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Chirality ubiquitously appears in nature, however, its quantification remains obscure owing to the lack of microscopic description at the quantum-mechanical level. We propose a way of evaluating chirality in terms of electric toroidal monopole, a practical entity of time-reversal even pseudoscalar (parity-odd) object reflecting relevant electronic wave functions. For this purpose, we analyze a twisted methane at the quantum-mechanical level, showing that the electric toroidal monopoles become a quantitative indicator for chirality. In the twisted methane, we clarify that the handedness of chirality corresponds to the sign of the expectation value of the electric toroidal monopole, and that the most important ingredient is the modulation of the spin-dependent imaginary hopping between the hydrogen atoms, while the relativistic spin-orbit coupling within the carbon atom is irrelevant for chirality., Comment: 11 pages, 8 figures, 5 tables

Transport property under metallic ferroaxial ordering in an external magnetic field is theoretically investigated. After presenting the relation between the magnetoconductivity tensor and ferroaxial moment from the symmetry viewpoint, we analyze the behavior of the unconventional Hall effect and magnetoconductivity for a general five $d$-orbital tight-binding model under the point group $C_{\rm 4h}$, where the ferroaxial moment is activated. We show that the crystalline electric field that arises from the symmetry reduction from $D_{\rm 4h}$ to $C_{\rm 4h}$ is essential for the ferroaxial-related magnetotransport, while the relativistic spin-orbit coupling is not required. We also compare the unconventional Hall effect driven by the ferroaxial moment with the conventional Hall effect, the latter of which does not require the ferroaxial moment. The present results provide characteristic transport properties in the ferroaxial systems, which can be observed in various candidate materials like Ca$_5$Ir$_3$O$_{12}$., Comment: 16 pages, 6 figures, 2 tables

We have developed a symmetry-adapted modeling procedure for molecules and crystals. By using the completeness of multipoles to express spatial and time-reversal parity-specific anisotropic distributions, we can generate systematically the complete symmetry-adapted multipole basis set to describe any of electronic degrees of freedom in isolated cluster systems and periodic crystals. The symmetry-adapted modeling is then achieved by expressing the Hamiltonian in terms of the linear combination of these bases belonging to the identity irreducible representation, and the model parameters (linear coefficients) in the Hamiltonian can be determined so as to reproduce the electronic structures given by the density-functional computation. We demonstrate our method for the modeling of graphene, and emphasize usefulness of the symmetry-adapted basis to analyze and predict physical phenomena and spontaneous symmetry breaking in a phase transition. The present method is complementary to de-facto standard Wannier tight-binding modeling, and it provides us with a fundamental basis to develop a symmetry-based analysis for materials science., Comment: 15 pages, 8 figures in main text + supplemental material

Microscopic origin of chirality and possible electric-field induced rotation and rotation-field induced electric polarization are investigated. By building up a realistic tight-binding model for elemental Te crystal in terms of symmetry-adopted basis, we identify the microscopic origin of the chirality and essential couplings among polar and axial vectors with the same time-reversal properties. Based on this microscopic model, we elucidate quantitatively that the inter-band process, driven by the nearest-neighbor spin-dependent imaginary hopping, is the key factor in the electric-field induced rotation and its inverse response. From the symmetry point of view, these couplings are characteristic common to any chiral material, leading to a possible experimental approach to achieve absolute enantioselection by simultaneously applied electric and rotation fields, or magnetic field and electric current, and so on, as a conjugate field of the chirality., Comment: 5figures

An electric ferro-axial moment, which is characterized by a nonzero expectation value of a time-reversal-even axial vector, exhibits distinct spatial-inversion and time-reversal properties from conventional ferroelectric, ferromagnetic, and ferro-magnetoelectric orders. Nevertheless, physical properties characteristic of the electric ferro-axial moment have been obscure owing to the absence of its conjugate electromagnetic fields. We theoretically investigate consequences of the presence of the ferro-axial moment on the basis of the symmetry and microscopic model analyses. We show that atomic-scale electric toroidal multipoles are the heart of the ferro-axial moment, which act as a nanometric rotator against external stimuli. Furthermore, we propose an intrinsic generation of a spin current parallel to an applied electric field in both metals and insulators. Our results not only provide a deep microscopic understanding of the role of the ferro-axial moment but also stimulate a new development for functional materials with use of the electric toroidal moment., Comment: 5 pages, 4 figures, 1table

A magnetic toroidal moment is a fundamental electronic degree of freedom in the absence of both spatial inversion and time-reversal symmetries and gives rise to novel multiferroic and transport properties. We elucidate essential model parameters of the nonlinear transport in the space-time ($\mathcal{PT}$) symmetric collinear antiferromagnetic metals accompanying a magnetic toroidal moment. By analyzing the longitudinal and transverse components of the second-order nonlinear conductivity on a two-dimensionally stacked zigzag chain based on the nonlinear Kubo formula, we show that an effective coupling between the magnetic toroidal moment and the antisymmetric spin-orbit interaction is an essential source of the nonlinear conductivity. Moreover, we find that the nonreciprocal longitudinal current and nonlinear transverse current in a multi-band system are largely enhanced just below the transition temperature of the antiferromagnetic ordering. We also discuss the relevance of the nonlinear conductivity to the linear magnetoelectric coefficient and conductivity. Our result serves as a guide for exploring microscopic essence and clarifying the parameter dependence of the nonlinear conductive phenomena in ferrotoroidal metals., Comment: 11 pages, 5 figures

We propose a systematic analysis method for identifying essential parameters in various linear and nonlinear response tensors without which they vanish. By using the Keldysh formalism and the Chebyshev polynomial expansion method, the response tensors are decomposed into the model-independent and dependent parts, in which the latter is utilized to extract the essential parameters. An application of the method is demonstrated by analyzing the nonlinear Hall effect in the ferroelectric SnTe monolayer for example. It is shown that in this example the second-neighbor hopping is essential for the nonlinear Hall effect whereas the spin-orbit coupling is unnecessary. Moreover, by analyzing terms contributing to the essential parameters in the lowest order, the appearance of the nonlinear Hall effect can be interpreted by the subsequent two processes: the orbital magneto-current effect and the linear anomalous Hall effect by the induced orbital magnetization. In this way, the present method provides a microscopic picture of responses. By combining with computational analysis, it stimulates further discoveries of anomalous responses by filling in a missing link among hidden degrees of freedom in a wide variety of materials., Comment: 23 pages, 10 figures

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., Comment: 10 pages, 4 figures, 3 tables, plus 27 pages supplemental materials (some corrections are made)

We investigate the nature of the time-reversal breaking pairing state in the hole-doped monolayer MoS$_{2}$ on the basis of the realistic three-orbital attractive Hubbard-like model with the atomic spin-orbit coupling. Due to the multi-band features arising from the Mo $d$ orbitals in the noncentrosymmetric crystal structure, the Lifshitz transition takes place upon hole doping. Across the Lifshitz transition point, the sign of the relative phase between the Cooper-pair components drastically changes, leading to the emergence of the time-reversal breaking phase with complex gap functions. It is shown that this intriguing pairing state is characterized by the finite momentum-space distributions of the orbital and spin angular momentum with three-fold rotational symmetry on the Fermi-surface pockets around K and K$'$ points. The present mechanism for the time-reversal breaking superconductivity can ubiquitously be applied to spin-orbit-coupled metals in noncentrosymmetric crystal structures., Comment: 5 pages, 3 figures

We theoretically investigate the Cooper-pair symmetry to be realized in hole-doped monolayer MoS$_2$ by solving linearized BCS gap equations on the three-orbital attractive Hubbard-like model in the presence of the atomic spin-orbit coupling. In hole-doped monolayer MoS$_2$, both spin-orbit coupling and the multi-orbital effects are more prominent than those of electron-doped system. Near the valence band edge, the Fermi surfaces are composed of three different types of hole pockets, namely, one mainly consisting of the almost spin-degenerate $|d_{z^{2}}\rangle$ orbital near $\Gamma$ point, and the others of the spin-split upper and lower bands near ${\rm K}$ and ${\rm K}'$ points arising from the $|d_{x^{2}-y^{2}}\rangle$ and $|d_{xy}\rangle$ orbitals. The number of relevant Fermi pockets increases with increase of the doping. At very low doping, the upper split bands of $|d_{x^{2}-y^{2}}\rangle$ and $|d_{xy}\rangle$ are concerned, yielding extremely low $T_{\rm c}$ due to small density of states of the split bands. For further doping, the conventional spin-singlet state (SS) appears in the $\Gamma$ pocket, which has a mixture of the spin-triplet (orbital-singlet) (ST-OS) and spin-singlet (orbital-triplet) (SS-OT) states in the K and K$'$ pockets. The ratio of the mixture depends on the relative strength of the interactions, and the sign of the exchange interactions. Moderately strong ferromagnetic exchange interactions even lead to the pairing state with the dominant ST-OS state over the conventional SS one. With these observations, we expect that the fascinating pairing with relatively high $T_{\rm c}$ emerges at high doping that involves all the three Fermi pockets., Comment: 8 pages, 4 figures