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There is a frequently mentioned concept in the field of topological insulator called "bulk-edge correspondence" or "bulk-boundary correspondence", which basically gives the relationship between the total number of edge states and the topological properties of all the bulk bands below the gap. Can anyone tell me what is the physics behind this "correspondence"?

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I take your question to mean "topological insulators" in the larger sense of all symmetry classes and all dimensions (which includes the quantum Hall effect case).

In this case, referring only to the quantum Hall case, the edge quantity and the bulk quantity physically mean the same thing: electrical conductivity in direction perpendicular to applied electric field. Thus, the physics is that this conductivity is the same. For systems with an edge, one interprets the current to be running along the edge, whereas for systems with no edge (bulk), one interprets the current to be running within the sample. In real-world samples, which of course have an edge, there is a certain amount of both types of currents (the ratio depends on the boundary conditions and "working function"), and anyway the quantized conductivity of either mechanism indeed agree.

The way to make the connection between the two quantized numbers is via the fact that we are assuming that in the part of space where both systems exist, their Hamiltonians are the same, so that the edge Hamiltonian may be considered a "restriction" of the bulk Hamiltonian up to some (spatially bounded) arbitrary boundary conditions.

In other symmetry classes and dimensions, it is not always clear (to me) what the quantized quantity refers to, but at least for $\mathbb{Z}_2$ time-reversal-invariant two-dimensional topological insulators, it refers to a kind of "time-reversal polarization", and for chiral insulators for chiral polarization, but these interpretations are vague.

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