Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators

We study the bulk-edge correspondence in topological insulators by taking Fu-Kane spin pumping model as an example. We show that the Kane-Mele invariant in this model is Z2 invariant modulo the spectral flow of a single-parameter family of 1+1-dimensional Dirac operators with a global boundary condition induced by the Kramers degeneracy of the system. This spectral flow is defined as an integer which counts the difference between the number of eigenvalues of the Dirac operator family that flow from negative to non-negative and the number of eigenvalues that flow from non-negative to negative. Since the bulk states of the insulator are completely gapped and the ground state is assumed being no more degenerate except the Kramers, they do not contribute to the spectral flow and only edge states contribute to. The parity of the number of the Kramers pairs of gapless edge states is exactly the same as that of the spectral flow. This reveals the origin of the edge-bulk correspondence, i.e., why the edge states can be used to characterize the topological insulators. Furthermore, the spectral flow is related to the reduced eta-invariant and thus counts both the discrete ground state degeneracy and the continuous gapless excitations, which distinguishes the topological insulator from the conventional band insulator even if the edge states open a gap due to a strong interaction between edge modes. We emphasize that these results are also valid even for a weak disordered and/or weak interacting system. The higher spectral flow to categorize the higher-dimensional topological insulators are expected.Comment: 9 page, accepted for publication in Nucl Phys

Higher Education on Buildings: Case Study in the North Dakota Region

Because of the growing demand for local skilled professionals to improve the health, energy efficiency, and sustainability of residential and commercial buildings in North Dakota, this case study reports the current situation of higher education relating to buildings in the state’s vicinity, including Minnesota, Montana, North Dakota, and South Dakota. In this region, 116 programs relating to buildings were found in 41 postsecondary institutions, and both their majors and courses were then studied with frequency lists. The frequency information was analyzed over nine sets of curriculum areas at both graduate and undergraduate levels for the four states. After the current state of buildings in North Dakota was investigated, strategies were then proposed to rectify current issues regarding higher education on buildings, including but not limited to forming a comprehensive and interdisciplinary program on buildings (e.g., architectural engineering), providing more graduate programs, developing more courses in areas that lack adequate coursework, and increasing student enrollment. These strategies will greatly promote the health, energy efficiency, and sustainability for new and existing buildings in the four-state region of Minnesota, Montana, North Dakota, and South Dakota