Theory of non-integer high-harmonic generation in a topological surface state

High harmonic generation is a versatile experimental technique for probing ultrafast electron dynamics. While in the past it has been employed typically in dielectrics and semiconductors, recently high harmonic generation was also observed from a topological surface [Schmid et al., Nature 593, 385 (2021)]. It has been found that harmonic orders in the intermediate range of 13-18 continuously shift when the carrier envelope phase (CEP) is varied. In this work, we adopt a minimal model of the topological surface state and calculate analytically the high-harmonic spectrum. We derive formulae describing the parametric dependencies of CEP shifts in high harmonics; in particular, we have a transparent result for the shift of the (peak) frequency ω when changing the CEP φ: dω/dφ=−2f¯′ω/ω0, where ω0 describes the fundamental driving frequency and f¯′ characterizes the chirp of the driving laser pulse. We compare the analytical formula to full-fledged numerical simulations finding only 17% average absolute deviation in dω/dφ. Our analytical result is fully consistent with experimental observations. It therefore provides the first understanding of the phenomenon of CEP shifts in solids based on analytically derived parametric dependencies

Low-scaling GW algorithm applied to twisted transition-metal dichalcogenide heterobilayers

The $GW$ method is widely used for calculating the electronic band structure of materials. The high computational cost of $GW$ algorithms prohibits their application to many systems of interest. We present a periodic, low-scaling and highly efficient $GW$ algorithm that benefits from the locality of the Gaussian basis and the polarizability. The algorithm enables $G_0W_0$ calculations on a MoSe$_2$/WS$_2$ bilayer with 984 atoms per unit cell, in 42 hours using 1536 cores. This is four orders of magnitude faster than a plane-wave $G_0W_0$ algorithm, allowing for unprecedented computational studies of electronic excitations at the nanoscale

Low-Scaling GW Algorithm Applied to Twisted Transition-Metal Dichalcogenide Heterobilayers

The GW method is widely used for calculating the electronic band structure of materials. The high computational cost of GW algorithms prohibits their application to many systems of interest. We present a periodic, low-scaling, and highly efficient GW algorithm that benefits from the locality of the Gaussian basis and the polarizability. The algorithm enables G0W0 calculations on a MoSe2/WS2 bilayer with 984 atoms per unit cell, in 42 h using 1536 cores. This is 4 orders of magnitude faster than a plane-wave G0W0 algorithm, allowing for unprecedented computational studies of electronic excitations at the nanoscale

Low-Scaling GW Algorithm Applied to Twisted Transition-Metal Dichalcogenide Heterobilayers

The GW method is widely used for calculating the electronic band structure of materials. The high computational cost of GW algorithms prohibits their application to many systems of interest. We present a periodic, low-scaling, and highly efficient GW algorithm that benefits from the locality of the Gaussian basis and the polarizability. The algorithm enables G0W0 calculations on a MoSe2/WS2 bilayer with 984 atoms per unit cell, in 42 h using 1536 cores. This is 4 orders of magnitude faster than a plane-wave G0W0 algorithm, allowing for unprecedented computational studies of electronic excitations at the nanoscale

Ultrafast atomic-scale scanning tunnelling spectroscopy of a single vacancy in a monolayer crystal

Defects in atomically thin semiconductors and their moiré heterostructures have emerged as a unique testbed for quantum science. Strong light–matter coupling, large spin–orbit interaction and enhanced Coulomb correlations facilitate a spin–photon interface for future qubit operations and efficient single-photon quantum emitters. Yet, directly observing the relevant interplay of the electronic structure of a single defect with other microscopic elementary excitations on their intrinsic length, time and energy scales remained a long-held dream. Here we directly resolve in space, time and energy how a spin–orbit-split energy level of an isolated selenium vacancy in a moiré-distorted WSe2 monolayer evolves under the controlled excitation of lattice vibrations, using lightwave scanning tunnelling microscopy and spectroscopy. By locally launching a phonon oscillation and taking ultrafast energy-resolved snapshots of the vacancy’s states faster than the vibration period, we directly measure the impact of electron–phonon coupling in an isolated single-atom defect. The combination of atomic spatial, sub-picosecond temporal and millielectronvolt energy resolution marks a disruptive development towards a comprehensive understanding of complex quantum materials, where the key microscopic elementary interactions can now be disentangled, one by one

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