Topological invariants of time reversal invariant superconductors

TL;DR

The paper addresses the topological classification of gapped time-reversal invariant lattice superconductors by mapping the mean-field Bogoliubov–de Gennes Hamiltonian to a Bloch form. It extends the band-insulator $Z_2$ framework to superconductors, showing that a $Z_2$ invariant in $2$D and four such invariants in $3$D classify TR-invariant gapped states, and that these invariants persist as long as the bulk gap remains open. A concrete $2$D tight-binding model with $h_{etaeta}(k)$ and $ abla_{eta}(k)$ demonstrates a nontrivial $Z_2$ index for a $oxed{p_x+ip_y}$–type state, in contrast to trivial $s$-wave or $d_{x^2-y^2}+id_{xy}$ states, and indicates associated edge modes and Majorana physics. In $3$D, the framework predicts surface-state phenomena analogous to TR-invariant topological insulators, including momentum-space monopole-charge structures and their implications for exotic superconducting surface states.

Abstract

The topological invariants of gapped time reversal invariant lattice superconductors are studied by mapping the superconducting mean field Hamiltonian to a Bloch Hamiltonian. There is a single $Z_2 $ invariant in two dimensions and four such invariants in three dimensions. We briefly discuss the properties of states with non-trivial topological invariants.