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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Localization and Criticality in Antiblockaded Two-Dimensional Rydberg Atom Arrays

Controllable Rydberg atom arrays have provided new insights into fundamental properties of quantum matter both in and out of equilibrium. In this work, we study the effect of experimentally relevant positional disorder on Rydberg atoms trapped in a 2D square lattice under antiblockade (facilitation) conditions. We show that the facilitation conditions lead the connectivity graph of a particular subspace of the full Hilbert space to form a 2D Lieb lattice, which features a singular flat band. Remarkably, we find three distinct regimes as the disorder strength is varied: a critical regime, a delocalized but nonergodic regime, and a regime with a disorder-induced flat band. The critical regime’s existence depends crucially upon the singular flat band in our model, and is absent in any 1D array or ladder system. We propose to use quench dynamics to probe the three different regimes experimentally.

2-dimensional systems↗
DOE OSTI2025

Unconventional Fractional Phases in Multiband Vortexable Systems

We study topological flat bands with distinct features that deviate from conventional Landau level behavior. We show that even in the ideal quantum geometry limit, moiré flat band systems can exhibit physical phenomena fundamentally different from Landau levels without lattices. In particular, we find new fractional quantum Hall states emerging from multiband vortexable systems, where multiple exactly flat bands appear at the Fermi energy. While the set of bands as a whole exhibits ideal quantum geometry, individual bands separately lose vortexability, and thus making them very different from a stack of Landau levels. At certain filling fractions, we find fractional states whose Hall conductivity deviates from the filling factor. Through careful numerical and analytical studies, we rule out all known mechanisms—such as fractional quantum Hall crystals or separate filling of trivial and topological bands—as possible explanations. Leveraging the exact solvability of vortexable systems, we use analytic Bloch wave functions to uncover the origin of these new fractional states, which arises from the commensurability between the moiré unit cell and the magnetic unit cell of an emergent effective magnetic field.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Corrugation-driven symmetry breaking in magic-angle twisted bilayer graphene

Abstract The discovery of unconventional superconductivity in magic-angle twisted bilayer graphene (tBLG) supported the twist-angle-induced flat band structure predictions made a decade earlier. Numerous physical properties have since been linked to the interlayer twist angle using the flat band prediction as a guideline. However, some key observations like the nematic phase and striped charge order behind the superconductivity are missing in this initial model. Here we show that a thermodynamically stable large out-of-plane displacement, or corrugation of the bilayer, induced by the interlayer twist, demonstrates partially filled states of the flat band structure, accompanied by a broken symmetry, in the magic-angle regime and the presence of symmetry breaking associated with the superconductivity in tBLG. The distinction between low and high corrugation can also explain the observed evolution of the vibrational spectra of tBLG as a function of twist angle. Our observation that large out-of-plane deformation modes enable partial filling of states near the Fermi energy may lead to a strategy for offsetting the effects of disorder in the local twist angle, which suppresses unconventional superconductivity and correlated insulator behavior in magic-angle tBLG.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Engineering 2D Square Lattice Hubbard Models in 90° Twisted GeX/SnX (X =S, Se) Moiré Superlattices

Because of the large-period superlattices emerging in moiré two-dimensional (2D) materials, electronic states in such systems exhibit low energy flat bands that can be used to simulate strongly correlated physics in a highly tunable setup. While many investigations have thus far focused on moiré flat bands and emergent correlated electron physics in triangular, honeycomb, and quasi-one-dimensional lattices, tunable moiré realizations of square lattices subject to strong correlations remain elusive. Here, in this work, we propose a feasible scheme to construct moiré square lattice systems by twisting two or more layers of 2D materials in a rectangular lattice by 90°. We demonstrate the concept with twisted GeX/SnX (X =S, Se) moiré superlattices and calculate their electronic structures from first principles. We show that the lowest conduction flat band in these systems can be described by a square lattice Hubbard model with parameters which can be controlled by varying the choice of host materials, number of layers, and external electric fields. In particular, twisted double bilayer GeSe realizes a square lattice Hubbard model with strong frustration due to the next-nearest-neighbor hopping that could host unconventional superconductivity, in close analogy to the Hubbard model for copper-oxygen planes of cuprate high-temperature superconductors. The presented scheme uses 90° twisted 2D materials with rectangular unit cells as a promising platform for realizing the physical phenomena of square lattice Hubbard models, establishing a new route for studying its rich phase diagram of magnetism, charge order, and unconventional superconductivity in a highly tunable setting.

2-dimensional systems↗
DOE OSTI2024

Strong electron–phonon coupling in magic-angle twisted bilayer graphene

The unusual properties of superconductivity in magic-angle twisted bilayer graphene (MATBG) have sparked considerable research interest. However, despite the dedication of intensive experimental efforts and the proposal of several possible pairing mechanisms, the origin of its superconductivity remains elusive. Here, by utilizing angle-resolved photoemission spectroscopy with micrometre spatial resolution, we reveal flat-band replicas in superconducting MATBG, where MATBG is unaligned with its hexagonal boron nitride substrate. These replicas show uniform energy spacing, approximately 150 ± 15 meV apart, indicative of strong electron–boson coupling. Strikingly, these replicas are absent in non-superconducting twisted bilayer graphene (TBG) systems, either when MATBG is aligned to hexagonal boron nitride or when TBG deviates from the magic angle. Calculations suggest that the formation of these flat-band replicas in superconducting MATBG are attributed to the strong coupling between flat-band electrons and an optical phonon mode at the graphene K point, facilitated by intervalley scattering. These findings, although they do not necessarily put electron–phonon coupling as the main driving force for the superconductivity in MATBG, unravel the electronic structure inherent in superconducting MATBG, thereby providing crucial information for understanding the unusual electronic landscape from which its superconductivity is derived.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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 OSTI2026

Quantitative Analysis of the Semiconductor–Electrolyte Interface Using Cyclic Voltammetry Measurements

Small changes in the chemical potential at a semiconductor interface can result in dramatic changes to the space-charge layer that underpins applications in the electronic and photovoltaic industries as well as in photoelectrochemical cells for fuel production. There has hence been great interest in techniques that directly probe the space-charge layer, yet many fail at the semiconductor–electrolyte interface due to the potential drop in the electric double-layer region of the electrolyte. This article demonstrates that photovoltages, obtained from straightforward cyclic voltammetry measurements, provide an experimental and quantitative approach for characterizing the semiconductor–electrolyte interface. Key parameters accessible through this approach include the flat-band potential ( E fb ), the fraction of the total potential that drops across the space-charge layer (γ sc ) and the electric double layer, as well as the surface recombination lifetime (τ s ). Here, we report photovoltage measurements for p -type Si(111) photoelectrodes in contact with electrolytes containing redox-active species with a range of known reduction potentials that exceed the 1.1 eV bandgap. In tetrabutylammonium [NBu 4 ] + electrolyte, the flat-band potential determined for hydrogen-terminated ( p -Si–H), methyl-terminated ( p -Si–CH 3 ), and chemically oxidized ( p -Si–cSiO x ) surfaces were −0.02, −0.31, and 0.30 V vs Fc +/0 , respectively, agreeing well with expected shifts arising from surface dipole modifications. The quantitative analysis also reveals that 67% of the applied bias drops across the space-charge layer for p -Si–H, 73% for p -Si–CH 3 , and only 44% for p -Si–cSiO x . The remaining potential drop is attributed to the interfacial surface layer, which consists of a molecular dipole or oxide overlayer, and the Helmholtz layer within the electrolyte. When the larger [NBu 4 ] + electrolyte was replaced with Li + , the flat-band position showed minimal changes, but the fraction of the potential drop across the space-charge layer increased significantly, consistent with the small cation altering the structure of the electric double layer.

electrolytes↗
DOE OSTI2025

Topological triviality of flat Hamiltonians

Landau levels play a key role in theoretical models of the quantum Hall effect. Each Landau level is degenerate, flat, and topologically nontrivial. Motivated by Landau levels, we study tight-binding Hamiltonians whose energy levels are all flat. Here, we demonstrate that in two dimensions, for such Hamiltonians, the flat bands must be topologically trivial. To that end, we show that the projector onto each flat band is necessarily strictly local. Our conclusions do not need the assumption of lattice translational invariance.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Diverse electronic topography in a distorted kagome metal LaTi 3 ⁢Bi 4

Recent reports on a family of kagome metals of the form 𝐿⁢𝑛⁢Ti 3 ⁢Bi 4 (𝐿⁢𝑛 = Lanthanide) have stoked interest due to the combination of highly anisotropic magnetism and a rich electronic structure. The electronic structure near the Fermi level is proposed to exhibit Dirac points and van Hove singularities (VHSs). Here, in this manuscript, we use angle-resolved photoemission spectroscopy measurements in combination with density functional theory calculations to investigate the electronic structure of an interesting kagome metal LaTi 3 ⁢Bi 4 . Our results reveal multiple VHSs with one VHS located in the vicinity of the Fermi level. We clearly observe two flat bands, which originate from the destructive interference of wave functions within the Ti kagome motif. These flat bands and VHSs originate from Ti 𝑑 orbitals and are very responsive to the polarization of the incident beam. We notice a significant anisotropy in the electronic structure, resulting from the breaking of sixfold rotational symmetry in this material. Our findings demonstrate this member of Ti based kagome materials as a promising platform to explore novel emerging phenomena in the wider 𝐿⁢𝑛⁢Ti 3⁢ Bi 4 (𝐿⁢𝑛 = lanthanide) family.

electronic structure↗
DOE OSTI2022

Strong Structural and Electronic Coupling in Metavalent PbS Moiré Superlattices

Moiré superlattices are twisted bilayer materials in which the tunable interlayer quantum confinement offers access to new physics and novel device functionalities. Previously, moiré superlattices were built exclusively using materials with weak van der Waals interactions, and synthesizing moiré superlattices with strong interlayer chemical bonding was considered to be impractical. Here, in this study, using lead sulfide (PbS) as an example, we report a strategy for synthesizing moiré superlattices coupled by strong chemical bonding. We use water-soluble ligands as a removable template to obtain free-standing ultrathin PbS nanosheets and assemble them into direct-contact bilayers with various twist angles. Atomic-resolution imaging shows the moiré periodic structural reconstruction at the superlattice interface due to the strong metavalent coupling. Electron energy loss spectroscopy and theoretical calculations collectively reveal the twist-angle-dependent electronic structure, especially the emergent separation of flat bands at small twist angles. The localized states of flat bands are similar to well-arranged quantum dots, promising an application in devices. This study opens a new door to the exploration of deep energy modulations within moiré superlattices alternative to van der Waals twistronics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Frustrated electron hopping from the orbital configuration in a two-dimensional lattice

Electron hopping on spatially periodic lattices gives rise to intriguing electronic behaviour. For example, hopping on the geometrically frustrated two-dimensional kagome, dice and Lieb lattices yields electronic band structures with both massless Dirac-like and perfectly dispersion-less, flat bands. As materials featuring the dice and Lieb lattice structures are scarce, an alternative approach proposes to leverage atomic orbitals to realize the characteristic electron hopping of geometrically frustrated lattices. This strategy promises to expand the list of candidate materials with frustrated electron hopping, but is yet to be shown in experiments. Here, in this study, we demonstrate frustrated hopping in the van der Waals intermetallic Pd 5 AlI 2 , emerging from the arrangement of atomic orbitals in a primitive square lattice. Using angle-resolved photoemission spectroscopy and quantum oscillation measurements, we reveal that the band structure of Pd 5 AlI 2 includes linear Dirac-like bands intersected at their crossing point by a locally flat band—an essential characteristic of frustrated hopping in Lieb and dice lattices. Moreover, this compound shows exceptional chemical stability, with its unusual bulk band structure and metallicity persisting in ambient conditions down to the monolayer limit. Hence, our results showcase a way to realize electronic structures characteristic of geometrically frustrated lattices in non-frustrated systems.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Reentrant Correlated Insulators in Twisted Bilayer Graphene at 25 T ( 2 π Flux)

Twisted bilayer graphene (TBG) is remarkable for its topological flat bands, which drive strongly interacting physics at integer fillings, and its simple theoretical description facilitated by the Bistritzer-MacDonald Hamiltonian, a continuum model coupling two Dirac fermions. Because of the large moiré unit cell, TBG offers the unprecedented opportunity to observe reentrant Hofstadter phases in laboratory-strength magnetic fields near 25 T. This Letter is devoted to magic angle TBG at 2π flux where the magnetic translation group commutes. Here we use a newly developed gauge-invariant formalism to determine the exact single-particle band structure and topology. We find that the characteristic TBG flat bands reemerge at 2π flux, but, due to the magnetic field breaking C 2z T, they split and acquire Chern number ±1. We show that reentrant correlated insulating states appear at 2π flux driven by the Coulomb interaction at integer fillings, and we predict the characteristic Landau fans from their excitation spectrum.

36 MATERIALS SCIENCE↗
NASA NTRS2001

A Preliminary Detection of Arcminute Scale Cosmic Microwave Background Anisotropy with the BIMA Array

We have used the Berkeley-Illinois-Maryland-Association (BIMA) array outfitted with sensitive cm-wave receivers to expand our search for minute scale anisotropy of the Cosmic Microwave Background (CMB). The interferometer was placed in a compact configuration to obtain high brightness sensitivity on arcminute scales over its 6.6' FWHM field of view. The sensitivity of this experiment to flat band power peaks at a multipole of 1 = 5530 which corresponds to an angular scale of -2'. We present the analysis of a total of 470 hours of on-source integration time on eleven independent fields which were selected based on their low IR contrast and lack of bright radio sources. Applying a Bayesian analysis to the visibility data, we find CMB anisotropy flat band power Q_flat = 6.1(+2.8/-4.8) microKelvin at 68% confidence. The confidence of a nonzero signal is 76% and we find an upper limit of Q_flat < 12.4 microKelvin at 95% confidence. We have supplemented our BIMA observations with concurrent observations at 4.8 GHz with the VLA to search for and remove point sources. We find the point sources make an insignificant contribution to the observed anisotropy.

Dawson, K. S.↗
DOE OSTI2021

Enhanced Coherence in Superconducting Circuits via Band Engineering

In superconducting circuits interrupted by Josephson junctions, the dependence of the energy spectrum on offset charges on different islands is 2e periodic through the Aharonov-Casher effect and resembles a crystal band structure that reflects the symmetries of the Josephson potential. We show that higher-harmonic Josephson elements described by a cos(2φ) energy-phase relation provide an increased freedom to tailor the shape of the Josephson potential and design spectra featuring multiplets of flat bands and Dirac points in the charge Brillouin zone. Flat bands provide noise-insensitive energy levels, and consequently, engineering band pairs with flat spectral gaps can help improve the coherence of the system. We discuss a modified version of a flux qubit that achieves, in principle, no decoherence from charge noise and introduce a flux qutrit that shows a spin-1 Dirac spectrum and is simultaneously quite robust to both charge and flux noise.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Aqueous TiO 2 Nanoparticles React by Proton-Coupled Electron Transfer

Redox reactions of aqueous colloidal TiO 2 4 nm nanoparticles (NPs) have been examined, including both citratecapped and uncapped NPs (c-TiO 2 and uc-TiO 2 ). Photoreduction gave stable blue colloidal c-TiO 2 R NPs with 10–60 electrons per particle. Equilibration of these reduced NPs with soluble redox reagents such as methylviologen (MV 2+ ) provided measurements of the colloid reduction potential as a function of pH. The potentials of c-TiO 2 from pH 2–9 varied linearly with pH, with a slope of –60 ± 5 mV/pH. Estimates of the potential at pH 12 were consistent with extrapolating that line to high pH. The reduction potentials did not correlate with the zeta potentials (ζ) or the surface charge of the NPs across this pH range. Similar reduction potentials were observed for c- and uc-TiO 2 at low pH even though they have quite different ζ potentials. These results show that the common surface-charging explanation of the pH dependence is not tenable in these systems. Oxidation of reduced c-TiO 2 R with the electron-transfer oxidant potassium triiodide (KI 3 ) occurred with a significant drop in pH, showing that protons were released when the electrons were removed from the NPs. Smaller pH drops were observed for the proton-coupled electron transfer (PCET) reagents O 2 (air) and 4-MeO-TEMPO (4-methoxy-2,2,6,6-tetramethylpiperine-1-oxy radical). The difference in the number of protons released with KI 3 vs O 2 and 4-MeO-TEMPO was roughly one proton per electron removed. Thus, the thermodynamically preferred reactivity of these colloidal TiO 2 NPs is PCET over the pH 2–13 range studied. The measured redox potentials refer to the chemical process TiO 2 + H + + e – → TiO 2 ·e – ,H + ; and therefore they do not correspond with an electronic energy such as a conduction band edge or flat band potential. The 1e – /1H + stoichiometry means that the TiO 2 reduction potentials correspond to a TiO 2 –H bond dissociation free energy (BDFE), determined to be 49 ± 2 kcal mol –1 . The PCET description is consistent with the pH dependence of E(TiO 2 /TiO 2 ·e – ,H + ), the release of protons upon oxidation, the lack of correlation with ζ potentials, the similarity of capped and uncapped NPs, and the small change in the potential and BDFE from the first to the last electron/proton pair (H atom) removed. Furthermore, this behavior is suggested to be the norm for redox-active oxide/water interfaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

When van der Waals Met Kagome: A 2D Antimonide with a Vanadium-Kagome Network

2D materials showcase unconventional properties emerging from quantum confinement effects. In this work, a “soft chemical” route allows for the deintercalation of K + from the layered antimonide KV 6 Sb 6 , resulting in the discovery of a new metastable 2D-Kagome antimonide K 0.1(1) V 6 Sb 6 with a van der Waals gap of 3.2 Å. The structure of K 0.1(1) V 6 Sb 6 was determined via the synergistic techniques, including X-ray pair distribution function analysis, advanced transmission electron microscopy, and density functional theory calculations. The K 0.1(1) V 6 Sb 6 compound crystallizes in the monoclinic space group C2/m (a = 9.57(2) Å, b = 5.502(8) Å, c = 10.23(2) Å, β = 97.6(2)°, Z = 2). The [V 6 Sb 6 ] layers in K 0.1(1) V 6 Sb 6 are retained upon deintercalation and closely resemble the layers in the parent compound, yet deintercalation results in a relative shift of the adjacent [V 6 Sb 6 ] layers. The magnetic properties of the K 0.1(1) V 6 Sb 6 phase in the 2–300 K range are comparable to those of KV 6 Sb 6 and another Kagome antimonide KV 3 Sb 5 , consistent with nearly temperature-independent paramagnetism. Electronic band structure calculation suggests a nontrivial band topology with flat bands and opening of band crossing afforded by deintercalation. Transport property measurements reveal a metallic nature for K 0.1(1) V 6 Sb 6 and a low thermal conductivity of 0.6 W K –1 m –1 at 300 K. Additionally, ion exchange in KV 6 Sb 6 via a solvothermal route leads to a successful partial exchange of K + with A + (A = Na, Rb, and Cs). Here, this study highlights the tunability of the layered structure of the KV 6 Sb 6 compound, providing a rich playground for the realization of new 2D materials.

2D↗
DOE OSTI2023

A p -orbital honeycomb-Kagome lattice realized in a two-dimensional metal-organic framework

The experimental realization of p-orbital systems is desirable because p-orbital lattices have been proposed theoretically to host strongly correlated electrons that exhibit exotic quantum phases. Here, we synthesize a two-dimensional Fe-coordinated bimolecular metal-organic framework which constitutes a honeycomb lattice of 1,4,5,8,9,12-hexaazatriphenylene molecules and a Kagome lattice of 5,15-di(4-pyridyl)-10,20-diphenylporphyrin molecules on a Au(111) substrate. Density-functional theory calculations show that the framework features multiple well-separated spin-polarized Kagome bands, namely Dirac cone bands and Chern flat bands, near the Fermi level. Using tight-binding modelling, we reveal that these bands are originated from two effects: the low-lying molecular orbitals that exhibit p-orbital characteristics and the honeycomb-Kagome lattice. This study demonstrates that p-orbital Kagome bands can be realized in metal-organic frameworks by using molecules with molecular orbitals of p-orbital like symmetry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
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↗