Glide reflection symmetry, Brillouin zone folding and superconducting pairing for the $P4/nmm$ space group

TL;DR

This paper investigates how glide reflection symmetry influences Brillouin zone folding and superconducting pairing in materials with the P4/nmm space group, highlighting symmetry constraints and effects of spin-orbit coupling.

Contribution

It provides a symmetry-based analysis of Brillouin zone folding and superconducting pairing constraints in P4/nmm materials, especially in iron-based superconductors.

Findings

Brillouin zone folding is justified by glide reflection symmetry.

Spin-orbit coupling can invalidate the folding procedure.

Superconducting pairing states must have even parity under glide reflection.

Abstract

Motivated by the studies of the superconducting pairing states in the iron-based superconductors, we analyze the effects of Brillouin zone folding procedure from a space group symmetry perspective for a general class of materials with the $P4/nmm$ space group. The Brillouin zone folding amounts to working with an effective one-Fe unit cell, instead of the crystallographic two-Fe unit cell. We show that the folding procedure can be justified by the validity of a glide reflection symmetry throughout the crystallographic Brillouin zone and by the existence of a minimal double degeneracy along the edges of the latter. We also demonstrate how the folding procedure fails when a local spin-orbit coupling is included although the latter does not break any of the space group symmetries of the bare Hamiltonian. In light of these general symmetry considerations, we further discuss the implications of the glide reflection symmetry for the superconducting pairing in an effective multi-orbital $t-J_{1}-J_{2}$ model. We find that the $P4/nmm$ space group symmetry allows only pairing states with even parity under the glide reflection and zero total momentum.