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We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-- MacDonald PDE model, which is periodic with respect to the bilayer's moir\'e pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes--Thomas estimates. We then provide extensive numerical computations, which clarify the range of validity of the Bistritzer--MacDonald model. 

Momentum space transformations for incommensurate two-dimensional electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer--MacDonald model [Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 12233--12237]. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. First, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straightforward algorithm for writing the complex momentum space model from the original real space model. Second, we implement this for twisted bilayer graphene with mechanical relaxation effects included. We also analyze the convergence rates of the approximations and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer--MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation. 

In this expository article, we present a systematic formal derivation of the Kubo formula for the linear-response current due to a time-harmonic electric field applied to non-interacting, spinless charged particles in a finite volume in the quantum setting. We model dissipation in a transparent way by assuming a sequence of scattering events occurring at random-time intervals modeled by a Poisson distribution. By taking the large-volume limit, we derive special cases of the formula for free electrons, continuum and tight-binding periodic systems, and the nearest-neighbor tight-binding model of graphene. We present the analogous formalism with dissipation to derive the Drude conductivity of classical free particles. 

We describe the low‐temperature optical conductivity as a function of frequency for a quantum‐mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heegertight‐bindingHamiltonian for noninteracting spinless electrons on a one‐dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo‐type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three‐dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.

 

Abstract The relativistic charge carriers in monolayer graphene can be manipulated in manners akin to conventional optics. Klein tunneling and Veselago lensing have been previously demonstrated in ballistic graphene pn-junction devices, but collimation and focusing efficiency remains relatively low, preventing realization of advanced quantum devices and controlled quantum interference. Here, we present a graphene microcavity defined by carefully-engineered local strain and electrostatic fields. Electrons are manipulated to form an interference path inside the cavity at zero magnetic field via consecutive Veselago refractions. The observation of unique Veselago interference peaks via transport measurement and their magnetic field dependence agrees with the theoretical expectation. We further utilize Veselago interference to demonstrate localization of uncollimated electrons and thus improvement in collimation efficiency. Our work sheds new light on relativistic single-particle physics and provide a new device concept toward next-generation quantum devices based on manipulation of ballistic electron trajectory.