Effective dynamics of 2D Bloch electrons in external fields derived from symmetry
Symmetry is a powerful tool in studying the dynamics of Bloch electrons in crystalline solids. Here, using a tight binding description, a systematic method is developed to derive the symmetry labels, called irreducible representations (IRs), characterizing the Bloch eigenstates in a crystal. Starting from a tight-binding (TB) approach, the TB basis functions are decomposed into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves. Each of these two subproblems is independent of the details of a particular crystal structure and it is largely independent of the relevant aspects of the other subproblem, hence permitting for each subproblem an independent universal solution. This IR labeling scheme is applied on two-dimensional materials including MoS2 and graphene. The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector. For many crystal structures, it is possible to define multiple distinct coordinate systems such that for wave vectors at the border of the Brillouin zone the IRs characterizing the Bloch states depend on the coordinate system, i.e., these IRs are not uniquely determined by the symmetry of a crystal structure. The different coordinate systems are related by a coordinate shift that results in a rearrangement of the IRs characterizing the Bloch states. This rearrangement is illustrated with the three coordinate systems for MoS2. The freedom to choose different distinct coordinate systems can simplify the symmetry analysis of the Bloch states. Given the IRs of the Bloch states in one coordinate system, a rearrangement lemma yields immediately the IRs of the Bloch states in the other coordinate systems. The rearrangement of the IRs in different coordinate systems does not affect observable physics such as the effective Hamiltonians for the dynamics of Bloch states in external fields. Using symmetry analysis in conjunction with the theory of invariants, a generic multiband Hamiltonian for monolayer MoS2 is constructed, which incorporates the effect of spin-orbit coupling, strain and external electric and magnetic fields.