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The way to understand these topological invariants from the bulk is through the entanglement structure of the system. Suppose you have an infinitely long spin chain in the Haldane phase. Now imagine partitioning it in two, so you have a left and a right half. These two halves will be entangled with one another, and by Schmidt decomposition the total system ...


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Within projected augmented wave (PAW) method it is incorrect to use the smooth pseudo wave function directly to obtain integrals. The same applies for ultra-soft pseudo potentials, but I will proceed discussing about PAW. The results can indeed be wrong. For example, for coin-metals, the d-band smooth wave functions $\tilde{\Psi}$ can have norms of 0.2-0.4, ...



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