Octupolar Order Research Articles

Signatures of hidden octupolar order from nonlinear Hall effects

Signatures of hidden octupolar order from nonlinear Hall effects

Probing octupolar hidden order via Janus impurities

Quantum materials with non-Kramers doublets are a fascinating venue to realize multipolar hidden orders. Impurity probes which break point group symmetries, such as implanted muons or substitutional impurities, split the non-Kramers degeneracy and exhibit a Janus-faced influence in such systems: they can destroy the very order they seek to probe. Here, we explore this duality in cubic osmate double perovskites which are candidates for exotic d-orbital octupolar order competing with quadrupolar states. Using ab initio computations, Landau theory, and Monte Carlo simulations, we show that Janus impurities induce local strain fields, nucleating quadrupolar puddles and suppressing the octupolar Tc. At the same time, strains mix the non-Kramers doublet with an excited magnetic triplet, creating parasitic dipole moments which directly expose the hidden octupolar order parameter. Our work unravels this Janus duality in recent impurity nuclear magnetic resonance (NMR) experiments, with important implications for uncovering hidden order in diverse multipolar materials.

Search for octupolar order in NpO2 by neutron powder diffraction

The magnetic structure of neptunium dioxide (NpO2) remains a mystery despite decades of research. We revisited the search for magnetic ordering in NpO2 by performing a powder neutron diffraction experiment at low temperatures and relatively large values of Q. Diffraction data were collected for 300 ​mK, 15 ​K, and 35 ​K to understand the phase transition near 25 ​K. However, no significant changes in the neutron diffraction pattern were identified. Using the octupolar magnetic form factor j6, the maximum theoretical expected value of the magnetic moment (3.27 μB), the calculated magnetic Bragg peak intensities would not be observable in neutron powder diffraction data. This value is much larger than the previously estimated magnetic moment of ∼0.1 μB. Supported by experimental results, these calculations suggest that unpolarized neutron powder diffraction is unsuitable for measuring octupolar ordering in NpO2.

Excitation of the σl_l = 90° resonance by the cavity rf accelerating fields

In rf linacs the longitudinal focalization is done by si-nusoidal forces and at high accelerating fields the zero-current longitudinal phase advance per longitudinal fo-cusing period σ0l_l can be high. The nonlinear compo-nents of the sinusoidal rf field (sextupolar, octupolar and higher order components) can then excite parametric resonances, including the 4th-order resonance (σl_l = 90°) when σ0l_l is higher than 90°, inducing strong longitudinal emittance growths and acceptance reductions. As pointed out in previous papers, the longitudinal beam dynamics is therefore complex, even when the space-charge forces are ignored. The parametric resonance excitation by the rf field is analyzed before discussing the additional effect of the space-charge forces, in particular to explain why the zero-current longitudinal phase advance per trans-verse focusing period σ0l_t is not a relevant parameter. Examples are given in the SPIRAL2 linac case.

A Comprehensive Multipolar Theory for Periodic Metasurfaces

AbstractOptical metasurfaces consist of 2D arrangements of scatterers, and they control the amplitude, phase, and polarization of an incidence field on demand. Optical metasurfaces are the cornerstone for a future generation of flat optical devices in a wide range of applications. The rapid advances in nanofabrication have made the versatile design and analysis of these ultra‐thin surfaces an ever‐growing necessity. However, a comprehensive theory to describe the optical response of periodic metasurfaces in closed‐form and analytical expressions has not been formulated, and prior attempts are frequently approximate. Here, a theory is developed that analytically links the properties of the scatterer, from which a metasurface is made, to its response via the lattice coupling matrix. The scatterers are represented by their polarizability or T matrix. Explicit expressions for the optical response up to octupolar order in both spherical and Cartesian coordinates are provided, for normal or oblique incidence. Several examples demonstrate that the proposed theoretical approach is a powerful tool for exploring the physics of metasurfaces and designing novel flat optics devices. Novel fully‐diffracting metagratings and particle‐independent polarization filters are proposed, and novel insights into bound states in the continuum, collective lattice resonances, and the response of Huygens’ metasurfaces under oblique incidence are provided.

Competing multipolar orders in a face-centered cubic lattice: Application to the osmium double perovskites

In 5$d^2$ Mott insulators with strong spin-orbit coupling, the lowest pseudospin states form a non-Kramers doublet, which carries quadrupolar and octupolar moments. A family of double perovskites where magnetic ions form a face-centered cubic (fcc) lattice was suggested to unveil an octupolar order offering a rare example in $d$-orbital systems. The proposed order requires a ferromagnetic (FM) octupolar interaction, since the antiferromagnetic (AFM) Ising model is highly frustrated on the fcc lattice. A microscopic model was recently derived for various lattices: for an edge-sharing octahedra geometry, AFM Ising octupolar and bond-dependent quadrupolar interactions were found when only dominant inter- and intraorbital hopping integrals were taken into account. Here we investigate all possible intra- and interorbital exchange processes and report that interference of two intraorbital exchanges generates a FM octupolar interaction. Applying the strong-coupling expansion results together with tight-binding parameters obtained by density functional theory, we estimate the exchange interactions for the osmium double perovskites, Ba$_2$BOsO$_6$ (B=Mg,Cd, and Ca). Using classical Monte Carlo simulations, we find that these systems are close to the phase boundary between AFM type-I quadrupole and FM octupole orders. We also find that exchange processes beyond second-order perturbation theory including virtual processes via pseudospin-triplet states may stabilize an octupolar order.

Search for Octupolar Order in Npo2 by Unpolarized Neutron Powder Diffraction

Search for Octupolar Order in Npo2 by Unpolarized Neutron Powder Diffraction

Ferro-octupolar Order and Low-Energy Excitations ind2Double Perovskites of Osmium

Conflicting interpretations of experimental data preclude the understanding of the quantum magnetic state of spin-orbit coupled $d^2$ double perovskites. Whether the ground state is a Janh-Teller-distorted order of quadrupoles or the hitherto elusive octupolar order remains debated. We resolve this uncertainty through direct calculations of all-rank inter-site exchange interactions and inelastic neutron scattering cross-section for the $d^2$ double perovskite series Ba$_2M$OsO$_6$ ($M$= Ca, Mg, Zn). Using advanced many-body first principles methods we show that the ground state is formed by ferro-ordered octupoles coupled by superexchange interactions within the ground-state $E_g$ doublet. Computed ordering temperature of the single second-order phase-transition is consistent with experimentally observed material-dependent trends. Minuscule distortions of the parent cubic structure are shown to qualitatively modify the structure of gaped magnetic excitations.

Octupolar order and Ising quantum criticality tuned by strain and dimensionality: Application to d -orbital Mott insulators

Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling driven by the interplay of two distinct intra-orbital hopping terms. Using classical Monte Carlo simulations on the face-centered cubic lattice, we obtain a ferrooctupolar transition temperature which is in good agreement with experiments on the osmate double perovskites. We also explore the impact of uniaxial strain and dimensional tuning via ultrathin films, which are shown to induce a transverse field on the Ising octupolar order. This suppresses $T_c$ and potentially allows one to access octupolar Ising quantum critical points. We discuss possible implications of our results for a broader class of materials which may host such non-Kramers doublet ions.

Two-fluid model for a fluid with tetrahedral-octupolar order.

We investigate the macroscopic dynamics of a two-fluid system with tetrahedral order. As all normal-fluid two-fluid systems one has-compared to a simple fluid-the velocity difference between the two subsystems and the concentration of one component as additional macroscopic variables. Depending on the type of system, the concentration can either be a conserved quantity or relax on a long, but finite timescale. Due to the existence of the tetrahedral order such a system breaks parity symmetry. Here we discuss physical systems without preferred direction in real space, meaning that our description applies to optically isotropic materials. We find a number of reversible as well as dissipative dynamic cross-coupling terms due to the additional octupolar order, when compared to a fluid mixture. As the most interesting cross-coupling term from an experimental point of view, we identify a dissipative cross-coupling between the relative velocity and the usual velocity gradients. Applying a shear flow in a plane, this dissipative cross-coupling leads to a velocity difference perpendicular to the shear plane. As a result one can obtain a spatially homogeneous oscillation of the relative velocity. In addition, this induced relative velocity can couple as a function of time and space to the concentration, which gives rise to an overdamped propagating soundlike mode, where the overdamping arises from the fact that velocity difference is a macroscopic variable and not strictly conserved. We also show that electric field gradients are connected with an analogous reversible cross-coupling and can lead in a planar shear geometry to an overdamped propagating mode as well.

Director Deformations, Geometric Frustration, and Modulated Phases in Liquid Crystals

This article analyzes modulated phases in liquid crystals, from the long-established cholesteric and blue phases to the recently discovered twist-bend, splay-bend, and splay nematic phases, as well as the twist-grain-boundary (TGB) and helical nanofilament variations on smectic phases. The analysis uses the concept of four fundamental modes of director deformation: twist, bend, splay, and a fourth mode related to saddle-splay. Each mode is coupled to a specific type of molecular order: chirality, polarization perpendicular and parallel to the director, and octupolar order. When the liquid crystal develops one type of spontaneous order, the ideal local structure becomes nonuniform, with the corresponding director deformation. In general, the ideal local structure is frustrated; it cannot fill space. As a result, the liquid crystal must form a complex global phase, which may have a combination of deformation modes, and may have a periodic array of defects. Thus, the concept of an ideal local structure under geometric frustration provides a unified framework to understand the wide variety of modulated phases.

The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet

We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxial D 3h spin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.

Nanoscale ordering of planar octupolar molecules for nonlinear optics at higher temperatures

We develop scenarios for orientational ordering of an in-plane system of small flat octupolar molecules at the low-concentration limit, aiming towards nonlinear-optical (NLO) applications at room temperatures. The octupoles interact with external electric poling fields and intermolecular interactions are neglected. Simple statistical-mechanics models are used to analyze the orientational order in the very weak poling limit, sufficient for retrieving the NLO signals owing to the high sensitivity of NLO detectors and measurement chains. Two scenarios are discussed. Firstly, the octupolar poling field is imparted by a system of point charges; the setup is subject to cell-related constraints imposed by mechanical strength and dielectric breakdown limit. The very weak octupolar order of benchmarking TATB molecules is shown to emerge at Helium temperatures. The second scenario addresses the dipoling of octupolar molecules with a small admixture of electric dipolar component. It requires a strong field regime to become effective at Nitrogen temperature range. An estimation of the nonlinear susceptibility coefficient matrix for both scenarios is done in the high-temperature (weak interaction) limit formalism. We argue that moderate modifications of the system like, e.g., an increase of the size of the octupole, accompanied by dipole-assisted octupoling, can increase the poling temperature above Nitrogen temperatures.

Symmetry aspects in the macroscopic dynamics of magnetorheological gels and general liquid crystalline magnetic elastomers

Abstract We investigate theoretically the macroscopic dynamics of various types of ordered magnetic fluid, gel, and elastomeric phases. We take a symmetry point of view and emphasize its importance for a macroscopic description. The interactions and couplings among the relevant variables are based on their individual symmetry behavior, irrespective of the detailed nature of the microscopic interactions involved. Concerning the variables we discriminate between conserved variables related to a local conservation law, symmetry variables describing a (spontaneously) broken continuous symmetry (e.g., due to a preferred direction) and slowly relaxing ones that arise from special conditions of the system are considered. Among the relevant symmetries, we consider the behavior under spatial rotations (e.g., discriminating scalars, vectors or tensors), under spatial inversion (discriminating e.g., polar and axial vectors), and under time reversal symmetry (discriminating e.g., velocities from polarizations, or electric fields from magnetic ones). Those symmetries are crucial not only to find the possible cross-couplings correctly but also to get a description of the macroscopic dynamics that is compatible with thermodynamics. In particular, time reversal symmetry is decisive to get the second law of thermodynamics right. We discuss (conventional quadrupolar) nematic order, polar order, active polar order, as well as ferromagnetic order and tetrahedral (octupolar) order. In a second step, we show some of the consequences of the symmetry properties for the various systems that we have worked on within the SPP1681, including magnetic nematic (and cholesteric) elastomers, ferromagnetic nematics (also with tetrahedral order), ferromagnetic elastomers with tetrahedral order, gels and elastomers with polar or active polar order, and finally magnetorheological fluids and gels in a one- and two-fluid description.

Antiferromagnetic long-range order in the 5d1 double-perovskite Sr2MgReO6

The double-perovskite A$_2$BB'O$_6$ with heavy transition metal ions on the ordered B' sites is an important family of compounds to study the interplay between electron correlation and spin-orbit coupling (SOC). Here we prepared high-quality Sr$_2$MgReO$_6$ powder and single-crystal samples and performed non-resonant and resonant synchrotron x-ray diffraction experiments to investigate its magnetic ground state. By combining the magnetic susceptibility and heat capacity measurements, we conclude that Sr$_2$MgReO$_6$ exhibits a layered antiferromagnetic (AF) order at temperatures below $\sim$ 55 K with a propagation vector q = (001), which contrasts the previously suspected spin glass state. Our works clarify the magnetic order in Sr$_2$MgReO$_6$ and demonstrate it as a candidate system to look for magnetic octupolar orders and exotic spin dynamics.

Field-Angle-Resolved Magnetic Excitations as a Probe of Hidden-Order Symmetry in CeB6

In contrast to magnetic order formed by electrons’ dipolar moments, ordering phenomena associated with higher-order multipoles (quadrupoles, octupoles, etc.) are more difficult to characterize because of the limited choice of experimental probes that can distinguish different multipolar moments. The heavy-fermion compound CeB6 and its La-diluted alloys are among the best-studied realizations of the long-range-ordered multipolar phases, often referred to as “hidden order.” Previously, the hidden order in phase II was identified as primary antiferroquadrupolar and field-induced octupolar order. Here, we present a combined experimental and theoretical investigation of collective excitations in phase II of CeB6. Inelastic neutron scattering (INS) in fields up to 16.5 T reveals a new high-energy mode above 14 T in addition to the low-energy magnetic excitations. The experimental dependence of their energy on the magnitude and angle of the applied magnetic field is compared to the results of a multipolar interaction model. The magnetic excitation spectrum in a rotating field is calculated within a localized approach using the pseudospin representation for the Γ8 states. We show that the rotating-field technique at fixed momentum can complement conventional INS measurements of the dispersion at a constant field and holds great promise for identifying the symmetry of multipolar order parameters and the details of intermultipolar interactions that stabilize hidden-order phases.6 MoreReceived 6 January 2020Revised 20 February 2020Accepted 5 March 2020DOI:https://doi.org/10.1103/PhysRevX.10.021010Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsHeavy-fermion systemsStrongly correlated systemsTechniquesInelastic neutron scatteringCondensed Matter, Materials & Applied Physics

Octupolar order in d -orbital Mott insulators

Motivated by experimental and theoretical interest in realizing multipolar orders in $d$-orbital materials, we discuss the quantum magnetism of $J\!=\!2$ ions which can be realized in spin-orbit coupled oxides with $5d^2$ transition metal ions. Based on the crystal field environment, we argue for a splitting of the $J\!=\!2$ multiplet, leading to a low lying non-Kramers doublet which hosts quadrupolar and octupolar moments. We discuss a microscopic mechanism whereby the combined perturbative effects of orbital repulsion and antiferromagnetic Heisenberg spin interactions leads to ferro-octupolar coupling between neighboring sites, and stabilizes ferro-octupolar order for a face-centered cubic lattice. This same mechanism is also shown to disfavor quadrupolar ordering. We show that studying crystal field levels via Raman scattering in a magnetic field provides a probe of octupolar order. We study spin dynamics in the ferro-octupolar state using a slave-boson approach, uncovering a gapped and dispersive magnetic exciton. For sufficiently strong magnetic exchange, the dispersive exciton can condense, leading to conventional type-I antiferromagnetic (AFM) order which can preempt octupolar order. Our proposal for ferrooctupolar order, with specific results in the context of a model Hamiltonian, provides a comprehensive understanding of thermodynamics, $\mu$SR, X-ray diffraction, and inelastic neutron scattering measurements on a range of cubic $5d^2$ double perovskite materials including Ba$_2$ZnOsO$_6$, Ba$_2$CaOsO$_6$, and Ba$_2$MgOsO$_6$. Our proposal for exciton condensation leading to type-I AFM order may be relevant to materials such as Sr$_2$MgOsO$_6$.

Unveiling hidden multipolar orders with magnetostriction

Broken symmetries in solids involving higher order multipolar degrees of freedom are historically referred to as “hidden orders” due to the formidable task of detecting them with conventional probes. In this work, we theoretically propose that magnetostriction provides a powerful and novel tool to directly detect higher-order multipolar symmetry breaking—such as the elusive octupolar order—by examining scaling behaviour of length change with respect to an applied magnetic field h. Employing a symmetry-based Landau theory, we focus on the family of Pr-based cage compounds with strongly correlated f-electrons, Pr(Ti,V,Ir)2(Al,Zn)20, whose low energy degrees of freedom are purely higher-order multipoles: quadrupoles {cal{O}}_{20,22} and octupole {cal{T}}_{xyz}. We demonstrate that a magnetic field along the [111] direction induces a distinct linear-in-h length change below the octupolar ordering temperature. The resulting “magnetostriction coefficient” is directly proportional to the octupolar order parameter, thus providing clear access to such subtle order parameters.

Improving the NRTidal model for binary neutron star systems

Accurate and fast gravitational waveform (GW) models are essential to extract information about the properties of compact binary systems that generate GWs. Building on previous work, we present an extension of the NRTidal model for binary neutron star (BNS) waveforms. The upgrades are: (i) a new closed-form expression for the tidal contribution to the GW phase which includes further analytical knowledge and is calibrated to more accurate numerical relativity data than previously available; (ii) a tidal correction to the GW amplitude; (iii) an extension of the spin-sector incorporating equation-of-state-dependent finite size effects at quadrupolar and octupolar order; these appear in the spin-spin tail terms and cubic-in-spin terms, both at 3.5PN. We add the new description to the precessing binary black hole waveform model IMRPhenomPv2 to obtain a frequency-domain precessing binary neutron star model. In addition, we extend the SEOBNRv4_ROM and IMRPhenomD aligned-spin binary black hole waveform models with the improved tidal phase corrections. Focusing on the new IMRPhenomPv2_NRTidalv2 approximant, we test the model by comparing with numerical relativity waveforms as well as hybrid waveforms combining tidal effective-one-body and numerical relativity data. We also check consistency against a tidal effective-one-body model across large regions of the BNS parameter space.

Dynamic interplay of nematic, magnetic, and tetrahedral order in ferromagnetic nematic phases

We consider the influence of tetrahedral (octupolar) order on ferromagnetic nematic liquid-crystalline phases. The presence of tetrahedral order leads to broken parity symmetry in an achiral liquid-crystalline system, in addition to broken time-reversal symmetry associated with the existence of a spontaneous magnetization. As a consequence, we find static as well as reversible and irreversible dynamic cross-coupling terms absent in usual ferromagnetic nematics. Several static and dynamic experiments are suggested to detect possible tetrahedral order. We predict that linear gradient terms in the generalized energy involving the ferromagnetic magnetization and the nematic director field lead to chiral domains of ambidextrous helicity. As a characteristic dissipative dynamic cross coupling we point out that the rotation of the magnetization can be driven by temperature and/or concentration gradients. Conversely, heat and concentration currents can be generated by rotations of the magnetization. As a characteristic example for reversible cross-coupling terms we analyze the consequences of the coupling between the molecular field of the nematic director and temperature and concentration gradients.