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Real magnetic and lattice deformation gauge fields have been investigated in honeycomb lattice of graphene. The coexistence of these two gauges will induce a gap difference between two valley points ( K  and  K ′) of system. This gap difference allows us to study the possible topological valley Hall current and valley polarization in the graphene sheet. In the absence of magnetic field, the strain alone could not generate a valley polarization when the Fermi energy coincides exactly with the Dirac points. Since in this case there is not any imbalance between the population of the valley points. In other words each of these gauges alone could not induce any topological valley-polarized current in the system at zero Fermi energy. Meanwhile at non-zero Fermi energies population imbalance can be generated as a result of the external strain even at zero magnetic field. In the context of Berry curvature within the linear response regime the valley polarization (both magnetic free polarization, Π 0 , and field dependent response function,  χ    α  ) in different values of gauge fields of lattice deformation has been obtained.

The role of the strain induced population imbalance in Valley polarization of graphene: Berry curvature perspective