Scattering formula for the topological quantum number of a disordered multimode wire

The topological quantum number $Q$ of a superconducting or chiral insulating wire counts the number of stable bound states at the end points. We determine $Q$ from the matrix $r$ of reflection amplitudes from one of the ends, generalizing the known result in the absence of time-reversal and chiral symmetry to all five topologically nontrivial symmetry classes. The formula takes the form of the determinant, Pfaffian, or matrix signature of $r$, depending on whether $r$ is a real matrix, a real antisymmetric matrix, or a Hermitian matrix. We apply this formula to calculate the topological quantum number of $N$ coupled dimerized polymer chains, including the effects of disorder in the hopping constants. The scattering theory relates a topological phase transition to a conductance peak, of quantized height and with a universal (symmetry class independent) line shape. Two peaks which merge are annihilated in the superconducting symmetry classes, while they reinforce each other in the chiral symmetry classes.

Figure 1

Superconducting wire (S) connected to a normal-metal lead (N) which is closed at one end. A bound state at the Fermi level can form at the NS interface, characterized by a unit eigenvalue of the product $r_{\mathrm{N}}{r}_{\mathrm{NS}}$ of two matrices of reflection amplitudes (indicated schematically by arrows).

Figure 2

Conductance $G$ (top panels) and topological quantum number $Q$ (bottom panels) in the superconducting class D (left panels) and the chiral class BDI (right panels). The black and blue curves are calculated from the Hamiltonian (4.2), for a single disorder realization in a wire with $N=5$ modes. The red dotted curve shows the universal line shape (4.7) of an isolated conductance peak. Energies ${\Delta }_0$ and $U_0$ are measured in units of $\hslash v_F/\delta L$ for $\delta L=L/10$.

Figure 3

Conductance (black dotted line, left axis) and topological quantum number (blue solid line, right axis) of $N=3$ coupled polymer chains (each containing $N_L=300$ sites). These curves are calculated from the reflection and transmission matrices, obtained from the Hamiltonian (5.1) for $t_0=1$, $\alpha =1$, and $t_{\mathrm{inter}}=0.1$, for a single realization of the random $\delta u_n$’s (having a Gaussian distribution with zero average and standard deviation $0.2$). The red dotted curve shows the universal line shape (4.7) of an isolated conductance peak.