We construct an effective low-energy Hamiltonian, which describes fermions dwelling on a deformed honeycomb lattice with dislocations and disclinations, and with arbitrary hopping parameters of the corresponding tight binding model. Despite the presence of dislocations and disclinations, the tight binding Hamiltonian preserves the connectivity number 3 at each lattice site. This construction is related to fermions with a two-dimensional gravity. The effective theory has a local SU(2) gauge invariance of the group of rotations. We reformulate the model by fermions interacting with the deformation as a fermion lattice model with a phonon field and calculate the response of the fermion currents to the external deformation or phonon field. This indicates a $Z_2$ anomaly. This can be detected experimentally.

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• Elastic deformation
• Electronic structure
• Phonons
• Graphene
• Gauge symmetries