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In graphene in the presence of strain the elasticity theory metric naturally appears. However, this is not the one experienced by fermionic quasiparticles. Fermions propagate in curved space, whose metric is defined by expansion of the effective Hamiltonian near the topologically protected Fermi point. We calculate the corresponding emergent gravitational fields. Next, we extend our consideration to the case, when the dislocations are present. We consider the situation, when the deformation is described by elasticity theory and calculate both torsion and emergent magnetic field carried by the dislocation. The dislocation carries singular torsion in addition to the quantized flux of emergent magnetic field. Both may be observed in the scattering of quasiparticles on the dislocation. Emergent magnetic field flux manifests itself in the Aharonov - Bohm effect while the torsion singularity results in Stodolsky effect. Besides, we extend our consideration to the tight-binding model with cubic symmetry that may be relevant for the description of a certain class of 3D Weyl semimetals. We take into account elastic deformations of the semimetal through the modification of hopping parameters. This modification results in the appearance of emergent gauge field and the coordinate dependent anisotropic Fermi velocity. The latter may also be interpreted as emergent gravitational field.

(Following “Emergent Horava gravity in graphene” by G.E. Volovik, M.A. Zubkov, arXiv:1305.4665 [cond-mat.mes-hall], 10.1016/j.aop.2013.11.003, Annals Phys. 340 (2014) 352-368, “Emergent geometry experienced by fermions in graphene in the presence of dislocations” by G.E. Volovik, M.A. Zubkov. arXiv:1412.2683 [cond-mat.mes-hall], 10.1016/j.aop.2015.03.005 Annals Phys. 356 (2015) 255-268, “Emergent gravity in the cubic tight-binding model of Weyl semimetal in the presence of elastic deformations” by Alberto Cortijo, M.A. Zubkov, arXiv:1508.04462 [cond-mat.mes-hall] 10.1016/j.aop.2016.01.006. Annals Phys. 366 (2016) 45-56.)