A generalized framework for straintronics in 2D quantum materials using group theory

In the era of 2D and quasi-2D quantum materials one needs to model strain at the level of the Hamiltonian as opposed to a semi-classical approach. Corrections to the electronic Hamiltonian due to strain arise from two sources: deformations of the lattice and changes in the hoppings. Here, we provide a general theory that takes into account the symmetry of the lattice and that of the bonds, and allows us to write down the strain corrections from all sources in any 2D lattice in terms of the band structure parameters like the velocity and the inverse mass tensor. We then use Group theory to identify when strain can be described as a scalar- and/or a vector-potential. We discuss the nature of the potentials that arise from in- and out-of-plane hoppings, allowing us to model multi-layer systems. We also show that, in general, one encounters multiple vector-potentials in different sectors of the Hilbert space, but one can derive a simpler effective low energy strained Hamiltonian via Hilbert space projections. We discuss several toy models of 2D systems and present strained bilayer graphene as a practical system that incorporates all of the above features. We identify a strain dependent energy scale in bilayer graphene above which we would no longer need to account for multiple vector-potentials. The generality of this formulation allows for a wider range of materials to be investigated for quantum transport straintronics.