The emergence of flat electronic bands and of the recently discovered strongly correlated and superconducting phases in twisted bilayer graphene crucially depends on the interlayer twist angle upon approaching the magic angle ߠ ெ ≈ 1.1°. Although advanced fabrication methods allow alignment of graphene layers with global twist angle control of about 0.1°, little information is currently available on the distribution of the local twist angles in actual magic angle twisted bilayer graphene (MATBG) transport devices. Here we map the local ߠ variations in hBN encapsulated devices with relative precision better than 0.002° and spatial resolution of a few moiré periods. Utilizing a scanning nanoSQUID-on-tip, we attain tomographic imaging of the Landau levels in the quantum Hall state in MATBG, which provides a highly sensitive probe of the charge disorder and of the local band structure determined by the local ߠ. We find a correlation between the degree of twist angle disorder and the quality of the typical MATBG transport characteristics. However, even state-of-the-art transport devices, exhibiting pronounced global MATBG features, such as multiple correlated insulator states, high-quality Landau fan diagrams, and superconductivity, display significant variations in the local ߠ with a span that can be close to 0.1°. Devices may even have substantial areas where no local MATBG behavior is detected, yet still display global MATBG characteristics in transport, highlighting the importance of percolation physics. The derived ߠ maps reveal substantial gradients and a network of jumps. We show that the twist angle gradients generate large unscreened electric fields that drastically change the quantum Hall state by forming edge states in the bulk of the sample, and may also significantly affect the phase diagram of correlated and superconducting states. The findings call for exploration of band structure engineering utilizing twist-angle gradients and gate-tunable built-in planar electric fields for novel correlated phenomena and applications.

An error occurred in the numerical examples presented in the original article, as the contribution from the ns → np transition was neglected in the calculation of the dipole Casimir-Polder (CP) shift and transition rates. This transition is relevant for rubidium as a result of differing quantum defects for the ns and np states. As seen in the corrected Fig. 1, the 43s → 43p transition contributes about 50% to the total level shift. As a result, the CP shifts of rubidium in Rydberg states are even larger than stated in the original article, roughly by a factor of 2, as can be seen from the corrected Fig. 2(a). The decay rates are affected in a similar way [cf. the corrected Fig. 2(b)]. Note that these quantitative changes do not affect any of the conclusions made in the original article regarding the physics of Rydberg atoms near surfaces.Also, in the text following Eq. (9), the Drude relation for the permittivity of a metal should read ε(ω) = 1 − ω 2 p /ω(ω + iγ ). We thank R. Fermani for bringing these issues to our attention. 38 39 40 41 42 44 45 46 47 0 0.2 0.4 0.6 0.8 1 43 Principal quantum number n Dipole Quadrupole FIG. 1. (Color online) Relative contributions from different transitions to the CP dipole and quadrupole level shift of the state 43s of 87 Rb. 0 5 10 15 20 −10 9

The dyadic Green's function of the inhomogeneous vector Helmholtz equation describes the field pattern of a single frequency point source. It appears in the mathematical description of many areas of electromagnetism and optics including both classical and quantum, linear and nonlinear optics, dispersion forces (such as the Casimir and Casimir-Polder forces) and in the dynamics of trapped atoms and molecules. Here, we compute the Green's function for a layered topological insulator. Via the magnetoelectric effect, topological insulators are able to mix the electric, E, and magnetic induction, B, fields and, hence, one finds that the T E and T M polarizations mix on reflection from/transmission through an interface. This leads to novel field patterns close to the surface of a topological insulator. 78.20.Ek, 78.67.Pt, 42.25.Gy whereR − is the reflection matrix at the d − surface. Solving for B and D givesandDe ikzd− =M −+ · e ikz|d−−z |R − +e ikz(d+−d−) e ikz|d+−z |R − ·R + , (70)

We calculate the Casimir-Polder frequency shift and decay rate for an atom in front of a nonreciprocal medium by using macroscopic quantum electrodynamics. The results are a generalization of the respective quantities for matter with broken time-reversal symmetry which does not fulfill the Lorentz reciprocity principle. As examples, we contrast the decay rates, the resonant and nonresonant frequency shifts of a perfectly conducting (reciprocal) mirror to those of a perfectly reflecting nonreciprocal mirror. We find different power laws for the distance dependence of all quantities in the retarded and nonretarded limits. As an example of a more realistic nonreciprocal medium, we investigate a topological insulator subject to a time-symmetry breaking perturbation.

We study the energy spectrum and quantum Hall effects of the twisted double bilayer graphene in uniform magnetic field. We investigate two different arrangements, AB-AB and AB-BA, which differ in the relative orientation but have very similar band structures in the absence of a magnetic field. For each system, we calculate the energy spectrum and quantized Hall conductivities at each spectral gap by using a continuum Hamiltonian that satisfies the magneto-translation condition. We show that the Hofstadter butterfly spectra of AB-AB and AB-BA stackings differ significantly, even though their zero magnetic field band structures closely resemble; the spectrum of AB-AB has valley degeneracy, which can be lifted by applying interlayer potential asymmetry, while the spectrum of AB-BA has no such degeneracy in any case. We explain the origin of the difference from the perspectives of lattice symmetry and band topology.