PDF Publication Title:
Text from PDF Page: 052
R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann On the level of effective Hamiltonian, the 1d edge Hamiltonian is obtained from the 2d bulk Hamiltonian by a partial Fourier transform along the direction with translational invariance in Sec. 2.4. Such correspondence realized by the partial Fourier transform holds for any dimension, so there exists a one-to-one correspon- dence between the bulk and edge Hamiltonians. For 3d topological insulators, the Z2 invariant is the mod 2 WZW term, which is derived from the Chern–Simons theory in Sec. 4.2.1. In other words, the bulk-edge correspondence on the level of action functional is given by the duality between the bulk Chern–Simons action and the boundary topological WZW term. If we consider quantum field theories, then the bulk is described by a topological quantum field theory (TQFT), and the boundary is characterized by a rational conformal field theory (RCFT). So the bulk-edge correspondence on the level of field theory is given by CS3/WZW2 as a correspondence between 3d TQFT and 2d RCFT. On the other hand, based on the gauge group G such as SU(2), there exists a one-to-one correspondence between the WZW models and Chern–Simons theories, both characterized by their level k ∈ Z. So on the level of group cohomology, the bulk-edge correspondence is given by the Dijkgraaf–Witten theory [27]. If the Dijkgraaf–Witten transgression map is τ : H4(BG,Z) → H3(G,Z), then τ is the correspondence between the Chern–Simons theory and WZW model associated with G, for the geometric construction on bundle gerbes, see [22]. The generalized index theorem in Sec. 2.4 can be viewed as a bulk-edge corre- spondence on the level of index theory. With the presence of a domain wall, two quantum Hall states characterized by distinct first Chern numbers are separated by a boundary created by the domain wall. The variation of the first Chern numbers across the domain wall is identified with the spectral flow of the edge Hamiltonian along the boundary. In this model, the index map of the extended Hamiltonian plays the role of a correspondence connecting the topological invariants from the bulk and the analytic behavior of the edge states. For 3d Z2 topological insulators, we have a similar picture as that in the above quantum Hall model. With the presence of the time reversal symmetry, two 3d bulk systems characterized by distinct Chern–Simons theories are separated by a boundary consisting of the fixed points of the time reversal symmetry. If the time reversal symmetry is encoded into a specific gauge transformation, then the variation of the bulk Chern–Simons actions is exactly the topological WZW term of this specific gauge transformation. By the global SU(2)-anomaly, the WZW term is indeed Z2-valued. On the other side, the Pfaffian formalism of the Kane–Mele invariant counts the parity of the spectral flow of the chiral edge state along the boundary, i.e. the fixed points. Now the correspondence, i.e. the index map, is given by the Witten index Tr(−1)F since it counts the parity of Majorana zero modes such as Dirac cones. Hence the story of the topological Z2 invariant can be interpreted as a bulk-edge correspondence realized by the Witten index. If we model the bulk and boundary spaces by C∗-algebras, then the bulk-edge correspondence can be realized by Kasparov module in bivariant K-theory. So on 1630003-52 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
PDF Search Title:
Notes on topological insulatorsOriginal File Name Searched:
RMP.pdfDIY PDF Search: Google It | Yahoo | Bing
Sulfur Deposition on Carbon Nanofibers using Supercritical CO2 Sulfur Deposition on Carbon Nanofibers using Supercritical CO2. Gamma sulfur also known as mother of pearl sulfur and nacreous sulfur... More Info
CO2 Organic Rankine Cycle Experimenter Platform The supercritical CO2 phase change system is both a heat pump and organic rankine cycle which can be used for those purposes and as a supercritical extractor for advanced subcritical and supercritical extraction technology. Uses include producing nanoparticles, precious metal CO2 extraction, lithium battery recycling, and other applications... More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)