Local quantities and topological invariants of hermitian matrices
[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This thesis contains developments of two related topics, both born out of a desire to understand the electronic eigenstates of crystal lattices with the fewest assumptions possible. The rst topic is the free-electron network model [1, 2], also known in some other sources as the free-electron theory of conjugated molecules , although as stated, here the model will be applied to lattices rather than molecules. The second, and larger, portion concerns the theory of the Berry phases, and the closely related Berry connections and curvatures, which we will refer to as the local quantities, in the sense that they are functions of the Bloch momentum k over the reciprocal space of a periodic system such as a crystal. The integrals of these local quantities over the rst Brillouin zone of a crystal lattice yields integer values called the Chern numbers, which are a type of topological invariant. Topological invariants determine the properties of topological insulators. An example is the Chern insulator phase [4, 5], which has protected electronic edge states circulating around the perimeter of a two-dimensional crystal. In particular, to calculate the local quantities we will use the characteristic adjoint matrix BH( ) of a matrix H, which has the property that any of its columns is an eigenvector of H for the appropriate eigenvalue [subscript] .
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