Optically Controlled Orbitronics on a Triangular Lattice

The propagation of electrons in an orbital multiplet dispersing on a lattice can support anomalous transport phenomena deriving from an orbitally induced Berry curvature. In striking contrast to the related situation in graphene, we find that anomalous transport for an $L=1$ multiplet on the primitive 2D triangular lattice is activated by easily implemented on site and optically tunable potentials. We demonstrate this for dynamics in a Bloch band where point degeneracies carrying opposite winding numbers are generically offset in energy, allowing both an anomalous charge Hall conductance with the sign selected by off-resonance coupling to circularly polarized light and a related anomalous orbital Hall conductance activated by layer buckling.

Figure 1

(a) Model for the propagation of an $L=1$ orbital multiplet on a triangular lattice. Here, $\widehat{T}$ is the hopping operator, ${\mathbf{E}}_c$ denotes a circularly polarized optical field, and $\mathbf{J}$ and ${\mathbf{J}}_o$ are the charge and orbital responses to an applied transverse $\mathbf{E}$ field. (b) Energy surfaces for bands that are even (odd) under $z$ reflection shown in yellow (green). Surfaces with opposite mirror eigenvalues intersect on line nodes (projected red and purple lines). Twofold degenerate point nodes are shown as black and blue points with a quadratic contact at the zone center (black) and linear band contacts the zone corners (blue). (c) The dispersion of the composite bands along symmetry directions in a model with $\mathcal{T}$ symmetry (dashed) and with a $\mathcal{T}$-breaking potential (solid). Blue points denote the intersection of the nodal lines with the plane of the figure, and red points are twofold degeneracies pinned to high-symmetry points. Simulation parameters are given in [20]. All energies are scaled relative to $W$, the bandwidth at $\mathrm{\Gamma }$.

Figure 2

Density plots of $\arg [d(\mathbf{k})]$ for tight-binding Hamiltonians of a scalar field on the honeycomb lattice (a) and for the mirror-even states on the $L=1$ triangular lattice (b). In (a), a branch cut connects point singularities at time-reversed points $K$ and $K^{\prime }$. In (b), time-reversed zone-corner singularities carry the same winding number, and connect to a compensating second-order node at $\mathrm{\Gamma }$.

Figure 3

Density plots of the occupation-weighted Berry curvature defined by the summand in Eq. (4) (a) and the occupation-weighted orbital Berry curvature defined by the summand in Eq. (8) (b) when band degeneracies are lifted by uniform local $\mathcal{T}$-breaking fields. In (a), $\mathcal{T}$ symmetry is broken by coherent coupling to an optical field. In (b), $z$-mirror even and odd sectors hybridize, replacing nodal lines by a quartet of linear point degeneracies. Simulation parameters are given in [20]. The color scales are given in units of $a^2$, where $a$ is the lattice constant.

Figure 4

(a) Band dispersion for a ribbon of the triangular lattice showing the projections of the bulk bands (gray) and confined edge modes inside the bulk gaps (red and green). (Inset) The $K$ point gaps host counterpropagating states on opposite edges of the ribbon. (b) The anomalous Hall conductance in the presence of a $\mathcal{T}$-breaking potential evolves from a particlelike to a holelike response as a function of band filling $\mu$. Plateaus occur when the chemical potential is tuned within the induced gaps at the $K$ and $\mathrm{\Gamma }$ points. (c) Breaking $z$-reflection symmetry activates an orbital Hall conductance describing a transverse angular-momentum current driven by an in-plane electric field. The orbital response is strong in two sharp spectral features where the orbital current is correlated/anticorrelated with the charge current. Simulation parameters are given in [20].