Abstract
It is shown that the application of theorems of induced representations method, namely, Frobenius reciprocity theorem, transitivity of induction theorem, and Mackey theorem on symmetrized squares, makes simplifying standard techniques in the theory of electron structure and constructing Cooper pair wavefunctions on the basis of one-electron solid-state wavefunctions possible. It is proved that the nodal structure of topological superconductors in the case of multidimensional irreducible representations is defined by additional quantum numbers. The technique is extended on projective representations in the case of nonsymmorphic space groups and examples of applications for topological superconductors UPt3 and Sr2RuO4 are considered.
Highlights
Induced representation method is a powerful grouptheoretical technique for the systems with subgroups of symmetry, such as symmetrical clusters and crystals
It follows from the spacegroup approach to the wavefunction of a Cooper pair [14, 15] that generally accepted direct relations between multiplicity and parity of a Cooper pair [16] are violated on symmetry axis in a BZ and on surfaces of a BZ in the case of nonsymmorphic space groups
Since we investigated products of Eu basis set by all one-dimensional IRs, our results include symmetry of OSP case and confirm that vertical lines of nodes are possible for OSP pairs of Eu symmetry
Summary
Induced representation method is a powerful grouptheoretical technique for the systems with subgroups of symmetry, such as symmetrical clusters and crystals. The induced representation method and its application to crystals and molecules have been described in two classical books of Bradley and Cracknell [1] and Altman [2]. Calculation of Cooper pair functions making use of the space group approach for UPt3 and some other superconductors [17,18,19,20,21] confirmed this statement. The advantages of induced representation method for normal vibrations and SALCs construction and for the investigations of Cooper pair symmetry in topological superconductors are reviewed and some calculations for topological superconductors are performed. The pair nodal structure dependence on additional quantum number of IR (irreducible representation) Eu of D4h group in x-y form is investigated
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