Scattering theory of topological insulators and superconductors

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonian-based approaches (surface versus bulk), it is numerically more efficient. In particular, is better suited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.

Figure 1

Sketch of the tight binding model used to attach leads in order to open-up the Hamiltonian $H\left(\mathit{k}\right)$ of (2.1). In 2D we introduce four leads shown as circles labeled by 1, $\overline{1}$, 2, and $\overline{2}$. The on-site terms (boxes) are connected by hoppings (lines). The additional trivial hoppings 1 and $-1$ are introduced such that the lead properties drop out when twisted periodic boundary conditions are applied. For the Mahaux-Weidenmüller formula (2.4), the three nodes form the on-site Hamiltonian $\tilde{H}$ which is then connected via the trivial hoppings to ideal leads.

Figure 2

Symmetry properties of $r\left(\mathit{k}\right)$ and $H\left(\mathit{k}\right)$ in the ten symmetry classes. Time-reversal symmetry is denoted by $\mathcal{T}$, particle-hole symmetry by $\mathcal{P}$. The signs at the top and left of the table denote either the absence ($\times$) of a corresponding symmetry, or the value of the squared symmetry operator. The entries of the table with a gray background have an additional chiral symmetry $\mathcal{C}$, which always has the form shown in the AIII entry of the table. In particular, we always chose a basis such that $r\left(\mathit{k}\right)=r^{†}\left(\mathit{k}\right)$ in the chiral symmetry classes. The way symmetry classes transform under our definition of $H_{d-1}$ [cf. (3.4)] is denoted by the arrows; the double arrow implies a doubling of degrees of freedom as in Eq. (3.4b). Going along an arrow, the symmetry of the reflection block $r\left(\mathbf{k}\right)$ (marked by a solid box) transforms into the symmetry of the reduced Hamiltonian (marked by a dashed box). In the chiral classes there is an additional symmetry (not marked by a box) which can be obtained from the other by combining it with the chiral symmetry, $H\left(\mathit{k}\right)=-{\tau }_{z}H\left(\mathit{k}\right){\tau }_z$ and $r\left(\mathit{k}\right)=r^{†}\left(\mathit{k}\right)$, respectively.

Figure 3

A system in $d$ dimensions consisting out of two parts with different Hamiltonians $H$ and $H^{\prime }$. Reflection blocks of the scattering matrix $r$ and $r^{\prime }$ are used to define the lower dimensional Hamiltonians $H_{d-1}$ and $H_{d-1}^{\prime }$. We prove the correspondence between topological invariants in $d$ and $d-1$ dimensions using the relation between the surface state at the interface between $H$ and $H^{\prime }$ and the edge state at the interface between $H_{d-1}$ and $H_{d-1}^{\prime }$.

Figure 4

The value of the chemical potential ${\mu }_c$ where the ensemble averaged topological invariant equals to 0.5, as a function of system size $L$. Dashes, solid line: topological invariant defined in terms of the scattering matrix, from Eq. (4.12). Circles, dotted line: topological invariant obtained from the Hamiltonian expression of Ref. 4. Lines represent fits as described in the text.

Figure 5

Top panel: Evolution of the poles (green dots) and the zeros (red dots) of $\det r\left(z\right)$ as a function of a parameter $\alpha$ which tunes through the topological phase transition in classes A, AII, and DIII in 2D. Shown is the complex plane with the unit circle $\left|z\right|=1$ indicated in blue. Time-reversal symmetry in AII and DIII implies that for every zero/pole at $z_0$ there is additionally one at $1/z_0$. In DIII there is additional particle-hole symmetry which additionally dictates zeros/poles at $z_0^{*}$ and $1/z_0^{*}$. The phase transition happens when at least one of the zeros crosses the unit circle. This event coincides with a change of the topological invariant $\mathcal{Q}$ (green) defined by Eqs. (4.12, 4.13, 4.14), as shown in the bottom panels.

Figure 6

Average topological invariant $\mathcal{Q}$ (4.12) and longitudinal conductance $G$ of a disordered quantum Hall sample for different aspect ratios as a function of the mixing angle $\alpha$.

Figure 7

Conductance and topological invariant (4.16) for a disordered 3D topological insulator in class AII.