Chirality, an object's asymmetry upon mirroring, is prevalent in nature and influences material properties like optical activity and piezoelectricity. When time-reversal symmetry is broken by magnetism, unconventional phenomena emerge, especially in chiral-lattice magnets with macroscopic magnetization, leading to nonreciprocal responses in quasiparticles. However, time-reversal symmetry breaking can also occur in antiferromagnets without magnetization, often described by magnetic multipole ordering. Magnetic quadrupoles, breaking both time-reversal and space-inversion symmetries, are of particular interest due to their role in magnetoelectric effects, nonreciprocal optical responses, and magnetotransport phenomena, and their potential link to superconductivity. This paper proposes that the combination of a Qxy-type magnetic quadrupole and chirality leads to unusual magnetization induction orthogonal to the applied field and reversible by chirality switching. This proposal is tested through magnetoelectric effect measurements in Pb(TiO)6Cu4(PO4)4, a chiral-lattice antiferromagnet with ferroic Qxy magnetic quadrupole order. The study aims to demonstrate the control of the Qxy sign using a magnetic field alone and validate the proposal through theoretical calculations and a phenomenological approach.

The study builds upon prior research demonstrating nonreciprocal phenomena in chiral-lattice magnets exhibiting macroscopic magnetization (ferromagnetic order) or under external magnetic fields. Previous work has shown nonreciprocal propagation and dichroism in photons, electrons, magnons, and phonons. Recent theoretical studies have also established that time-reversal-broken antiferromagnetic states can be fully described by magnetic multipole and toroidal multipole ordering, with magnetic quadrupoles as fundamental components. These quadrupoles have garnered attention for their roles in linear magnetoelectric effects, nonreciprocal optical responses, and magnetotransport. However, the combination of magnetic quadrupoles and chirality to generate novel phenomena remained largely unexplored before this study. The authors cite relevant works on magneto-chiral dichroism, electrical magnetochiral anisotropy, and magnetoelectric effects in antiferromagnets.

The research employed a combination of experimental measurements and theoretical calculations. Single crystals of Pb(TiO)6Cu4(PO4)4 (PbTCPO) with both left-handed (C-) and right-handed (C+) chirality were grown and characterized. The crystal structure of PbTCPO, consisting of Cu4O12 square cupolas, was analyzed to understand the chirality and its impact on the magnetic structure. Electric polarization (Py) measurements were conducted under various conditions of magnetic field (H) and electric field (E) to probe the magnetoelectric effects. The magnetic field was applied along the X-axis (Hx) and sometimes tilted towards the Z-axis (Hz) to introduce a small Z-component. Ferroelectric hysteresis loops (Py vs. Ey) were measured at different temperatures and magnetic field strengths to investigate the magnetic quadrupole domains and their switching behavior. The sign of Py in finite Hx directly probed the sign of the orthogonal magnetization (Mz). To study the effects of chirality, measurements were conducted on both C- and C+ crystals. Theoretical calculations using a chiral spin-1/2 model, incorporating the Dzyaloshinskii-Moriya interaction and a staggered rotation of the DM vector to account for chirality, were performed using cluster mean-field theory. The calculations aimed to reproduce the experimental observations and provide a microscopic understanding of the chirality-induced orthogonal magnetization. Additionally, a cluster multipole analysis and a phenomenological approach based on Landau theory were employed to analyze the experimental results and understand the coupling among magnetic quadrupoles, octupoles, and chirality.

The study's key findings include the experimental demonstration of magnetic quadrupole domain control solely by a magnetic field in PbTCPO. The sign of the magnetic quadrupoles was successfully reversed by switching the chirality. The measurements of ferroelectric hysteresis loops revealed a spontaneous Py that increased linearly with Hx, indicating a linear magnetoelectric effect. Furthermore, a bias electric field (ΔE) was observed when the magnetic field was slightly tilted from the X-axis, indicating the presence of orthogonal magnetization (Mz). The sign of ΔE was reversed by switching the chirality, supporting the proposed relationship between Mz, chirality, and magnetic quadrupoles. The theoretical calculations based on a chiral spin-1/2 model successfully reproduced the experimental observations, confirming the existence of the chirality-induced orthogonal magnetization. The cluster multipole analysis identified the O2-type magnetic octupole as the dominant contributor to the second-order nonlinear magnetization. The phenomenological model based on Landau theory explained the cross-control of magnetic quadrupoles and octupoles via the application of electric and magnetic fields, highlighting the crucial role of chirality in mediating this coupling. Finally, the study demonstrated the achievement of a single magnetic quadrupole domain state using only a tilted magnetic field, eliminating the need for an electric field.

The findings directly address the research question by demonstrating the control of magnetic quadrupole domains in a chiral antiferromagnet using a magnetic field alone. This unconventional control mechanism is a consequence of the chirality-induced coupling between magnetic quadrupoles and octupoles. The significance of the results lies in their potential impact on the manipulation of antiferromagnetic domains, particularly in metallic antiferromagnets where electric field control is challenging due to screening effects. The study opens new avenues for exploring and utilizing chiral-lattice time-reversal-broken antiferromagnets in applications requiring precise control of magnetic properties. The observation of switchable magnetic quadrupole and octupole orders suggests PbTCPO as a distinct type of multiferroic.

This study successfully demonstrated the control of magnetic quadrupole domains in a chiral antiferromagnet using a tilted magnetic field, highlighting the crucial role of chirality and the emergent magnetic octupoles. The findings provide a novel method for manipulating antiferromagnetic domains, especially in metallic systems, opening avenues for spintronics and other applications. Future research could focus on exploring similar phenomena in other chiral antiferromagnets, investigating the potential for electric-current control, and exploring external chirality induction in nonchiral materials.

The study focused on a specific material, Pb(TiO)6Cu4(PO4)4, and the generalizability of the findings to other materials needs further investigation. The theoretical model employed approximations that may not fully capture all aspects of the complex magnetic interactions. Further studies are warranted to fully delineate the mechanism of chirality-mediated interaction.