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Dirac Cone Metric and the Origin of the Spin Connections in Monolayer Graphene

We show that the modulation of the hopping amplitudes in the honeycomb lattice of the monolayer graphene uniquely defines a metric which corresponds to the shape of the Dirac cone. The spin connection of this effective metric field can be obtained from the microscopic tight-binding Hamiltonian exactly, completing the analogy between the sublattice pseudospin travelling in the monolayer graphene with ripples and strain fields, and the real spin $1/2$ fermion travelling in a curved space. The effective metric as seen by the sublattice pseudospin is different from the real space metric as defined by the two-dimensional manifold of the graphene monolayer. All relevant terms of the effective gauge field from the microscopic model is calculated exactly for a unimodular effective metric.