Origin of magic angles revealed
The angles at which flat bands emerge to produce superconductivity in twisted bilayer graphene are shown to be fundamentally connected to quantum Hall wave functions, as revealed in our recent paper published as Editors' Suggestion in Physical Review Letters.
Graphene is a deviceful atomically-thin hexagonal carbon crystal which is a host to many exotic quantum phenomena. Two graphene sheets rotated (”twisted”) by a small relative angle form a long-periodic Moire pattern – similar to those geometric patterns which appear in photography, typography and textile industry. Moire patterns induce a long-periodic effective potential in twisted bilayer graphene (TBG), which can enlarge the crystalline unit cell in hundreds times. In the past year (March 2018- March 2019), the twisted bilayer graphene (TBG) has boosted the condensed matter community after reported discovery of correlated insulation and unconventional superconductivity when tuned to the magic angles. The excitement of TBG-like systems is well-motivated, as it is a new and physically rich system, precisely controllable with the twist degree of freedom, which is host to several exotic physical phenomena such as anomalous quantum Hall effect, unconventional superconductivity, magic-angle band flatness and quasicrystal behavior at larger twists. Currently, the nature of superconductivity in this system is still being debated, however it is clear that the flat bands emerging at the magic angles play the fateful role. Until recently, even the very origin of the magic angles was still unclear, and many people in community think that the appearance of the magic angles is nothing more than a lucky engineering of material properties.
Motivated by recent observations of unconventional superconductivity in graphene bilayers twisted to the so-called "magic angles" (the angles at which emerging flat bands play the fateful role for superconductivity), we discover the fundamental origin of the magic angles and absolutely flat bands. To our surprise, the flatness is not just a matter of engineering material properties, but have deep hidden connections to Quantum Hall wave functions - a surprising “interdisciplinary” connection in rapidly evolving topological physics. At the heart of our work is the model with the perfectly flat bands which induce the robust periodicity of the magic angles. In fact, the magic angles in our model follow a remarkable sequence with robust asymptotic periodicity, which was never reported before. We present both numerical and analytical evidence that the bands can be done perfectly flat at magic angles. This flatness is not just a matter of engineering material properties, but have deep hidden connections to Quantum Hall wave functions and maps into the lowest Landau Level in Quantum Hall effect on torus. As we show in our paper, the principal magic angle can be calculated very precisely and its value is the same as reported in experiments.
Our paper is the first to provide a dramatically new insight concerning precisely how the magic angles appear in the twisted bilayer graphene, and provides deeper understanding of the twistronics effects, and a powerful analytic tool for future calculations. This is a new twist for geometric engineering of material properties.
G. Tarnopolsky, A J Kruchkov, A Vishwanath Phys. Rev. Lett. 122, 106405 – Published 15 March 2019