Spin reorientation and 'quasi-nematic' order in the re-entrant tetragonal phase of Ba$_{0.76}$K$_{0.24}$Fe$_2$As$_2$

Clarifying the phase diagram and the nature of the competing or coexisting orders between magnetism, nematicity and superconductivity is of primary importance for the understanding of the mechanism of iron-based superconductors. Here, we investigate the re-entrant tetragonal phase in Ba$_{0.76}$K$_{0.24}$Fe$_{2}$As$_{2}$ in detail by DC magnetisation, resistivity, thermal expansion, thermal conductivity and thermo-electrical measurements. The thermal expansion indicates that the transition into the re-entrant phase is incomplete, but becomes more complete in high magnetic fields. The magnetization provides strong evidence that the spin alignment in the re-entrant C$_{4}$ phase is out-of-plane. The Nernst coefficient shows a large negative value in the stripe-type spin density wave (SDW) state owing to the Fermi surface reconstruction associated with nematic order. At the transition into the re-entrant C$_{4}$ tetragonal phase it hardly changes, suggesting that a similar ‘quasi-nematic’ electronic order could be present in this phase, in a novel form that preserves the tetragonal crystal symmetry. We propose a chequerboard type of charge or orbital ordering that replaces the nematic order of the stripe-type SDW phase below the C$_{4}$ re-entrant transition.