Abstract

The overlap matrix of electronic energy eigenstates, sometimes referred to as the form factor, determines the quantum geometric tensor of electrons in solids. Here, we show that the variation in the overlap of two eigenstates with opposite momenta can be directly observed via quasiparticle interference (QPI) imaging. We study the QPI in twisted bilayer graphene using a real-space tight-binding model combined with the kernel polynomial method. The resulting QPI patterns, which are largely independent of whether the two graphene layers are commensurate or incommensurate, reveal all intralayer and interlayer interference processes. While the intralayer interference signals resemble those of monolayer graphene, the interlayer interference — which vanishes at large twist angles — displays a chiral structure that reverses between the two layers and between the valence and conduction bands. Furthermore, the QPI patterns explicitly demonstrate the approximate translational symmetries and valley charge conservation in twisted bilayer graphene, validating the topological obstruction to constructing the Wannier orbitals of states at the Dirac cones. Using a continuum model of twisted bilayer graphene, we show that all characteristics of the observed QPI patterns can be explained by the form factor of eigenstates projected onto a single layer. Our results provide fundamental insights into the electronic spectrum and wave functions of twisted bilayer graphene, and establish QPI as an experimental probe for the form factor of back-scattering states.

Abstract

The overlap matrix of electronic energy eigenstates, sometimes referred to as the form factor, determines the quantum geometric tensor of electrons in solids. Here, we show that the variation in the overlap of two eigenstates with opposite momenta can be directly observed via quasiparticle interference (QPI) imaging. We study the QPI in twisted bilayer graphene using a real-space tight-binding model combined with the kernel polynomial method. The resulting QPI patterns, which are largely independent of whether the two graphene layers are commensurate or incommensurate, reveal all intralayer and interlayer interference processes. While the intralayer interference signals resemble those of monolayer graphene, the interlayer interference — which vanishes at large twist angles — displays a chiral structure that reverses between the two layers and between the valence and conduction bands. Furthermore, the QPI patterns explicitly demonstrate the approximate translational symmetries and valley charge conservation in twisted bilayer graphene, validating the topological obstruction to constructing the Wannier orbitals of states at the Dirac cones. Using a continuum model of twisted bilayer graphene, we show that all characteristics of the observed QPI patterns can be explained by the form factor of eigenstates projected onto a single layer. Our results provide fundamental insights into the electronic spectrum and wave functions of twisted bilayer graphene, and establish QPI as an experimental probe for the form factor of back-scattering states.

Abstract

The overlap matrix of electronic energy eigenstates, sometimes referred to as the form factor, determines the quantum geometric tensor of electrons in solids. Here, we show that the variation in the overlap of two eigenstates with opposite momenta can be directly observed via quasiparticle interference (QPI) imaging. We study the QPI in twisted bilayer graphene using a real-space tight-binding model combined with the kernel polynomial method. The resulting QPI patterns, which are largely independent of whether the two graphene layers are commensurate or incommensurate, reveal all intralayer and interlayer interference processes. While the intralayer interference signals resemble those of monolayer graphene, the interlayer interference — which vanishes at large twist angles — displays a chiral structure that reverses between the two layers and between the valence and conduction bands. Furthermore, the QPI patterns explicitly demonstrate the approximate translational symmetries and valley charge conservation in twisted bilayer graphene, validating the topological obstruction to constructing the Wannier orbitals of states at the Dirac cones. Using a continuum model of twisted bilayer graphene, we show that all characteristics of the observed QPI patterns can be explained by the form factor of eigenstates projected onto a single layer. Our results provide fundamental insights into the electronic spectrum and wave functions of twisted bilayer graphene, and establish QPI as an experimental probe for the form factor of back-scattering states.

Abstract

The overlap matrix of electronic energy eigenstates, sometimes referred to as the form factor, determines the quantum geometric tensor of electrons in solids. Here, we show that the variation in the overlap of two eigenstates with opposite momenta can be directly observed via quasiparticle interference (QPI) imaging. We study the QPI in twisted bilayer graphene using a real-space tight-binding model combined with the kernel polynomial method. The resulting QPI patterns, which are largely independent of whether the two graphene layers are commensurate or incommensurate, reveal all intralayer and interlayer interference processes. While the intralayer interference signals resemble those of monolayer graphene, the interlayer interference — which vanishes at large twist angles — displays a chiral structure that reverses between the two layers and between the valence and conduction bands. Furthermore, the QPI patterns explicitly demonstrate the approximate translational symmetries and valley charge conservation in twisted bilayer graphene, validating the topological obstruction to constructing the Wannier orbitals of states at the Dirac cones. Using a continuum model of twisted bilayer graphene, we show that all characteristics of the observed QPI patterns can be explained by the form factor of eigenstates projected onto a single layer. Our results provide fundamental insights into the electronic spectrum and wave functions of twisted bilayer graphene, and establish QPI as an experimental probe for the form factor of back-scattering states.