Ballistic electrons in solids can have mean free paths far larger than the smallest features patterned by lithography. This has allowed development and study of solid-state electron-optical devices such as beam splitters and quantum point contacts, which have informed our understanding of electron flow and interactions. Recently, high-mobility graphene has emerged as an ideal two-dimensional semimetal that hosts unique chiral electron-optical effects due to its honeycomb crystalline lattice. However, this chiral transport prevents simple use of electrostatic gates to define electron-optical devices in graphene. Here, we present a method of creating highly-collimated electron beams in graphene based on collinear pairs of slits, with absorptive sidewalls between the slits. By this method, we achieve beams with angular width 18 degrees or narrower, and transmission matching semiclassical predictions.

Arthur W. Barnard Alex Hughes Aaron L. Sharpe Kenji Watanabe Takashi Taniguchi David Goldhaber-Gordon

Geometric electron optics may be implemented in solid state when transport is ballistic on the length scale of a device. Currently, this is realized mainly in 2D materials characterized by circular Fermi surfaces. Here we demonstrate that the nearly perfectly hexagonal Fermi surface of PdCoO2 gives rise to highly directional ballistic transport. We probe this directional ballistic regime in a single crystal of PdCoO2 by use of focused ion beam (FIB) micro-machining, defining crystalline ballistic circuits with features as small as 250nm. The peculiar hexagonal Fermi surface naturally leads to electron self-focusing effects in a magnetic field, well below the geometric limit associated with a circular Fermi surface. This super-geometric focusing can be quantitatively predicted for arbitrary device geometry, based on the hexagonal cyclotron orbits appearing in this material. These results suggest a novel class of ballistic electronic devices exploiting the unique transport characteristics of strongly faceted Fermi surfaces.

Maja D. Bachmann Aaron L. Sharpe Arthur W. Barnard Carsten Putzke Markus König Seunghyun Khim David Goldhaber-Gordon Andrew P. Mackenzie Philip J. W. Moll

Point identification of causal effects requires strong assumptions that are unreasonable in many practical settings. However, informative bounds on these effects can often be derived under plausible assumptions. Even when these bounds are wide or cover null effects, they can guide practical decisions based on formal decision theoretic criteria. Here we derive new results on optimal treatment regimes in settings where the effect of interest is bounded. These results are driven by consideration of superoptimal regimes; we define regimes that leverage an individual's natural treatment value, which is typically ignored in the existing literature. We obtain (sharp) bounds for the value function of superoptimal regimes, and provide performance guarantees relative to conventional optimal regimes. As a case study, we consider a commonly studied Marginal Sensitivity Model and illustrate that the superoptimal regime can be identified when conventional optimal regimes are not. We similarly illustrate this property in an instrumental variable setting. Finally, we derive efficient estimators for upper and lower bounds on the superoptimal value in instrumental variable settings, building on recent results on covariate adjusted Balke-Pearl bounds. These estimators are applied to study the effect of prompt ICU admission on survival.

Julien D. Laurendeau Aaron L. Sarvet Mats J. Stensrud

We present transport measurements of bilayer graphene with 1.38{\deg} interlayer twist and apparent additional alignment to its hexagonal boron nitride cladding. As with other devices with twist angles substantially larger than the magic angle of 1.1{\deg}, we do not observe correlated insulating states or band reorganization. However, we do observe several highly unusual behaviors in magnetotransport. For a large range of densities around half filling of the moir\'e bands, magnetoresistance is large and quadratic. Over these same densities, the magnetoresistance minima corresponding to gaps between Landau levels split and bend as a function of density and field. We reproduce the same splitting and bending behavior in a simple tight-binding model of Hofstadter's butterfly on a square lattice with anisotropic hopping terms. These features appear to be a generic class of experimental manifestations of Hofstadter's butterfly and may provide insight into the emergent states of twisted bilayer graphene.

Joe Finney Aaron L. Sharpe Eli J. Fox Connie L. Hsueh Daniel E. Parker Matthew Yankowitz Shaowen Chen Kenji Watanabe Takashi Taniguchi Cory R. Dean Ashvin Vishwanath Marc Kastner David Goldhaber-Gordon

Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle. Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer-MacDonald model to include uniaxial heterostrain. We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies. Coupled to a Boltzmann magnetotransport calculation, this reproduces previously-unexplained non-saturating $B^2$ magnetoresistance over broad ranges of density near filling $\nu=\pm 2$, and predicts subtler features that had not been noticed in the experimental data. In contrast to these distinctive signatures in longitudinal resistivity, the Hall coefficient is barely influenced by strain, to the extent that it still shows a single sign change on each side of the charge neutrality point -- surprisingly, this sign change no longer occurs at a van Hove point. The theory also predicts a marked rotation of the electrical transport principal axes as a function of filling even for fixed strain and for rigid bands. More careful examination of interaction-induced nematic order versus strain effects in twisted bilayer graphene could thus be in order.

Xiaoyu Wang Joe Finney Aaron L. Sharpe Linsey K. Rodenbach Connie L. Hsueh Kenji Watanabe Takashi Taniguchi M. A. Kastner Oskar Vafek David Goldhaber-Gordon

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp l \Bbb HP^k\sharp m \bar{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\sharp l CaP^2\sharp m \bar{CaP^2})=Y(M),$$ where $CaP^2$ is the Cayley plane.

Chanyoung Sung

Sharp estimates for C - and L - norms of functions that are conjugate with functions from the classes W^rH_\omega of periodic functions having prescribed concave majorant of moduli of continuity, as well as sharp estimates for the differences of such functions are obtained.

Vladislav Babenko

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties.

L. Barbieri-Viale A. Bertapelle

We establish unconditional sharp upper bounds of the $k$-th moments of the family of quadratic Dirichlet $L$-functions at the central point for $0 \leq k \leq 2$.

Peng Gao

We study the Second Main Theorem in non-archimedean Nevanlinna theory, giving an improvement to the non-archimedean Second Main Theorems of Ru and An in the case where all the hypersurfaces have degree greater than one and all intersections are transverse. In particular, under a transversality assumption, if f is a nonconstant non-archimedean analytic map to P^n and D_1,..,D_q are hypersurfaces of degree d, we prove the defect relation \sum_{i=1}^q\delta_f(D_i)\leq n-1+1/d, which is sharp for all positive integers n and d.

Aaron Levin