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Register a new userGround State and Hidden Symmetry of Magic Angle Graphene at Even Integer Filling
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In magic angle twisted bilayer graphene, electron-electron interactions play a central role resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question and potentially linked to the relatively high temperature superconductivity observed in the same devices. Here we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC exhibits a pattern of alternating circulating currents which triples the graphene unit cell leading to an orbital magnetization density wave. Although translation and time reversal symmetry are broken, a combined `Kramers time reversal symmetry is preserved. Our analytic arguments are built on first identifying an approximate ${rm U}(4) times {rm U}(4)$ symmetry, resulting from the remarkable properties of the tBG band structure, which helps select a low energy manifold of states, which are further split to favor the K-IVC. This low energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry lowering perturbations can stabilize other insulators and the semi-metallic state, and discuss the ground state at half filling and a comparison with experiments.
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We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the Moire flat bands of twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structures and interactions and focus on the charge neutrality point, where experiments have variously reported either insulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remains unbroken, we find three types of self-consistent solutions with competitive ground state energy (i) insulators that break $C_2 {mathcal T}$ symmetry, including valley Chern insulators (ii) spin or valley polarized insulators and (iii) rotation $C_3$ symmetry breaking semimetals whose gaplessness is protected by the topology of the Moire flat bands. We find that the relative stability of these states can be tuned by weak strains that break $C_3$ rotation. The nematic semimetal and also, somewhat unexpectedly, the $C_2 {mathcal T}$ breaking insulators, are stabilized by weak strain. These ground states may be related to the semi-metallic and insulating behaviors seen at charge neutrality, and the sample variability of their observation. We also compare with the results of STM measurements near charge neutrality.
Motivated by the appearance of a `reflection symmetry in transport experiments and the absence of statistical periodicity in relativistic quantum field theories, we propose a series of relativistic composite fermion theories for the compressible states appearing at filling fractions $ u=1/2n$ in quantum Hall systems. These theories consist of electrically neutral Dirac fermions attached to $2n$ flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to $ u=1/2n$, and we show that such states can be related by the observed reflection symmetry, at least at mean field level. We further argue that the lowest Landau level limit requires that the Dirac fermions be tuned to criticality, whether or not this symmetry extends to the compressible states themselves.
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. At a non-zero density away from an integer filling the excitations interact. The system on the small mass side has a large bandwidth and forms a Fermi liquid. On the large mass side the bandwidth is narrow, the compressibility is negative and the Fermi liquid is likely unstable. This explains the observed sawtooth features in compressibility, the Landau fans pointing away from charge neutrality as well as their degeneracies. By providing a description of the charge itineracy in the normal state this framework sets the stage for superconductivity at lower temperatures.
We present a systematic study of the low-energy collective modes for different insulating states at integer fillings in twisted bilayer graphene. In particular, we provide a simple counting rule for the total number of soft modes, and analyze their energies and symmetry quantum numbers in detail. To study the soft mode spectra, we employ time dependent Hartree-Fock whose results are reproduced analytically via an effective sigma model description. We find two different types of low-energy modes - (i) approximate Goldstone modes associated with breaking an enlarged U(4)$times$U(4) symmetry and, surprisingly, a set of (ii) nematic modes with non-zero angular momentum under three-fold rotation. The modes of type (i) include true gapless Goldstone modes associated with exact symmetries in addition to gapped pseudo-Goldstone modes associated with approximate symmetries. While the modes of type (ii) are always gapped, we show that their gap decreases as the Berry curvature grows more concentrated. For realistic parameter values, the gapped soft modes of both types have comparable gaps of only a few meV, and lie completely inside the mean-field bandgap. The entire set of soft modes emerge as Goldstone modes of a different idealized model in which Berry flux is limited to a solenoid, which enjoys an enlarged U(8) symmetry. Furthermore, we discuss the number of Goldstone modes for each symmetry-broken state, distinguishing the linearly vs quadratically dispersing modes. Finally, we present a general symmetry analysis of the soft modes for all possible insulating Slater determinant states at integer fillings that preserve translation symmetry, independent of the energetic details. The resulting soft mode degeneracies and symmetry quantum numbers provide a fingerprint of the different insulting states enabling their experimental identification from a measurement of their soft modes.
Spontaneous symmetry breaking plays a pivotal role in many areas of physics, engendering a variety of excitations from sound modes in solids to pions in nuclear physics. Equally important excitations are solitons, nonlinear configurations of the symmetry breaking field, which can enjoy exceptional stability as in the Skyrme model of nuclear forces. Here we argue that similar models may describe magic angle graphene, a remarkable new material . When the angle between two sheets of graphene is near the magic angle of $sim 1^circ$, insulating behavior is observed, which gives way to superconductivity on changing the electron density. We propose a unifying description of both the order underlying the insulator as well as the superconductor. While the symmetry breaking condensate leads to the ordered phase, topological solitons in the condensate - skyrmions - are shown to be bosons that carry an electric charge of 2e. Condensation of skyrmions leads to a superconductor whose pairing strength, symmetry and other properties are inferred. More generally, we show how topological textures can mitigate Coulomb repulsion to pair electrons and provide a new route to superconductivity. Our mechanism potentially applies to much wider class of systems but crucially invokes certain key ingredient such as inversion symmetry present in magic angle graphene. We discuss how these insights not only clarify why certain correlated moire materials do not superconduct, they also point to promising new platforms where robust superconductivity is anticipated.
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