Relationship between embedding-potential eigenvalues and topological invariants of time-reversal invariant band insulators

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The embedding potential defined on the boundary surface of a semi-infinite crystal relates the value and normal derivative of generalized Bloch states propagating or decaying toward the interior of the crystal. It becomes Hermitian when the electron energy $\ensuremath{\epsilon}$ is located in a projected bulk band gap at a given wave vector $\mathbf{k}$ in the surface Brillouin zone (SBZ). If one plots the real eigenvalues of the embedding potential for a time-reversal invariant insulator in the projected bulk band gap along a path $\ensuremath{\epsilon}={\ensuremath{\epsilon}}_{0}(\mathbf{k})$ passing between two time-reversal invariant momentum (TRIM) points in the SBZ, then, they form Kramers doublets at both end points. We will demonstrate that the ${Z}_{2}$ topological invariant, $\ensuremath{\nu}$, which is either 0 or 1, depending on the product of time-reversal polarizations at the two TRIM points, can be determined from the two different ways these eigenvalues are connected between the two TRIM points. Furthermore, we will reveal a relation, $\ensuremath{\nu}=P$ mod 2, where $P$ denotes the number of poles that the embedding potential exhibits along the path. We also discuss why gapless surface states crossing the bulk band gap inevitably occur on the surface of topological band insulators from the view point of the embedding theory.