable at the transition from an axion insulator to a trivial insulator. We derive this conclusion from general arguments, from classical percolation theory, and from the numerical study of a 3D quantum network model simulating a disordered axion insulator through a layer construction. We find the localization length critical exponent near the delocalization transition to be ν = 1.42 ± 0.12. We further show that this delocalization transition is stable even to weak breaking of the average inversion symmetry, up to a critical strength. Finally, we also quantitatively map our quantum network model to an effective Hamiltonian and we find its low-energy k ∙ p expansion.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k · p two-band model

We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011)] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functions of the BM model, which first elucidates why the BM K M -point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θ M ≈ 1°). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a Γ M -point centered model of six plane waves, we analytically understand the reason for the small Γ M -point gap between the active and passive bands in the isotropic limit w 0 = w 1 . Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the Γ M point, where no symmetry forces them to be. Furthermore, moving away from the Γ M and K M points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite the first magic angle is defined by only requiring a vanishing K M -point Dirac velocity. We derive analytically a connected “magic manifold” w 1 = $2\sqrt{1 + w^{2}_{0}}$ $-\sqrt{2 + 3w^2_0}$, on which the bands remain extremely flat as w 0 is tuned between the isotropic (w 0 = w 1 ) and chiral (w 0 = 0) limits. We analytically show why going away from the isotropic limit by making w 0 less (but not larger) than w 1 increases the Γ M -point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic Γ M point k ∙ p two-band model that reproduces the TBG band structure and eigenstates within a certain w 0 , w 1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band Γ M point k ∙ p model in the full w 0 , w 1 parameter range.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. II. Stable symmetry anomaly

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with lattice models. Previous works [e.g., Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019); J. Ahn, S. Park, and B.-J. Yang, Phys. Rev. X 9, 021013 (2019); H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Phys. Rev. B 99, 195455 (2019); J. Kang and O. Vafek, Phys. Rev. X 8, 031088 (2018); M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018); J. Liu, J. Liu, and X. Dai, Phys. Rev. B 99, 155415 (2019); L. Zou, H. C. Po, A. Vishwanath, and T. Senthil, Phys. Rev. B 98, 085435 (2018)] found that the two flatbands in TBG possess a fragile topology protected by the C 2z T symmetry. Song et al. [Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, Phys. Rev. Lett. 123, 036401 (2019)] also pointed out an approximate particle-hole symmetry (P) in the continuous model of TBG. In this paper, we numerically confirm that P is indeed a good approximation for TBG and show that the fragile topology of the two flatbands is enhanced to a P-protected stable topology. This stable topology implies 4l + 2 (l ϵ N) Dirac points between the middle two bands. The P-protected stable topology is robust against arbitrary gap closings between the middle two bands and the other bands. We also show that, remarkably, this P-protected stable topology, as well as the corresponding 4l + 2 Dirac points, cannot be realized in lattice models that preserve both C 2z T and P symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flatbands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries

We derive the explicit Hamiltonian of twisted bilayer graphene (TBG) with Coulomb interaction projected into the flat bands and study the symmetries of the Hamiltonian. First, we show that all projected TBG Hamiltonians can be written as positive semidefinite Hamiltonians, an example of which was found in work by Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. We then prove that the interacting TBG Hamiltonian exhibits an exact U(4) symmetry in the exactly flat band (nonchiral-flat) limit. We further define, besides a first chiral limit where the AA stacking hopping is zero, a second chiral limit where the AB/BA stacking hopping is zero. In the first chiral-flat limit (or second chiral-flat limit) with exactly flat bands, the TBG is enhanced to have an exact U(4) × U(4) symmetry, whose generators are different between the two chiral limits. While in the first chiral limit and in the nonchiral case these symmetries have been found in work by Bultinck et al. [Phys. Rev. X 10, 031034 (2020)], for the eight lowest bands, we here prove that they are valid for projection into any 8 n max particle-hole symmetric TBG bands, with n max > 1 being the practical case for small twist angles < 1°. Furthermore, in the first or second chiral-nonflat limit without flat bands, an exact U(4) symmetry still remains. We also elucidate the link between the U(4) symmetry presented here and the similar but different U(4) of Kang and Vafek [Phys. Rev. Lett. 122, 246401 (2019)]. Additionally, we show that our projected Hamiltonian can be viewed as the normal-ordered Coulomb interaction plus a Hartree-Fock term from passive bands, and exhibits a many-body particle-hole symmetry which renders the physics symmetric around charge neutrality. We also provide an efficient parametrization of the interacting Hamiltonian. The existence of two chiral limits with an enlarged symmetry suggests a possible duality of the model yet undiscovered.

2-dimensional systems↗

DOE OSTI2021

Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram

Here, we derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4) × U(4) symmetry, and we find that the exact ground states at integer filling – 4 ≤ $\textit{ν}$ ≤ 4 relative to charge neutrality are Chern insulators of Chern numbers $ν_C$ = 4 – |$\textit{ν}$ |, 2 – |$\textit{ν}$ |, $\cdots$, |$\textit{ν}$ | – 4, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find $\textit{ν}$ = 0 , ± 2 has exact ground states of Chern number 0, while $\textit{ν}$ = ±1, ± 3 has perturbative ground states of Chern number $ν_C$ = ± 1, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second-order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number $ν_C$ = 0 (0 < |$ν_C$| < 4 – |$\textit{ν}$ |) at integer fillings $\textit{ν}$ are fully (partially) intervalley coherent, while the insulator states with Chern number |$ν_C$| = 4 – |$\textit{ν}$ | are valley polarized. However, for 0 < |$ν_C$| ≤ 4 – |$\textit{ν}$ |, the fully intervalley coherent states are highly competitive (0.005 meV/electron higher). At nonzero magnetic field |$\textit{B}$ | > 0, a first-order phase transition for $\textit{ν}$ = ± 1, ± 2 from Chern number $ν_C$ = sgn ($\textit{νB}$)(2 – |$\textit{ν}$ |) to $ν_C$ = sgn ($\textit{νB}$)(4 – |$\textit{ν}$ |) is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: Charge gap, Goldstone modes, and absence of Cooper pairing

We find exact analytic expressions for the energies and wave functions of the charged and neutral excitations above the exact ground states (at rational filling per unit cell) of projected Coulomb Hamiltonians in twisted bilayer graphene. Our exact expressions are valid for any form of the Coulomb interaction and any form of A A and A B / B A tunneling. The single charge excitation energy is a convolution of the Coulomb potential with a quantum geometric tensor of the TBG bands. The neutral excitations are (high-symmetry group) magnons, and their dispersion is analytically calculated in terms of the form factors of the active bands in TBG. The two-charge excitation energy and wave functions are also obtained, and a sufficient condition on the graphene eigenstates for obtaining a Cooper pair from Coulomb interactions is obtained. For the actual TBG bands at the first magic angle, we can analytically show that the Cooper pair binding energy is zero in all such projected Coulomb models, implying that either phonons and/or nonzero kinetic energy are needed for superconductivity. Since Vafek and Kang [Phys. Rev. Lett. 125, 257602 (2020)] showed that the kinetic energy bounds on the superexchange energy are less 10 -3 in Coulomb units, the phonon mechanism becomes then very likely. If nonetheless the superconductivity is due to kinetic terms which render the bands nonflat, one prediction of our theory is that the highest T c would not occur at the highest DOS.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling

Using exact diagonalization, we research the projected Hamiltonian with the Coulomb interaction in the eight flat bands of first magic angle twisted bilayer graphene. Employing the U(4) [U(4) × U(4)] symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent that allows for study around ν = ±3,±2,±1 fillings. In the first chiral limit w 0 /w 1 = 0, where w 0 (w 1 ) is the A A (AB) stacking hopping, we find that the ground states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge ±1 excitations found in Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] are the lowest charge excitations up to system sizes 8 × 8 (for restricted Hilbert space) in the chiral-flat limit. We also find that the flat metric condition (FMC) used by Bernevig et al. [Phys. Rev. B 103, 205411 (2021)], Song et al. [Phys. Rev. B 103, 205412 (2021)], Bernevig et al. [Phys. Rev. B 103, 205413 (2021)], Lian et al. [Phys. Rev. B 103, 205414 (2021)], and Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] for obtaining a series of exact ground states and excitations holds in a large parameter space. For ν = -3, the ground state is the spin and valley polarized Chern insulator with ν C = ±1 at w 0 /w 1 ≲ 0.9 (0.3) with (without) FMC. At ν = -2, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when w 0 /w 1 ≳ 0.5 t where t ϵ [0,1] is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [Lian. et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [Lian et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. For ν = -3 with/without FMC, when w 0 /w 1 is large, the finite-size gap Δ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at ν = -3 suggests a competition among (nematic) metal, momentum M M (π) stripe and K M -CDW orders at large w 0 /w 1 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field

Here, we derive the Hamiltonian for trilayer moiré systems with the Coulomb interaction projected onto the bands near the charge neutrality point. Motivated by the latest experimental results, we focus on the twisted symmetric trilayer graphene (TSTG) with a mirror symmetry with respect to the middle layer. We provide a full symmetry analysis of the noninteracting Hamiltonian with a perpendicular displacement field coupling the band structure made otherwise of the twisted bilayer graphene (TBG) and the high-velocity Dirac fermions, and we identify a hidden nonlocal symmetry of the problem. In the presence of this displacement field, we construct an approximate single-particle model, akin to the tripod model for TBG, capturing the essence of noninteracting TSTG. We also derive more quantitative perturbation schemes for the low-energy physics of TSTG with displacement field, obtaining the corresponding eigenstates. This allows us to obtain the Coulomb interaction Hamiltonian projected in the active band TSTG wave functions and derive the full many-body Hamiltonian of the system. We also provide an efficient parametrization of the interacting Hamiltonian. Finally, we show that the discrete symmetries at the single-particle level promote the U (2) × U (2) spin-valley symmetry to enlarged symmetry groups of the interacting problem under different limits. The interacting part of the Hamiltonian exhibits a large U (4) × U (4) × U (4) × U (4) symmetry in the chiral limit. Moreover, by identifying a symmetry which we dub spatial many-body charge conjugation, we show that the physics of TSTG is symmetric around charge neutrality.

2-dimensional systems↗

DOE OSTI2021

Open momentum space method for the Hofstadter butterfly and the quantized Lorentz susceptibility

Here we develop a generic k ∙ p open momentum space method for calculating the Hofstadter butterfly of both continuum (moiré) models and tight-binding models, where the quasimomentum is directly substituted by the Landau level (LL) operators. By taking a LL cutoff (and a reciprocal lattice cutoff for continuum models), one obtains the Hofstadter butterfly with in-gap spectral flows. For continuum models such as the moiré model for twisted bilayer graphene, our method gives a sparse Hamiltonian, making it much more efficient than existing methods. The spectral flows in the Hofstadter gaps can be understood as edge states on a momentum space boundary, from which one can determine the two integers (t ν , s ν ) of a gap ν satisfying the Diophantine equation. The spectral flows can also be removed to obtain a clear Hofstadter butterfly. While t ν is known as the Chern number, our theory identifies s ν as a dual Chern number for the momentum space, which corresponds to a quantized Lorentz susceptibility γ xy = eBs ν .

2-dimensional systems↗

DOE OSTI2020

Strongly correlated Chern insulators in magic-angle twisted bilayer graphene

Interactions between electrons and the topology of their energy bands can create unusual quantum phases of matter. Most topological electronic phases appear in systems with weak electron–electron interactions. The instances in which topological phases emerge only as a result of strong interactions are rare and mostly limited to those realized in intense magnetic fields. The discovery of flat electronic bands with topological character in magic-angle twisted bilayer graphene (MATBG) has created a unique opportunity to search for strongly correlated topological phases. Here we introduce a local spectroscopic technique using a scanning tunnelling microscope to detect a sequence of topological insulators in MATBG with Chern numbers C = ±1, ±2 and ±3, which form near filling factors of ±3, ±2 and ±1 electrons per moiré unit cell, respectively, and are stabilized by modest magnetic fields. One of the phases detected here (C = +1) was previously observed when the sublattice symmetry of MATBG was intentionally broken by a hexagonal boron nitride substrate, with interactions having a secondary role. We also demonstrate that strong electron–electron interactions alone can produce not only the previously observed phase, but also other unexpected Chern insulating phases in MATBG. The full sequence of phases that we observe can be understood by postulating that strong correlations favour breaking time-reversal symmetry to form Chern insulators that are stabilized by weak magnetic fields. Our findings illustrate that many-body correlations can create topological phases in moiré systems beyond those anticipated from weakly interacting models.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

Landau level of fragile topology

Here, we study the Hofstadter butterfly and Landau levels of the twisted bilayer graphene (TBG). We show that the nontrivial fragile topology of the lowest two bands near the charge neutral point makes their Hofstadter butterfly generically connected with higher bands, closing the gap between the first and second conduction (valence) bands at a certain magnetic flux per unit cell. We also develop a momentum space method for calculating the TBG Hofstadter butterfly, from which we identify three phases where the Hofstadter butterflies of the lowest two bands and the higher bands are connected in different ways. We show this leads to a crossing between the $\textit{ν}$ = 4 Landau fan from the charge neutral point and the zero-field band gap at one flux per moiré unit cell, which corresponds to a magnetic field 25$θ^2$ T (twist angle $\textit{θ}$ in degrees). This provides an experimentally testable feature of the fragile topology. In general, we expect it to be a generic feature that the Hofstadter butterfly of the topological bands are connected with the Hofstadter spectra of other bands. We further show the TBG band theory with Zeeman splitting being the most sizable splitting could result in Landau fans at the charge neutral point and half fillings near the magic angle, and we predict their variations under an in-plane magnetic field.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗