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We decipher the microscopic mechanism of the formation of tilt in the two-dimensional Dirac cone of 8P mmn borophene sheet. With the aid of ab initio calculations, we identify relevant low-energy degrees of freedom on the 8P mmn 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 8P mmn lattice (ridge sites). Local chemical bonds formed between the low-and high-energy sublattices provide the required off-diagonal coupling between the two sectors. Elimination of high-energy ridge sites gives rise to a remarkably large effective further neighbor hoppings on the coarse grained (honeycomb) lattice of inner sites that determine the location and tilt of the Dirac cone. This insight based on real space renormalization of the 8P mmn lattice enables us to design atomic scale substitutions that can lead to desired change in the tilt of the Dirac cone. We furthermore encode the process of renormalization into an effective tight-binding model on a parent honeycomb lattice that facilitates numerical modeling of various effects such as disorder/interactions/symmetry-breaking for tilted Dirac cone fermions of 8P mmn structure. The tilt parameters determine a spacetime metric, and therefore the ability to vary the tilt over distances much larger than the atomic separations opens up a paradigm for fabricating arbitrary solid-state spacetimes.

Two dimensional quantum materials possessing Dirac cones in their spectrum are fascinating due to their emergent low-energy Dirac fermions. In 8Pmmn borophene the Dirac cone is furthermore tilted, which is a proxy for spacetime geometry, since the future light-cone depends on the underlying metric. Therefore it is important to understand the microscopic origin of the tilt. Here, based on ab-initio calculations, we decipher the atomistic mechanism of the formation of tilt. First, nearest-neighbor hopping on a buckled honeycomb lattice forms an upright Dirac cone. Then, the difference in the renormalized anisotropic second-neighbor hopping, formed by virtual hoppings on one-dimensional chains of atoms, tilts the Dirac cone. We construct an accurate tight-binding model on honeycomb graph for analytical investigation, and we find that substitution of certain boron atoms by carbon provides a way to change the tilt of the cone.