Electric Toroidal Quadrupoles in the Spin-Orbit-Coupled Metal Cd_{2}Re_{2}O_{7}

Spin-Derived Electric Polarization and Chirality Density Inherent in Localized Electron Orbitals

In solid state physics, any phase transition is commonly observed as a change in the microscopic distribution of charge, spin, or current. However, there is an exotic order parameter inherent in the localized electron orbitals that cannot be primarily captured by these three fundamental quantities. This order parameter is described as the electric toroidal multipoles connecting different total angular momenta under the spin-orbit coupling. The corresponding microscopic physical quantity is the spin current tensor on an atomic scale, which induces spin-derived electric polarization aligned circularly and the chirality density of the Dirac equation. Here, elucidating the nature of this exotic order parameter, we obtain the following general consequences that are not restricted to localized electron systems; chirality density is indispensable to unambiguously describe electronic states and it is a species of electric toroidal multipoles, just as the charge density is a species of electric multipoles. Furthermore, we derive the equation of continuity for chirality and discuss its relation to chiral anomaly and optical chirality. These findings link microscopic spin currents and chirality in the Dirac theory to the concept of multipoles and provide a new perspective for quantum states of matter.

Successive phase transitions of the spin-orbit-coupled metal Cd2Re2O7probed by high-resolution synchrotron x-ray diffraction

The 5dpyrochlore oxide superconductor Cd₂Re₂O₇(CRO) has attracted significant interest as a spin-orbit-coupled metal (SOCM) that spontaneously undergoes a phase transition to an odd-parity multipole phase by breaking the spatial inversion symmetry due to the Fermi liquid instability caused by strong spin-orbit coupling. Despite the significance of structural information during the transition, previous experimental results regarding lattice deformation have been elusive. We have conducted ultra-high resolution synchrotron radiation x-ray diffraction experiments on a high-quality CRO single crystal. The temperature-dependent splitting of the 0 0 16 and 0 0 14 reflections, which are allowed and forbidden, respectively, in the high-temperature cubic phase I (space groupFd-3m), has been clearly observed and reveals the following significant facts: inversion symmetry breaking and tetragonal distortion occur simultaneously atT_s1= 201.5(1) K; the previously believed first-order transition between phase II (I-4m2) and phase III (I4₁22) atT_s2∼120 K consists of two close second-order transitions atT_s2= 115.4(1) K andT_s3∼ 100 K; there is a new orthorhombic phase XI (F222) in between. The order parameters (OPs) of these continuous transitions are uniquely represented by a two-dimensional irreducible representationE_uof theO_hpoint group, and the OPs of phase XI are a linear combination of those of phases II and III. Each phase is believed to correspond to a distinct odd-parity multipole order, and the complex successive transitions observed may be the result of an electronic phase transition that resolves the Fermi liquid instability in the SOCM.

Generalization of microscopic multipoles and cross-correlated phenomena by their orderings

The generalization of the atomic-scale multipoles is discussed. By introducing the augmented multipoles defined in the hybrid orbitals or in the site/bond-cluster, any of electronic degrees of freedom can be expressed in accordance with the crystallographic point group. These multipoles are useful to describe the cross-correlated phenomena, band-structure deformation, and generation of effective spin-orbit coupling due to antiferromagnetic ordering in a systematic and comprehensive manner. Such a symmetry-adapted multipole basis set could be a promising descriptor for materials design and informatics.

Harnessing interpretable and unsupervised machine learning to address big data from modern X-ray diffraction

The information content of crystalline materials becomes astronomical when collective electronic behavior and their fluctuations are taken into account. In the past decade, improvements in source brightness and detector technology at modern X-ray facilities have allowed a dramatically increased fraction of this information to be captured. Now, the primary challenge is to understand and discover scientific principles from big datasets when a comprehensive analysis is beyond human reach. We report the development of an unsupervised machine learning approach, X-ray diffraction (XRD) temperature clustering (X-TEC), that can automatically extract charge density wave order parameters and detect intraunit cell ordering and its fluctuations from a series of high-volume X-ray diffraction measurements taken at multiple temperatures. We benchmark X-TEC with diffraction data on a quasi-skutterudite family of materials, (CaxSr[Formula: see text])3Rh4Sn13, where a quantum critical point is observed as a function of Ca concentration. We apply X-TEC to XRD data on the pyrochlore metal, Cd2Re2O7, to investigate its two much-debated structural phase transitions and uncover the Goldstone mode accompanying them. We demonstrate how unprecedented atomic-scale knowledge can be gained when human researchers connect the X-TEC results to physical principles. Specifically, we extract from the X-TEC-revealed selection rules that the Cd and Re displacements are approximately equal in amplitude but out of phase. This discovery reveals a previously unknown involvement of [Formula: see text] Re, supporting the idea of an electronic origin to the structural order. Our approach can radically transform XRD experiments by allowing in operando data analysis and enabling researchers to refine experiments by discovering interesting regions of phase space on the fly.

A quantum liquid of magnetic octupoles on the pyrochlore lattice

Spin liquids are highly correlated yet disordered states formed by the entanglement of magnetic dipoles1. Theories define such states using gauge fields and deconfined quasiparticle excitations that emerge from a local constraint governing the ground state of a frustrated magnet. For example, the '2-in-2-out' ice rule for dipole moments on a tetrahedron can lead to a quantum spin ice2-4 in rare-earth pyrochlores. However, f-electron ions often carry multipole degrees of freedom of higher rank than dipoles, leading to intriguing behaviours and 'hidden' orders5-6. Here we show that the correlated ground state of a Ce3+-based pyrochlore, Ce2Sn2O7, is a quantum liquid of magnetic octupoles. Our neutron scattering results are consistent with a fluid-like state where degrees of freedom have a more complex magnetization density than that of magnetic dipoles. The nature and strength of the octupole-octupole couplings, together with the existence of a continuum of excitations attributed to spinons, provides further evidence for a quantum ice of octupoles governed by a '2-plus-2-minus' rule7-8. Our work identifies Ce2Sn2O7 as a unique example of frustrated multipoles forming a 'hidden' topological order, thus generalizing observations on quantum spin liquids to multipolar phases that can support novel types of emergent fields and excitations.