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Defects which appear in heterostructure junctions involving topologicalinsulators are sources of gapless modes governing the low energy properties ofthe systems, as recently elucidated by Teo and Kane [Physical Review B82,115120 (2010)]. A standard approach for the calculation of topologicalinvariants associated with defects is to deal with the spatial inhomogeneityraised by defects within a semiclassical approximation. In this paper, wepropose a full quantum formulation for the topological invariantscharacterizing line defects in three-dimensional insulators with no symmetry byusing the Green's function method. On the basis of the full quantum treatment,we demonstrate the existence of a nontrivial topological invariant in thetopological insulator-ferromagnet tri-junction systems, for which asemiclassical approximation fails to describe the topological phase. Also, ourapproach enables us to study effects of electron-electron interactions andimpurity scattering on topological insulators with spatial inhomogeneity whichgives rise to the Axion electrodynamics responses.

Green's Function Method for Line Defects and Gapless Modes in Topological Insulators : Beyond Semiclassical Approach