Tunning the tilt of a Dirac cone by atomic manipulations: application to 8Pmmn borophene

We decipher the microscopic mechanism of the formation of tilt in the two-dimensional Dirac cone of $8Pmmn$ borphene. In our ab-initio calculations, we identify relevant low-energy degrees of freedom on the $8Pmmn$ lattice and find that these atomic orbitals reside on an effective honeycomb lattice (inner sites), while the high-energy degrees of freedom reside on the rest of the $8Pmmn$ lattice (ridge sites). Integrating out the high-energy atomic orbitals, gives rise to remarkably large effective further neighbor hoppings on the coarse grained "honeycomb graph" of inner sites that determine the location and tilt of the Dirac cone. Molecular orbitals -- that can be modified by atomic manipulations -- are responsible for the creation of further neighbor edges on the honeycomb graph that controls the tilt of the resulting Dirac cone. This leads to an effective tight-binding model on a parent honeycomb graph that facilitates numerical modeling of various effects such as disorder/interactions/symmetry-breaking for tilted Dirac cone fermions. Since the tilt is a proxy to spacetime metric, our result offers a robust perspective on the fabrication of desired emergent spacetime structure and synthesis of geometric forces at ambient conditions that are likely to enhance our control on the movement of electrons in electric/optical devices.