Magnetic Groups and Their Corepresentations

Abstract

A review is given of the theory of magnetic groups and of their unitary corepresentations. Particular application is made to magnetic space groups, this part of the work being set in the framework of little-group theory. The symmetry problems in physics which lead to magnetic groups are analyzed and various applications of the theory to such problems are pointed out. Finally a method is given for obtaining the Kronecker products of corepresentations of magnetic groups, and an example is presented in which the unitary subgroup of the magnetic group is the point group C2,.

Topological Insulators from Group Cohomology

“…The arguments in H 2 are defined as follows. (a) The first argument, G ∘ , is a magnetic point group [36] consisting of those space-time symmetries that (i) preserve a spatial point and (ii) map the circle cðk ∥ Þ to itself. For d ¼ 3, the possible magnetic point groups comprise the 32 classical point groups [37] without time reversal (T), 32 classic point groups with T, and 58 groups in which T occurs only in combination with other operations and not by itself.…”

“…In this section, we identify the relevant W symmetries and show their corresponding group (G π;k z ) to be an extension of the ordinary group (G ∘ ) by quasimomentum translations, where G ∘ corresponds purely to space-time transformations; the inequivalent extensions are classified by the second cohomology group, which we also introduce here. In crystals, G ∘ would be a magnetic point group [36] …”

Topological magnetic crystalline insulators and corepresentation theory

“…For example, in a TR invariant system, the double degeneracy due to Kramers' theorem of TR symmetry is regarded as an "additional degeneracy" that is not included in the conventional representation theory [38]. In order to understand this "additional degeneracy", Wigner first developed the so-called "co-representation" theory [43,44]. The co-representation is different from the conventional representation because the multiplication rule is modified due to anti-unitary operators.…”

“…To answer this question, we develop a theory for topological phases in magnetic crystals, dubbed "topological magnetic crystalline insulators (TMCIs)", based on the "co-representation" theory of magnetic groups in this paper [38,43]. It is well-known that the degeneracy of energy states is related to irreducible representations of a crystalline symmetry group in the representation theory.…”

Second-Order Phase Transitions in Magnetic Ca3Al2(SiO4)(Oh10) and Non Magnetic Cubic ZnO

“…In order to determine the phase transitions, we calculated the coreps using Wigner [5] and Bradley-Davies [6] theories. We listed them in Table V. Consequently, using Luybarskii [7] and our method we have found particular active coreps of phonons which bring the crystals to lower magnetic symmetries listed in Table VI.…”