Hindawi Advances in Mathematical Physics Volume 2019, Article ID 4873914, 10 pages https://doi.org/10.1155/2019/4873914 Research Article Induced Representation Method in the Theory of Electron Structure and Superconductivity V. G. Yarzhemsky 1,2 1 Kurnakov Institute of General and Inorganic Chemistry, 31 Leninsky Prospect, 119991 Moscow, Russia 2Moscow Institute of Physics and Technology, Dolgoprudny, 9 Institutskiy Lane, 141700 Moscow, Russia Correspondence should be addressed to V. G. Yarzhemsky; [email protected] Received 30 November 2018; Accepted 11 March 2019; Published 9 April 2019 Academic Editor: Francesco Toppan Copyright © 2019 V. G. Yarzhemsky. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 defned by additional quantum numbers. Te 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. 1. Introduction useofthespacegroupapproachforUPt3 and some other superconductors [17–21] confrmed this statement. It was Induced representation method is a powerful group- also shown that transitivity of induction theorem results theoretical technique for the systems with subgroups of in additional quantum numbers for nanoclusters [22] and symmetry, such as symmetrical clusters and crystals. Tere Cooper pairs [23–25]. are two essential subgroups, namely, local symmetry group of In the present work, the advantages of induced represen- an atom in a cluster or crystal and the wavevector group (little tation method for normal vibrations and SALCs construction group) in a BZ (Brillouin zone). Te induced representation and for the investigations of Cooper pair symmetry in topo- method and its application to crystals and molecules have logical superconductors are reviewed and some calculations been described in two classical books of Bradley and for topological superconductors are performed. Te pair Cracknell [1] and Altman [2]. On the basis of this theory, it nodal structure dependence on additional quantum number was recognized that the standard technique for construction of IR (irreducible representation) �� of �4ℎ group in �-� form of normal vibrations [3–6] and SALCs (symmetry adapted is investigated. linear combinations) in clusters [6, 7] can be signifcantly simplifed [8–11]. Te Mackey theorem on symmetrized 2. Induced Representations squares [12] and its new form for solid-state wavefunctions of Bradley and Davis [13] made the application of Pauli Consider a fnite group � and its subgroup �.Tewhole exclusion principle to solid-state wavefunctions possible, symmetry group is decomposed into lef cosets with respect easy, and straightforward [14]. It follows from the space- to �: group approach to the wavefunction of a Cooper pair [14, 15] � that generally accepted direct relations between multiplicity �=∑���. (1) and parity of a Cooper pair [16] are violated on symmetry axis �=1 in a BZ and on surfaces of a BZ in the case of nonsymmorphic Tere are two physical cases in which this decomposition space groups. Calculation of Cooper pair functions making is of great use. Te atoms in a symmetric nanostructure are 2 Advances in Mathematical Physics � located at points with local symmetry ,whichmaycoincide Ｔ1 with the full group �, or with any of its subgroup, including a subgroup consisting of an identity element only. Te action Ｒ1 Ｓ of the lef coset representatives in (1) on initial atom results in 1 an orbit of this atom. Te second case is the band theory of z solids [1, 2], where subgroup � is the wavevector group. Te action of the lef coset representatives �� on the wavevector Ｒ4 Ｓ � results in a star {�} of this vector. Te number � of atoms, 2 Ｓ4 Ｒ2 forming the orbit of an atom and the number n of prongs in y x a star of the wavevector, is given by the relation Ｔ Ｔ4 2 |�| Ｒ3 �= , |�| (2) Ｓ3 wherethemodulussignstandsforthenumberofele- Ｔ3 ments of a group. � Consider IR � of subgroup � and its basis set {��,�= Figure 1: Basis set for vibrations in tetrahedral molecule. ��, ��,and � �� are atomic displacements. 1, ...��},where�� stands for the dimension of � . According to the theorem on induced representation [1, 2], a set of functions ���� (� = 1...�, � = 1,...��,) is invariant in the � �� ↑ group and forms the basis of induced representation the equilibrium position are the components of the vector. � ,givenbytheformula Under the operations of the subgroup �,thesecoordinates � � � −1 −1 are transformed by IRs � of this subgroup. Te set of 3� (� ↑�)(�)��,�] =� (�� ���) �(�� ���,�), (3) �] displacements of � atoms forming an orbit of atom A1 must be transformed by the representation (reducible) of the entire where group �. Te displacements of other atoms are obtained by −1 1, � �� ∈� the action of lef coset representatives �� (see formula (1)) on −1 { if � � �(�� ���,�)={ (4) the basis set of the frst atom. Let the atom A1 be transformed 0, �−1�� ∉�. { if � � into the atom A� by the element ��,andthesameelement transforms the basis {�1,�1,�1} into basis {��,��,��}.Ten, In a general case, the induced representation (3) is � under the action of the elements of the symmetry group reducible in � and is decomposed into the sum of its IRs Γ −1 of this atom, �� =����� ,thedisplacements{��,��,��} of � as follows: � are transformed by the conjugate representation �� (ℎ�)= �� ↑�=∑� Γ�, ��(�−1ℎ � ) � (5) � � � of this group. Tis means that under the action � the whole symmetry group G basis set of displacements of an � orbit is transformed as a basis of the induced representation where �� stand for frequencies of appearance of IRs Γ . � � ↑�. Note that its matrix depends on the choice of lef For the decomposition of induced representation (3), one coset representatives, but its decomposition into IRs of � can apply Frobenius reciprocity theorem [27], according to does not. To calculate the set of IRs in the decomposition which the frequency �� of IR Γ� of the whole group in the � of the whole matrix of displacements, one does not need to decomposition of induced representation � ↑�equals the � construct it. Tis decomposition can be done making use frequency of appearance of IR � in the decomposition of IR � of the Frobenius reciprocity theorem (i.e., by summation of Γ subduced to �: characters on the subgroup in formula (7)). � � � � Consider CH4 molecule of �� symmetry. Te basis set �(Γ |� ↑�)=�(� |Γ ↓�). (6) of atomic displacements is shown in Figure 1. Te frst atom � Tus, according to Frobenius reciprocity theorem, the H1 is in -axis and directions of the axis on it are the same as those on central atom. Te basis sets on other atoms are values of �� in the right-hand side of (5) are given by the formula obtained by C2 rotations about corresponding axis. Local symmetry group of atom H1 is �3V. Te stretching vibrations 1 �(Γ� |�� ↑�)= ∑ �(�� (ℎ))�∗ (Γ� (ℎ)), (�-component) of atom A1 displacements belong to IR A1 of � (7) | | ℎ∈� �3V and bending vibrations (�-and�-components) belong to IR � of �3V. Te decomposition of the induced represen- where � stand for characters of representations. tations can be easily done by Frobenius reciprocity theorem (7) and making use of characters of �3V and of T� on �3V, 3. Molecular Vibrations and SALCs presented in Table 1. Making use of formula (7), one obtains that �1 ↑�� =�1 +�2 and �↑�� =�+�1 +�2.Toobtainall Consider normal vibration construction for symmetrical vibration modes, one should add vibrations of central atom, molecules. Te displacements {�1,�1,�1} of an atom A1 from which belong to IR T2,andsubtracttherepresentationof Advances in Mathematical Physics 3 Table 1: Part of charactertable of group �� on its subgroup �3V (top �4V of the frst atom. Tese orbitals are called �-and�- � � � part) and characters of IRs 1 and of group 3V.(bottompart). orbitals, respectively. Under the action of the elements of �ℎ group, these functions are transformed independently Group/subgroup IR Element (number in class)/character by induced representations �1 ↑�ℎ and �↑�ℎ,whose �(1) � (2) � (3) 3 V dimensions are 6 and 12, respectively. Te characters of these �1 111induced representations are the same as those constructed � � �2 11-1for -and -orbitals in Oℎ symmetry explicitly [7]. It follows � � � 2-10from Frobenius reciprocity theorem that there is no need to construct these characters and it is sufcient to check the �1 30-1 orthogonality of characters of IRs A1 and E of group C4� with �2 301 characters of IRs of the whole group �ℎ. Tese characters are � 1 111presented in Table 2. It is seen from Table 2 that the character �3V � 2-10of IR A1 of �4V group is not orthogonal to the characters of � � � � � ↑� = IRs 1�, �,and 1u of ℎ.Tus,weobtainthat 1 ℎ � +� +� � 1� � 1u. Similar analysis for IR E of 4V results in pz1 the fact that �-orbitals belong to IRs �1�, �1�, �2�,and�2�. Tese sets of orbitals are the same as those obtained for �- py1 and �-orbitals, making use of explicit construction of six- and � px1 twelve- dimensional matrix [7]. We see that IR 1� appears pz5 twice in the whole basis set constructed from p-orbitals of px5 one orbit of atoms. Tese basis sets are labeled by physical � px3 quantum number (i.e., IR A1 or E of local group 4V). In py6 z py5 the case when the atoms in symmetrical clusters are on the p z6 pz3 planes of symmetry, it is possible to distinguish basis sets of y p repeatingIRsbytheproperchoiceofintermediategroup[22]. y2 x py3 px6 Tis technique is useful for Cooper pairs [23] and will be considered in the next section.