A whole series of expressions for four species of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal) is provided as a complete basis set to describe arbitrary single-centered spinful electron systems. A compact formula to calculate matrix elements of these multipoles is also derived. A visualization method of an electronic state characterized in terms of multipoles is proposed. The complete basis set is useful to narrow down a candidate order parameter of electron systems in phase transition, to describe a property of cross-correlated phenomena, to analyze spectra of x-ray scattering in magnetically ordered state, and so on. We demonstrate a usage of the complete basis set by taking monopole and toroidal dipole orderings, and the mutual relationship among three distinct magnetic dipoles (orbital, spin angular momenta and anisotropic dipole) in a spin-orbit coupled system as prime examples.
We propose an efficient microscopic design procedure of electronic band structures having intrinsic spin and momentum dependences in spin-orbit-coupling free antiferromagnets. Our bottom-up design approach to creating desired spin-split and reshaped band structures could result in further findings of practical spin-orbit-coupling free materials exhibiting a giant spin-dependent and/or nonreciprocal transport, magneto-electric and elastic responses as a consequence of such band structures. We establish a systematic guideline to construct symmetric/antisymmetric spin-split and antisymmetrically …
We investigate microscopic aspects of multipole ordering in f-electron systems with emphasis on the effect of lattice structure. For the purpose, first we construct f-electron models on three kinds of lattices, simple cubic (sc), bcc, and fcc, by including f-electron hopping through (ffσ) bonding in a tight-binding approximation on the basis of a j-j coupling scheme. Then, an effective model is derived in the strong-coupling limit for each lattice structure with the use of second-order perturbation theory with respect to (ffσ). By applying mean-field theory to such effective models, we find di…
The microscopic conditions for the emergence of nonreciprocal magnon excitations in noncentrosymmetric magnets are theoretically investigated. We show that asymmetric magnon excitations appear when a bond magnetic toroidal dipole becomes active, which is defined as a parallel component between the Dzyaloshinskii-Moriya vector and the averaged spin moments at the ends of the bonds. Depending on magnetic structures accompanying the bond magnetic toroidal dipoles, the higher-rank magnetic toroidal multipoles can also be activated in a magnetic cluster, which describes various angle dependences of…
A systematic description of multipole degrees of freedom is discussed on the basis of the Stevens' operator-equivalent technique. The generalized Stevens' multiplicative factors are derived for all of the electric and the magnetic multipoles relevant to f-electron systems. With extensive use of the Stevens' factors, we express the spatial dependences of the electric and the magnetic fields, and the electric and the magnetic charge densities of localized f electrons. The latter is utilized to draw wave functions including their magnetic profile in addition to their shape with the charge density…
We discuss multipole properties of filled skutterudites containing heavy lanthanide Ln from a microscopic viewpoint on the basis of a seven-orbital Anderson model. For Ln=Gd, in contrast to naive expectation, quadrupole moments remain in addition to main dipole ones. For Ln=Ho, we find an exotic state governed by octupole moment. For Ln=Tb and Tm, no significant multipole moments appear at low temperatures, while for Ln=Dy, Er, and Yb, dipole and higher-order multipoles are dominant. We briefly discuss possible relevance of these multipole states with actual materials.
It is shown that $f$-electron multipole is naturally defined as spin-charge one-electron density operator in the second-quantized form with the use of tensor operator on the analogy of multipole expansion of electromagnetic potential from charge distribution in electromagnetism. Due to this definition of multipole, it is possible to determine multipole state from a microscopic viewpoint on the basis of the standard linear response theory for multipole susceptibility. In order to discuss multipole properties of filled skutterudites, we analyze a seven-orbital impurity Anderson model by employin…
We study resonant x-ray scattering (RXS) at Np M_{4,5} edges in the triple-\textbf{k} multipole ordering phase in NpO_{2}, on the basis of a localized electron model. We derive an expression for RXS amplitudes to characterize the spectra under the assumption that a rotational invariance is preserved in the intermediate state of scattering process. This assumption is justified by the fact that energies of the crystal electric field and the intersite interaction is smaller than the energy of multiplet structures. This expression is found useful to calculate energy profiles with taking account of…
We investigate effects of multipole moments in f-electron systems both from phenomenological and microscopic viewpoints. First, we discuss significant effects of octupole moment on the magnetic susceptibility in a paramagnetic phase. It is found that even within mean-field approximation, the magnetic susceptibility deviates from the Curie-Weiss law due to interactions between dipole and octupole moments. Next, we proceed to a microscopic theory for multipole ordering on the basis of a j-j coupling scheme. After brief explanation of a method to derive multipole interactions from the $f$-electro…
In order to clarify exotic multipole properties of filled skutterudites, we evaluate multipole susceptibility for $n$=1$\sim$5, where $n$ is the local $f$-electron number, on the basis of a multiorbital Anderson model constructed using the $j$-$j$ coupling scheme. For $n$=1, magnetic fluctuations dominate over low-temperature electronic properties, while for $n$=2 and 4, electronic states are dominated by both magnetic and quadrupole fluctuations. For $n$=3 and 5, octupole fluctuations are found to be significant, depending on the crystalline electric field potential. We discuss possible relev…
Toward the understanding of incommensurate $Γ_3$ quadrupole ordering in PrPb$_3$, we develop a microscopic theory of multipole ordering in $f^2$-electron systems from an itinerant picture on the basis of a $j$-$j$ coupling scheme. For this purpose, we introduce the $Γ_7$-$Γ_8$ Hubbard model on a simple cubic lattice with the effective interactions that induce local $Γ_3$ states. By evaluating multipole susceptibility in a random phase approximation, we find that the hybridization between $Γ_7$ and $Γ_8$ orbitals plays a key role in the emergence of $Γ_3$ quadrupole ordering. We also emphasize …
Magnetic diffraction of polarized neutrons by the cubic Laves compound UAl2 in a magnetic field has unveiled weak Bragg spots that are nominally forbidden. On the one hand, they can be viewed as magnetic analogues of the basis-forbidden (2, 2, 2) reflection in diamond-type structures that has been painstakingly and frequently investigated over almost a century. Alternatively, the pattern of weak intensities can be assigned to Dirac multipoles imbedded in field-induced magnetic charge. To this end, a published diffraction pattern is successfully confronted with intensities calculated from the a…
We study the electric quadrupole (E2) contribution to resonant x-ray scattering (RXS). Under the assumption that the rotational invariance is preserved in the Hamiltonian describing the intermediate state of scattering, we derive a useful expression for the RXS amplitude. One of the advantages the derived expression possesses is the full information of the energy dependence, lacking in all the previous studies using the fast collision approximation. The expression is also helpful to classify the spectra into multipole order parameters which are brought about. The expression is suitable to inve…
The odd-parity multipole is an emergent degree of freedom, leading to spontaneous inversion symmetry breaking. The odd-parity multipole order may occur by forming staggered even-parity multipoles in a unit cell. We focus on a locally noncentrosymmetric bilayer Rashba system, and study an odd-parity electric octupole order caused by the antiferro stacking of local electric quadrupoles. Analyzing the forward scattering model, we show that the electric octupole order is stabilized by a layer-dependent Rashba spin-orbit coupling. The roles of the spin-orbit coupling are clarified on the basis of t…
An attractive proposal for the hidden order (HO) in the heavy electron compound URu2Si2 is an itinerant multipole order of high rank. It is due to the pairing of electrons and holes centered on zone center and boundary, respectively in states that have maximally different total angular momentum components. Due to the pairing with commensurate zone boundary ordering vector the translational symmetry is broken and a HO quasiparticle gap opens below the transition temperature T_HO. Inelastic neutron scattering (INS) has demonstrated that for T<T_HO the collective magnetic response is dominated by…
By using a density matrix renormalization group method, we investigate the ground-state properties of a one-dimensional three-orbital Hubbard model on the basis of a j-j coupling scheme. For $B_4^0 \ne 0$, where $B_4^0$ is a parameter to control cubic crystalline electric field effect, one orbital is itinerant, while other two are localized. Due to the competition between itinerant and localized natures, we obtain orbital ordering pattern which is sensitive to $B_4^0$, leading to a characteristic change of $Γ_{3g}$ quadrupole state into an incommensurate structure. At $B_4^0 = 0$, all the thre…
Structural and magnetic transitions in a double perovskite hosting 5d1 Re ions are discussed on the basis of recently published high-resolution x-ray diffraction patterns [D. Hirai, et al., Phys. Rev. Res. 2, 022063(R) (2020)]. A reported structural transition below room temperature, from cubic to tetragonal symmetry, appears not to be driven by T2g-type quadrupoles, as suggested. A magnetic motif at lower temperature is shown to be composed of two order parameters, associated with propagation vectors k = (0, 0, 1) and k = (0, 0, 0). Findings from our studies, for structural and magnetic prope…
We provide a prescription to construct a microscopic model for heavy lanthanide systems such as Yb and Tm compounds by exploiting a $j$-$j$ coupling scheme. Here we consider a situation with a large spin-orbit coupling, in which $j$=5/2 sextet is fully occupied, while $j$=7/2 octet is partially occupied, where $j$ denotes total angular momentum. We evaluate crystalline electric field potentials and Coulomb interactions among the states of the $j$=7/2 octet to construct a local Hamiltonian in the $j$-$j$ coupling scheme. Then, it is found that the local $f$-electron states composed of the $j$=7…
Non-resonant inelastic x ray scattering (NIXS) experiments have been performed to probe the 5d-5f electronic transitions at the uranium O(4,5) absorption edges in uranium dioxide. For small values of the scattering vector q, the spectra are dominated by dipole-allowed transitions encapsulated within the giant resonance, whereas for higher values of q the multipolar transitions of rank 3 and 5 give rise to strong and well-defined multiplet structure in the pre-edge region. The origin of the observed non-dipole multiplet structures is explained on the basis of many-electron atomic spectral calcu…
Odd-Parity Crystalline Electric Field and Multipoles in Icosahedral Quasicrystal and Approximant: Magnetic Toroidal and Electric Dipoles in Ce-Based Systems
Kyushu Institute of Technology（九州工业大学）；Yokohama National University（横滨国立大学）
We present a new type of basis set which is local, compact, and orthogonal. The basis functions, called orthlets, are centered at the sites of a lattice and are specifically adapted to represent the system being studied. The adaptability includes the ability to have singular behavior within an orthlet, allowing a single orthlet to represent a function in the vicinity of a singularity.
We present a new kind of basis function for discretizing the Schrödinger equation in electronic structure calculations, called a gausslet, which has wavelet-like features but is composed of a sum of Gaussians. Gausslets are placed on a grid and combine advantages of both grid and basis set approaches. They are orthogonal, infinitely smooth, symmetric, polynomially complete, and with a high degree of locality. Because they are formed from Gaussians, they are easily combined with traditional atom-centered Gaussian bases. We also introduce diagonal approximations which dramatically reduce the com…
