We derive the effective charge and current density operators for the strong-coupling limit of a single-band Mott insulator in the presence of spin-orbit coupling and show that the spin-orbit contribution to the effective charge density leads to novel mechanisms for multiferroic behavior. In some sense, these mechanisms are the electronic counterpart of the ionic-based mechanisms, which are have been proposed for explaining the electric polarization induced by spiral spin orderings. The new electronic mechanisms are illustrated by considering cycloidal and proper screw magnetic orderings on saw-tooth and Kagome lattices. As for the isotropic case, geometric frustration is crucial for achieving this purely electronic coupling between spin and charge degrees of freedom.

In this paper we perform a systematic literature review of the diverse and somewhat fragmented current state of research on firms’ internationalization and innovation. We analyze 207 key works from 1989 through 2020 and synthesize them into an internationalization process framework that conceptually maps key internationalization-related antecedents and moderators that influence innovation behaviors and outcomes. Through an internationalization process framework, we categorize existing relevant studies into three key stages: (a) the pre-internationalization stage, (b) the internationalization entry stage, and (c) the post-internationalization stage. Furthermore, we review how firms’ various strategic decisions and operations in different stages influence their innovations by elaborating the moderating role of external country/region institutions and firm internal characteristics. Building on this review, we provide suggestions for future research to advance the developments of this domain.

Proposing a general formulism in terms of local coordinates representing the tilting of the ligands' octahedra, we evaluate the electric polarization in a chain of transition metal ions with unpolar octahedron rotation. We find the orbital ordering produced by the ligands's rotation and the spin order, together, determines the polarization features, manifesting that nonvannishing polarization appears in collinear spin order and the direction of polarization is no more restricted in the plane of spin rotation in cycloidal ordering. PACS numbers: 75.85.+t, 75.10.Pq, 75.30.Gw, Multiferroics refers to materials in which the spontaneous ferroelectricity and magnetism coexist, which makes it possible to control magnetization through electric field or vice versa [1][2][3][4][5][6]. Since the discovery of spindriven multiferroics TbMnO 3 [7] and TbMn 2 O 5 [8], it has absorbed more and more attentions. In spiral magnetic insulator, a spin-current mechanism [9] which is also referred as "inverse Dzyaloshinskii-Moriya (DM) interaction" [10,11] predicts that the electric polarization is closely related to the vector spin chirality. This mechanism fulfills in spiral multiferroics for systems with high symmetry. Moreover in collinear spin order system the ferroelectricity is induced by inverse GoodenoughKanamori mechanism where the exchange interaction is responsible for the presence of ferroelectricity in Ising chain magnet [12] and E-type spin order [13][14][15]. However in E-type materials, the magnitude of electric polarization measured in experiments [16,17] is about two order of magnitude smaller than some other theoretical predictions [13][14][15].We notice in orthorhombic(o) o-RMnO 3 (R = Ho, Er, Tm, Yb, and Lu) with E-type spin order that the tilting of the ligands' octahedra of transition metal ions always takes place. In recently found ferroaxials [18][19][20], both the spin order and special crystal structure are responsible for the inversion symmetry broken. In CaMn 7 O 12 "propeller-like" structure for which the tilted ligands' octahedra take shape, the axial vector due to the relative rotation between nearby octahedra (we simply call octahedron rotation hereafter) is essential in the inversion symmetry breaking [21]. There has been a phenomenological explanation [19,21] for the relationships between the electric polarization and the axial vector as well as spin orders, but there is not yet a microscopic mechanism to clarify the role that such an octahedron rotation plays in conventional spin-driven multiferroics. It is therefore obligatory, from a microscopic point of view, to elucidate the effect of octahedron rotation on spin induced ferroelectricity.In this letter we investigate how does an occurrence of the unpolar octahedron rotation affect the final electric polarization in a class of multiferroics. Starting from microscopic model, we derive an explicit dependence of the electric polarization on the spin order and the configurations of octahedron rotation, then discuss the special features by comparing w...