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

Coupled Simulation Of Hydraulic Fracturing, Production, And Refracturing For Unconventional Reservoirs

Horizontal well drilling and multi-stage hydraulic fracturing are two key techniques for the development of unconventional reservoir. However, the production from tight formation is associate with fast depletion of reservoir. When oil price is low, drilling new horizontal wells is not profitable. Creating secondary fractures from existing hydraulic fractured wells, i.e., refracture is an alternative method to increase stimulated reservoir volume (SRV) and gain additional production from existing hydraulic fractured wells. To optimize refracturing well selection and operation, it’s of economic importance to acquire knowledge from initial hydraulic fracturing operation, production history, and refracturing design perspectives. This initiated the idea of this research to develop an integrated hydraulic fracturing, production, and refracturing model. This research work mainly comprises of three sections. In the first section, hydraulic fracturing models were built using XSite software, a lattice-based simulator, to analyze the effect of changing rock properties and in-situ stresses on fracture propagation in a layered reservoir. The challenge was to quantify degree of fracture containment using the hydraulic fracturing simulator. To overcome this fracture aperture contours were obtained to quantify fracture containment with two proposed penetration parameters. The modeling results suggest that brittle rocks favor vertical migration of hydraulic fracture, while increasing minimum horizontal stress tends to inhibit vertical growth of hydraulic fracture and lead to containment at layer interface. In the Second part of this study, an innovative integrated multi-stage hydraulic fracturing and production model was built for a shale gas reservoir. The challenge was to utilize distributed fracture data presented from the lattice-based hydraulic fracturing simulator for history matching in the reservoir simulator. To identify fracture geometry, a moving tip clustering and linear regression clustering algorithms were developed to discretize distributed fracture data points using multiple crack segments. The former algorithm is prone to capture fracture with microcracks that contribute to SRV, thus contributing to higher simulated production. The latter algorithm mainly captures the major fracture path without consideration of microcracks. The modeling results also suggest that gas slippage, matrix shrinkage, and fracture closure play important roles in shale gas production. In the third section, an innovative hydraulic fracturing, production, refracturing, and post-refracturing production model was developed. The challenge in this part was to simulate refracture propagation based on existing fracture geometry and pore pressure distribution with higher accuracy and efficiency. A model was built by simulating the fracture and refracture propagation in XSite and modeling reservoir depletion and post refracturing reservoir depletion in the continuum mechanism based simulator. The results suggest the propagation of refractures is driven by proppant and depletion induced stress shadow and contributes to larger SRV and higher hydrocarbon production. The proposed algorithms and integrated models can potentially be applied in the field for better refracturing design to enhance ultimate recovery of oil and gas

The Sneaker Value: A Disquisition Into Sneaker Resale Market

The secondary market for sneakers has proliferated in recent years. In the secondary market of sneakers, it is typical for the resale price of sneakers to be higher than the retail price. Using the theory of planned behavior as a theoretical framework, this study will examine how the economic, hedonic, symbolic, and social values of sneakers influence consumer attitudes and subjective norms. In addition, this study will examine the relationship between social media hype and perceived behavioral control. Finally, this paper examines how TPB components (subjective norms, perceived behavioral control, and attitudes) influence consumers\u27 purchase intentions. To achieve this research objective, 510 valid questionnaires were collected using a questionnaire method to explore the correlation between these variables. The results of the hierarchical linear regression showed that economic, hedonic, and social values positively correlate with consumers\u27 attitudes, and hedonic and symbolic values positively correlate with consumers\u27 subjective norms. The results also show that perceived behavioral control and subjective normative attitudes positively correlate with consumers\u27 purchase intentions

Diagrammatic Approach to Four-Boson Systems in Effective Field Theory

The diagrammatic approach is used to study four-boson systems using two- and three-boson contact interactions from a leading-order effective field theory (EFT). In order to determine at what order a four-boson force is needed in an EFT, an renormalization group (RG) analysis, e.g., a study of the cutoff dependence, must be performed for four-boson observables. While existing four-boson calculations are limited below a threshold cutoff, $\Lambda_{t}$, above which deeply bound trimers appear, the method used in this calculation works at higher cutoffs by systematically incorporating the trimer poles in the four-boson calculation. Tetramer binding energies for cold $^4$He atoms are computed numerically and it is necessary to go above $\Lambda_{t}$ in order to obtain convergence. No four-boson force is needed at LO, as argued by previous studies, but more rigorously demonstrated here by the convergence at much higher cutoffs. Tetramer binding energies for cold $^4$He atoms agree with existing calculations. Tetramer binding energies and decay widths are also computed near the unitary limit. These results in the unitary limit also agree with existing calculations. The method used in this paper may be extended to study four-nucleon systems and the cutoff dependence of four-nucleon observables at high cutoffs.Comment: 42 pages, 14 page

Violence in South African adolescents:a road to recovery

South African adolescents are at risk for exposure to violence victimisation and a significant proportion of them engage in violence perpetration. The experiences of victimisation and perpetration can take a significant toll on adolescents’ psycho-social development and lead to long-term impact on health and functioning in adulthood. Given the pressing situation of violence in South Africa and the importance to promote positive functioning in adolescents so they grow up to be responsible individuals and drive the country’s socio-economic development, a needs assessment was conducted. Specifically, quantitative methods were used to examine the risk and protective factors associated with adolescents’ experiences of victimisation and perpetration, as well as the negative impact of victimisation on adolescents’ psychological functioning and health risk behaviours. Moreover, qualitative methods were used to explore the protective factors that enhance resilience in adolescents living in high-violence communities. The results may inform the development of interventions to prevent the experiences of victimisation and perpetration in South African adolescents, and help them thrive and attain positive developmental outcomes