Session S09: Topology in Twisted Multilayer Graphene

Chiral model of twisted bilayer graphene realized in a monolayer

When graphene is subject to a superlattice potential with a nearly commensurate $sqrt{3} imessqrt{3}$ supercell, the resulting moir'e physics exactly maps to the chiral model of twisted bilayer graphene, except with half as many degrees of freedom. In this talk, we describe this mapping, characterize the interacting physics at integer filling, and determine the impact of symmetry-allowed deviations from the ideal chiral model.   

Competition between valley symmetry breaking and intervalley coherence, and their respective routes to Kohn-Luttinger superconductivity in twisted bilayer graphene

We study the tight competition between valley symmetry breaking and intervalley coherence at integer filling fractions of magic angle twisted bilayer graphene under hydrostatic pressure, finding that different electronic states are very close in energy reflecting the emergent U(4) symmetry. We apply a self-consistent real-space Hartree-Fock approximation that accounts for the screened long-range Coulomb interaction as well as the on-site Hubbard interaction U, being able to include all remote bands of the electronic spectrum. In this way, we show that Kramers intervalley coherence is the dominant symmetry breaking pattern at the charge neutrality point. On the other hand, at integer filling -2 (+2) the ground state displays preferentially valley symmetry breaking (intervalley coherence). We investigate this asymmetry between the electron and hole-doped regimes, concluding that valley symmetry breaking and intervalley coherence imply different routes to superconductivity relying on a Kohn-Luttinger instability of the electron system.   

Particle-hole asymmetric correlated phases in twisted bilayer graphene

Twisted bilayer graphene has emerged as a paradigmatic platform for exploring the interplay between strong interactions and topological obstructions. While the majority of previous studies had used the effective (k.p) model of Bistritzer and MacDonald, the present work [2] aims at a more precise microscopic understanding using faithful tight-binding modeling of the twisted bilayer graphene. Here, based on the Wannier orbitals obtained by Carr et al. [1], we construct an extended Hubbard model with 8 orbitals, and perform a Hartree-Fock (HF) analysis to explore its phase diagram across  commensurate fillings from -3 to 3. In addition to the previously recognized role of the valley degree of freedom, we find that the orbitals also play a central role in determining the ground state at all fillings. In particular, a nearly-degenerate pair of insulating states are obtained at charge neutrality, both exhibiting orbital polarization. Doping the system away from the neutrality point result in insulating quantum anomalous hall states at fillings -1 and +2, while symmetry-broken metallic states are obtained at all other charged fillings. A universal feature away from charge-neutrality is the distinction between correlated states obtained at fillings , which results from the particle-hole asymmetry in the single-particle band structure. Thus, our extended Hubbard model provides a suitable microscopic starting point for examining the strongly correlated phenomena in twisted bilayer graphene.

[1] S. Carr, S. Fang, H. C. Po, A. Vishwanath, and E. Kaxiras, Phys. Rev. Res.1, 033072 (2019).

[2] R. Hou, S. Sur, L. K. Wagner and A. H. Nevidomskyy, to appear (2023)   

A novel technique to extract features of flat bands in a moiré Superlattice

 

Abstract: Twisted multilayer graphene systems with nearly flat bands are ideal platforms for studying strongly correlated physics. Exotic phases of matter such as correlated insulators at integer fillings, superconductivity, and anomalous quantum hall have been observed in such systems. However, a straightforward technique to directly extract detailed flat band features is still lacking. We propose a novel technique for the study of flat bands using a decoupled monolayer  graphene on a magic-angle twisted monolayer bilayer graphene heterostructure. We not only extract the width of the flat bands by measuring the chemical potentials, but also directly map the evolution of the electron-electron correlation as a function of the external displacement field. Through our measurements, we observe that the correlation maximizes where the flat band width is minimum. This new device configuration with an additional decoupled monolayer graphene can pave a way to study the flat bands which can further improve the understanding of the electronic band structure.   

Helical trilayer graphene: a moiré platform for strongly-interacting topological bands – Part 1

Topological bands in two dimensional materials can be promoted by broken xy-inversion (C_2z) symmetry. Existing examples include materials where the constituent layers break C_2z on their own. Here, we explore helical trilayer graphene (HTG), three graphene layers, each preserving C_2z on their own, twisted in sequence by the same angle. Unlike alternating-twist trilayer graphene, this forms two moiré patterns with different orientations. Although HTG is globally C_2z-symmetric, lattice relaxation leads to large periodic domains in which C_2z is broken on the moiré scale. Using magnetotransport, we observe the anomalous Hall effect – a clear signature of topological bands. At a magic angle of θ_m ≈ 1.8°, we uncover a robust phase diagram of correlated and magnetic states. Our results bring to light the importance of local symmetries on length scales comparable to the inter-particle distance, n^-1/2.   

Helical trilayer graphene: a moiré platform for strongly-interacting topological bands – Part 2

The combination of strong electronic correlations and non-trivial band topology is fertile ground for exotic electronic phenomena. Here, we explore topological flat bands that emerge in magic angle helical trilayer graphene (HTG), which comprises three layers of graphene with successive layers rotated in the same direction by the same relative twist angle, θ_m ≈ 1.8°. We find correlated states and anomalous Hall effect at multiple integer and fractional electron fillings per moiré unit cell. At the filling of one electron per moiré unit cell, a time-reversal symmetric phase appears beyond a critical electric displacement field, indicating a topological phase transition. Finally, hysteresis upon sweeping carrier density points to first-order phase transitions across a spatial mosaic of Chern domains separated by a network of topological gapless edge states. We establish HTG as an ideal platform for exploring orbital magnetism with Chern domain walls, and a promising system for realizing exotic strongly interacting topological phases.   

Helical trilayer graphene: a moiré platform for strongly-interacting topological bands – Part 3

Magic-angle helical trilayer graphene (HTG), a heterostructure consisting of three graphene layers with equal consecutive twists, relaxes into an array of commensurate domains with narrow and topological moir’e bands. We perform a detailed analysis of the interacting phase diagram at integer fillings, combining analytical strong-coupling analysis and Hartree-Fock numerics. We uncover a rich phase diagram of multiple topological symmetry-broken Chern insulators with |C| as large as 6. In contrast to other magic-angle graphene systems, we find that HTG remains in the “strong-coupling” regime, in which interactions dominate over kinetic energy, at all fillings. For experimentally-accessible displacement fields, these states undergo continuous topological phase transitions to more complex orders, including charge density waves and Kekul’e-ordered phases. Finally, we discuss connections to recent experimental studies in HTG. Our results demonstrate the robust capability of HTG to host gate-tunable topological and symmetry-broken correlated phases.   

Field-stabilized Chern insulator and possible nematicity in magic-angle Helical trilayer graphene

Helical trilayer graphene (HTG) consists of three layers of graphene with successive layers twisted by the same relative angle, resulting in two moire patterns with different orientations. Although HTG is globally C_2z-symmetric, lattice relaxations form large periodic domains where C_2z is broken on the moire scale, resulting in a spatial mosaic of Chern domains. Because domains have valley-contrasting Chern numbers, a network of topological gapless 1D states forms at their boundaries. When the graphene layers are twisted at a magic angle of 1.8 degrees, these topological bands become flat and a rich phase diagram of correlated states emerges, as was recently uncovered. 

Here, we will explore new physics in HTG that arise in finite fields. First, we discuss a special scenario where the network of gapless edge states is no longer topologically protected, leading to the apparent formation of a robustly quantized field-induced Chern insulator. Second, we discuss evidence of possible nematic order at specific integer filling states. These additional findings further highlight HTG as a platform for exploring both strongly correlated and topological phases.   

Strong correlations and isospin symmetry breaking in a supermoiré lattice

In multilayer moiré heterostructures, the interference of multiple twist angles ubiquitously leads to tunable ultra-long-wavelength patterns known as supermoiré lattices. However, their impact on the system's many-body electronic phase diagram remains unknown. In this talk, we present local compressibility measurements of twisted trilayer graphene revealing numerous incompressible states resulting from supermoiré-lattice-scale isospin symmetry breaking driven by strong interactions. By using the supermoiré lattice occupancy as a probe of isospin symmetry, we observe an unexpected doubling of the miniband filling near ν=−2, possibly indicating a hidden phase transition or normal-state pairing proximal to the superconducting phase. Our work establishes supermoiré lattices as a tunable parameter for designing novel quantum phases and an effective tool for unraveling correlated phenomena in moiré materials.   

Pseudomagnetic field in twisted bilayer graphene

Twisted bilayer graphene stands at the forefront of extensive research in condensed matter physics due to its intriguing electronic properties, such as unconventional insulating and superconducting phases. Many of these phenomena are still not understood and this is possible due to the fact that most of the used theoretical models ignore the role played by atoms in determining the electronic properties. In particular, atomic displacement due to the competition between the out-of-plane van der Waal  interaction and the in-plane elastic forces can couple to the electronic degrees of freedom in the form of a pseudo-magnetic field. In this theoretical work, we examine the effect of the pseudomagnetic field on the properties of the system. We show that the pseudomagnetic field shifts the magic angle and therefore produces a “magic range” that is expected to vary from sample to sample.   

Observation of flat band in twisted double bilayer graphene using nanoARPES

The strong correlation physics leads to observing superconductivity and insulating states in twisted bilayer graphene at the magic angle ^[1,2]. Similarly, twisted double bilayer graphene (tDBG) has been predicted to host remarkably flat bands at narrow regions around the magic angle (θ ~ 1.3 ˚) ^[3]. A direct method to understand this correlated electron physics is the evolution of band topology by tuning the density of states. We use in-operando angle-resolved photoemission spectroscopy with nanometer scale (nanoARPES) spatial resolution to study the tDBG device with the twist angle ~2.6 ± 0.2˚. A remarkable change in the band topology can be observed with the tuning of the charge carrier density using back gate voltages. Moiré miniband with flat dispersions over the whole mini-Brillouin zone is observed at higher gate voltage (Vg ≥ 6V). This flat band is separated from the dispersive Dirac bands, leading to multiple moiré hybridization. The application of external electric fields provides a control mechanism over the bandwidth and flat-band structure and unveils the electron−electron interaction phenomena at different filling factors with in situ electrostatic gating.

[1] Cao, Y. et al. Nature 556, 80–84 (2018).

[2] Cao, Y. et al. Nature 556, 43–50 (2018).

[3] Shen, C., Chu, Y., Wu, Q. et al. Nat. Phys. 16, 520–525 (2020).   

Orbital magnetism and Fermi surface reconstructions near half filling in twisted bilayer graphene

Magic-angle twisted bilayer graphene (MATBG) exhibits a wide variety of correlated phases, spanning from insulating to superconducting and magnetic states, favored by the flat bands. The degeneracy among closely competing ground states can be lifted by polarizing spin and valley degrees of freedom; hence, the four-fold degeneracy of the low-energy electrons has a significant impact on the underlying mechanism governing the correlated phases at different band fillings. The overall phase diagram of MATBG is remarkably sensitive to external perturbations such as carrier density, electromagnetic field, pressure, temperature, and dielectric environments. Despite this unprecedented tunability, a complete understanding of the observed phases has remained elusive. In our recent study, we conducted magneto-transport measurements on MATBG proximitized by a layer of tungsten diselenide, thereby introducing finite spin-orbit coupling into the system. Our findings unveiled an anomalous Hall effect in the vicinity of half-filling (ν = 2), accompanied by an abrupt switching of magnetization that can be fine-tuned by varying the carrier density. Such a reversal of hysteresis suggests a ferromagnetic ground state that is orbital in nature. Additionally, near ν = 2, we observed a series of Lifshitz transitions in the zero-magnetic field limit, indicating Fermi surface reconstructions. As we increased the magnetic field, a perfectly quantized Chern insulator was observed exactly at ν = 2. These intriguing results collectively suggest the presence of valley-polarized ground states near ν = 2, which are stabilized by the inclusion of spin-orbit coupling.

 

 

Funding acknowledgment:

U.C. acknowledges funding from SERB via SPG/2020/ 000164 and WEA/2021/000005.   

Disorder broadening of Hofstadter bands in the honeycomb lattice

For a large range of twist angles, above the magic angle, the Hofstadter bands of twisted bilayer and trilayer graphene are dual to the Hofstadter bands of the honeycomb lattice. Here, we calculate the broadening of the Hofstadter bands of a honeycomb lattice in the presence of disorder within the self-consistent Born approximation (SCBA). In this study, the disorder is treated by the method of impurity averaging with the disorder fluctuation having white noise correlations. We calculate the local density of states which can be compared with scanning tunneling microscopy (STM) measurements of Hofstadter bands in twisted bilayer graphene.   

Imaging the many-body wavefunctions in magic-angle graphene

Magic-angle twisted bilayer graphene (MATBG) hosts a set of flat electronic bands that produce a wide variety of correlated ground states including superconductors, correlated insulators, and magnetic topological phases. Our understanding of these phases has thus far been hampered by the lack of microscopic, atomic-scale information on them. In this talk, I will discuss how we use high-resolution scanning tunneling microscopy (STM) measurements to study the wavefunctions of these correlated phases. STM images reveal distinct symmetry-breaking patterns with a sqrt(3) x sqrt(3) superperiodicity on the graphene atomic lattice. To understand these patterns, we develop a symmetry-based analysis to visualize the images in terms of a set of complex-valued order parameters. At the correlated insulators at fillings ±2 electrons per moiré unit cell, comparison with theoretical candidates reveals a close match with the proposed incommensurate Kekulé spiral order in samples with typical values of interlayer strain, while in ultralow-strain samples, our data have local symmetries resembling the time-reversal symmetric intervalley coherent phase. These symmetry-based techniques can be further applied to other phases in MATBG, graphene systems, and potentially other material systems.   

Perpendicular electronic transport and moiré-induced resonance in twisted interfaces of 3D graphite

We theoretically study a perpendicular electronic transport in twisted three-dimensional (3D) systems using the effective continuum model and the recursive Green’s function method [1].

The perpendicular electric conduction in twisted systems was probed in a recent experiment [2] while the transmission across the twisted interface has not been well examined theoretically. In this study, we develop a formulation for the electronic transport in the twisted 3D system, which is a pair of 3D materials stacked with a twist angle, by using the effective continuum model and recursive Green’s function method.

We apply the method to twisted graphite (rotationally-stacked graphite pieces), which is one of the simplest twisted 3D systems.

In the twisted graphite, we found that the perpendicular conductivity depends on the twist angle non-monotonously, and it cannot be explained by a simple picture based on the Fermi-surface overlap. We reveal that the anomalous twist-angle dependence is due to the Fano resonance by an interface-localized state, which is a remnant of the flat state of magic-angle twisted bilayer graphene. The existence of the interface-localized state is confirmed by calculating the local density of states using the recursive Green’s function method.

[1] T. Tani, T. Kawakami, M. Koshino, arXiv:2308.03993

[2] A. Inbar et al., Nature 614, 682 (2023).