Broken relativistic symmetry groups, toroidal moments and superconductivity in magnetoelectric crystals

Abstract

A connection between creation of toroidal moments and breaking of the relativistic crystalline group associated to a given crystal, is presented in this paper. Indeed, if magnetoelectric effects exist, the interaction between electrons and elementary magnetic cells appears in such a way that the resulting local polarization and magnetization break the local relativistic crystalline symmetry. Therefore, Goldstone bosons, also associated to toroidal moments, are created and, as a consequence, corresponding toroidal phases in crystals. The list of the Shubnikov groups compatible with this kind of phases is given and possible consequences in superconductor theory in magnetoelectric crystals are examined.Key Words: toroidal moments, relativistic crystalline symmetries, symmetry breaking, anyons.PACS 1990: 05.30.L -Anyons and parastatistics. 61.50.E -Crystal symmetry, models and space groups, crystalline systems and classes. .80 -Magnetomechanical and magnetoelectric effects, magnetostriction. * E-mail: jacques.rubin@inln.cnrs.fr -IntroductionThe aim of this paper is to present a possible link between magnetoelectric effects and toroidal moments via the concept of relativistic crystalline groups [1,2,3,4,5] and to see how this link can be applied to derive a possible origin of superconductivity in electric and magnetic crystals. These crystals are characterized by their magnetic groups which are subgroups of the Shubnikov group O(3)1 ′ . Among the 122 magnetic groups, only 106 are compatible with the existence of a linear or quadratic magnetoelectric effect [6]. Let us recall that, in these groups, the time inversion 1 ′ appears in addition to or in combination with orthogonal transformations of the Euclidean space.In the present paper, we will consider relativistic symmetry group theory in crystals. Therefore, we need an extension from the Shubnikov group O(3)1 ′ to the group O(1, 3) in the Minkowski space. More particularly, our attention will be devoted to transformations of O(1, 3) leaving invariant polarization and magnetization vectors and generating a subgroup of the relativistic point group associated to the magnetic group G of a given crystal, namely the normalizerThis subgroup may not be identified to the magnetic group if G leaves invariant a particular non-zero velocity vector, i.e., if G ′ and G ⊆ G ′ are isotropy groups of O(1, 3). If such a vec-1 tor exists and G ′ = G, one strictly speaks about the relativistic crystalline symmetry G ′ . We can represent, as in figure 1, the list of the magnetic groups for which one can have in a crystal a spontaneous magnetization, a spontaneous polarization or an invariant non-zero velocity vector [15]. Thus, as it can be seen in the figure 1, only 31 magnetic groups are compatible with the existence of a relativistic crystalline symmetry (see the groups contained into the lowest circle). The invariant non-zero velocity vectors can be linked with toroidal moments from the point of view of magnetic symmetries as it has been already shown in previous papers [7,...

Magnetic and resonant X-ray scattering investigations of strongly correlated electron systems

“…In particular, a possible link between the magnetoelectric effects and toroidal moments in Shubnikov group can be applied to derive a possible origin of superconductivity in electric and magnetic crystals [33].…”

Relativistic crystalline symmetry breaking and anyonic states in magnetoelectric superconductors

“…In the present paper, we consider relativistic symmetry group theory in crystals [11]. Therefore, we need, first, an extension from the Shubnikov group O(3)1 ′ to the group O(1, 3) in the Minkowski space, and second and more particularly, transformations of O(1, 3) leaving invariant polarization and magnetization vectors, and generating a subgroup of the normalizer N (G) of G in O (1,3). This subgroup G ′ may not be identified with the magnetic group if G leaves invariant a particular non-vanishing velocity vector.…”

“…Then, to finish with this description, we present the list of magnetic groups compatible with such process of creation of toroidal moments [1] (let us remark that among the 16 compatible groups tabulated in [1], only 12 are associated to a non-trivial O(2) action of the normalizer; that is why four of them, namely the groups 2, 3, 4, 6, are not indicated in the list below):…”

“…

There exists a connection between the creation of toroidal moments (TM) and the breaking of the one-cell relativistic crystalline symmetry (RCS) associated to any given crystal [1] into which non-trivial magnetoelectric coupling effects (ME) exist [2] . Indeed, in this kind of crystals, any interaction between a charge carrier and an elementary magnetic cell can breaks the RCS of this previous given cell by varying, in the simplest case, the continuous defining parameters of the initial RCS.This breaking can be associated to a change of the initial Galilean proper frame of any given carrier to an "effective" one, into which the RCS of the interacting cell is kept.

…”

“…We can speak of a kind of "inverse" kineto-magnetoelectric effect [3]. The magnetic groups compatible with such process have been computed [1]. Moreover, one can notice that the TM's break the P and T symmetries but not the PT one as in anyons theories [4,5].…”