Scattering formula for the topological quantum number of a disordered multi-mode wire

TL;DR

This paper derives a universal scattering formula for the topological quantum number in disordered multi-mode wires, applicable across all symmetry classes, and demonstrates its use in analyzing phase transitions and conductance peaks.

Contribution

It generalizes the scattering formula for topological invariants to all symmetry classes, including disorder effects, and links topological phase transitions to measurable conductance features.

Findings

The formula uses determinant, Pfaffian, or signature of the reflection matrix depending on symmetry class.

Applied to disordered polymer chains to compute topological invariants.

Revealed universal conductance peak line shapes associated with topological phase transitions.

Abstract

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.