Index theoretic characterization of d-wave superconductors in the vortex state

TL;DR

This paper uses index theory to analyze the low energy spectrum of d-wave superconductors in vortex states, revealing differences between singly and doubly quantized vortices and their topological properties.

Contribution

It introduces an index theorem for the zero energy modes in vortex lattices and compares the spectral properties of different vortex quantizations in d-wave superconductors.

Findings

Zero energy modes are bounded by an index theorem.

Doubly degenerate Dirac-like spectrum near zero modes.

Topological quantum numbers differ between vortex types.

Abstract

We employ index theoretic methods to study analytically the low energy spectrum of a lattice d-wave superconductor in the vortex lattice state. This allows us to compare singly quantized $hc/2e$ and doubly quantized $hc/e$ vortices, the first of which must always be accompanied by $Z_2$ branch cuts. For an inversion symmetric vortex lattice and in the presence of particle-hole symmetry we prove an index theorem that imposes a lower bound on the number of zero energy modes. Generic cases are constructed in which this bound exceeds the number of zero modes of an equivalent lattice of doubly quantized vortices, despite the identical point group symmetries. The quasiparticle spectrum around the zero modes is doubly degenerate and exhibits a Dirac-like dispersion, with velocities that become universal functions of $\Delta_0/t$ in the limit of low magnetic field. For weak particle-hole symmetry breaking, the gapped state can be characterized by a topological quantum number, related to spin Hall conductivity, which generally differs in the cases of the $hc/2e$ and $hc/e$ vortex lattices.