Evidence for antiferromagnetism coexisting with charge order in the trilayer cuprate HgBa$_2$Ca$_2$Cu$_3$O$_{8+ δ}$

Abstract: Multilayered cuprates possess not only the highest superconducting temperature transition but also offer a unique platform to study disorder-free CuO$<em>2$ planes and the interplay between competing orders with superconductivity. Here, we study the underdoped trilayer cuprate HgBa$_2$Ca$_2$Cu$_3$O$</em>{8+ \delta}$ and we report the first quantum oscillation and Hall effect measurements in magnetic field up to 88 T. A careful analysis of the complex spectra of quantum oscillations strongly supports the coexistence of an antiferromagnetic order in the inner plane and a charge order in the outer planes. The presence of an ordered antiferromagnetic metallic state that extends deep in the superconducting phase is a key ingredient that supports magnetically mediated pairing interaction in cuprates.