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Twisted double-layer chiral spin structures in anti-ferromagnetic delafossite PdCrO$_2$

We predict that the anti-ferromagnetic (AFM) magnetic structure of delafossite PdCrO$_2$ has a staggered chirality with the double-layer periodicity along the $z$-axis with intriguing magnetic anisotropy. We study the electronic and magnetic structures of PdCrO$_2$ by carrying out density functional theory (DFT) calculation. The calculated ground state turns out to exhibit a 120$^\circ$ AFM ordering with staggered chirality. The result is consistent with the previous neutron experiments, but our calculations determine and provide more detailed information on the high symmetric easy-plane and local-easy-axis structure. We investigate the origin of staggered chirality as well as the easy-plane and local-easy-axis directions. Based on the DFT and effective spin interaction models, we demonstrate the complex magnetic ordering requires novel four-spin interactions, which cannot be described by the usual exchange interaction. Also, we show Pd electronic bands near the Fermi level pose a tiny degeneracy breaking, which arises from the magnetic anisotropy with different easy-plane and local-easy-axis. We expect that the twisting and rotating modes of easy-planes in the magnetic double layers can be the critical factor to understand a hidden magnetic property of PdCrO$_2$ and its related macroscopic property like the previously reported anomalous Hall effect.