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Register a new userHow to tune the tilt of a Dirac cone by atomic manipulations?
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We decipher the microscopic mechanism of the formation of tilt in the two-dimensional Dirac cone of $8Pmmn$ borophene sheet. With the aid of $ab~ initio$ 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). 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 $8Pmmn$ 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 $8Pmmn$ 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.
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