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For two-dimensional materials with periodic boundary conditions, we can solve the Bloch states and substitute them into the definition of Chern number, as shown in the picture: definition of Chern number

In the case of open boundary conditions, however, Bloch theory is no longer valid. Then how can we calculate the Chern number in this case?

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    $\begingroup$ For open boundary conditions, the Bloch theory still applies. As long as the system size goes to infinity, boundary condition should not matter anyway. $\endgroup$
    – Everett You
    Commented Apr 23, 2017 at 17:25
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    $\begingroup$ There is something call bulk-boundary correspondence. One of the defining property of topological insulators. $\endgroup$
    – L.K.
    Commented Apr 23, 2017 at 17:25
  • $\begingroup$ @EverettYou What if the system is finite? $\endgroup$
    – physics_lovers
    Commented Apr 23, 2017 at 17:26
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    $\begingroup$ @physics_lovers Finite system has no definition of Chern number. Because phases of matter (including topological phases) are only defined in the thermodynamic limit. $\endgroup$
    – Everett You
    Commented Apr 23, 2017 at 17:45
  • $\begingroup$ Just to add to what has already been said, the Chern number is a property of the bulk of a system which is why it is not defined for a finite system. $\endgroup$
    – honey.mustard
    Commented May 5, 2017 at 17:09

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This is the whole point of topological invariants; to determine the surface states(the states that their wave function is localised on the surface of the crystal). So instead of explicitly calculating these states for finite crystal one can use a trick.

This is called bulk-boundary correspondence, even if your crystal is finite you assume that it is infinite, and calculate the Chern number(for even dimension) or winding number (for odd dimension).

Then those number will give you an information of the surface states of your finite crystal.

This is bulk-boundary correspondence, the information of the finite crystal can be obtained from the infinite crystal.

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You can use transfer matrix approach for 1D case in case of finite chain and derive other invariants that may give same value for the edge state as given by the Chern number (see this in case of Hall effect Yasuhiro Hatsugai Phys. Rev. B 48, 11851). The Chern number are evaluated using the bulk i.e., system with the periodic boundary condition. The number of edge states calculated from the finite size is same as the Chern number calculated using the bulk. This established the bulk-boundary correspondence as clarified by @physshyp. However, the proof of this correspondence is non-trivial.

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