Impurity-bound states and Green's function zeros as local signatures of topology

Abstract

We show that the local in-gap Green's function of a band insulator G 0 ( ,k ,r ⊥ = 0), with r ⊥ the position perpendicular to a codimension-1 or codimension-2 impurity, reveals the topological nature of the phase. For a topological insulator, the eigenvalues of this Green's function attain zeros in the gap, whereas for a trivial insulator the eigenvalues remain nonzero. This topological classification is related to the existence of in-gap bound states along codimension-1 and codimension-2 impurities. Whereas codimension-1 impurities can be viewed as soft edges, the result for codimension-2 impurities is nontrivial and allows for a direct experimental measurement of the topological nature of two-dimensional insulators. Introduction. The topological characterization of condensed states of matter has emerged as a prominent research interest over the last few decades. The flourishing of the quantum Hall effect (QHE) [1] in particular elucidated many connections between physical signatures and topological invariants [2], which supplement the order parameters of the usual symmetry-breaking Landau-Ginzburg paradigm. More recently, however, it became apparent that topological order can also arise by virtue of symmetry, and in particular the very common and robust time-reversal (TR) symmetry is sufficient to establish the existence and stability of topological insulators [3][4][5]. This is quantified via a Z 2 invariant, and results in gapless helical edge states or chiral Dirac fermions localized at the perimeter of the sample in two and three dimensions, respectively. The topological insulator has proven extremely rich both experimentally and theoretically [6,7]. The concept has been generalized to a periodic table describing various discrete symmetries and dimensions [8][9][10]. Lattice symmetries can similarly lead to further topological distinctions, resulting in crystalline topological insulators [11], for which a general classification has been provided [12].One may ask whether the topology of band insulators has some local signature. In fact, in this paper, we will show that even the fully local in-gap Green's function contains information about the band topology, which is then directly accessible by experiments. The natural way this insight arises is through the study of impurities [14][15][16], similar to how the space group classification can be probed using lattice defects [17][18][19][20][21][22][23][24]. Consider a codimension-1 impurity line or surface in an insulator. In the limit where the impurity strength diverges, V → ∞, such an impurity acts like a real edge, which, following the bulk-boundary correspondence, should host zero gap metallic bound states in the topological phase. For finite V the codimension-1 impurity surface can thus be viewed as a soft edge. The codimension-2 impurity lines or points do not host gapless states in the strong V limit, so a priori there is no reason to expect they probe topology. However, we will see that they in fact inherit the topological structure of the soft edges....

Ab initio investigation of impurity-induced in-gap states in Bi2Te3 and Bi2Se3

“…Indeed, based on scanning tunneling spectroscopy (STS) measurements and a phenomenological model for the topological surface state, Sessi et al [31] showed that in-gap states lead to a local filling of the band gap in the presence of ferromagnetic order. These in-gap states consist of sharp resonances in the density of states lying within the bulk band gap [32,33] and had been interpreted as local signatures of topology [34]. Interestingly, similar resonances were also observed from experiment and first-principles calculations for 3d impurities deposited on a Cu(111) surface [35,36].…”

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

“…Finally, a lattice of magnetic atoms placed on top of a superconductor may be a route towards the realization of a rich variety of topological phases [16]. In principle, there is also a relation to the studies about so-called topological Anderson insulators [17,18], disorderdriven topological superconductivity [19,20], and impurity bound states as a signature of a topologically nontrivial phase [21,22].…”

Topological invariants from zero modes in systems of topological insulators (TI) and superconductors (TS)

“…Meanwhile, the topological defects such as magnetic vortex lines and dislocations in crystals can provide a π flux to certain wave-number electronic states, and host the zero modes. The interplay of topological defects and gapless modes was therefore investigated, which led to the emerging of the periodic table for defect classifications [10][11][12].…”

“…There are other ways to link the topological number to zero modes, such as zero eigenvalues of electronic local in-gap Green's functions in the presence of impurities [10], and Majorana zero modes hosted by a vortex line in a topological superconductor [13]. In addition, the topological line defects like dislocations in 3D TI can serve as the probes of weak TI states [14,15].…”