Boundary Green functions of topological insulators and superconductors

Topological insulators and superconductors are characterized by their gapless boundary modes. In this paper, we develop a recursive approach to the boundary Green function, which encodes this nontrivial boundary physics. Our approach describes the various topologically trivial and nontrivial phases as fixed points of a recursion and provides direct access to the phase diagram, the localization properties of the edge modes, as well as topological indices. We illustrate our approach in the context of various familiar models such as the Su-Schrieffer-Heeger model, the Kitaev chain, and a model for a Chern insulator. We also show that the method provides an intuitive approach to understand recently introduced topological phases which exhibit gapless corner states.

Figure 1

Schematic representation of the quasi-one-dimensional Hamiltonian with $N$ unit cells.

Figure 2

Tight-binding model for a second-order topological insulator on a square lattice. Solid and dashed lines in the $x$ and $y$ directions denote alternating nearest-neighbor hoppings $t_{x,1},t_{x,2}$ and $t_{y,1},t_{y,2}$. The hopping amplitudes for the bonds colored in brown include a relative minus sign. The effective Hamiltonian of the highlighted boundary parallel to the $x$ direction is topologically equivalent to an SSH model.

Figure 3

Wave function (modulus squared) of the single-particle excitation closest to zero energy for the second-order topological insulator in Eq. (92), evaluated on a finite-size lattice. Darker color indicates larger magnitude of the wave functions. The parameters are chosen as (a) $t_{x,1}=0.7,t_{y,1}=0.6$; (b) $t_{x,1}=1.2,t_{y,1}=1.5$; (c) $t_{x,1}=1.2,t_{y,1}=0.7$; and (d) $t_{x,1}=0.7,t_{y,1}=1.5$. We choose $t_{x,2}=t_{y,2}=1$ in all cases.