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. px2

pz2 px4 4. Additional Quantum Numbers for py4 Induced Representations

pz4 When the atoms are on the planes of symmetry, SALCs contain repeating IRs [22]. Tolabel repeating IRs, it is possible Figure 2: Basis set for SALC of atomic orbitals in octahedral to use the transitivity property of induction, which means that molecule. the two ways of induction, from � directly into � and via an intermediate subgroup �, result in the same IRs: � � � ↑�∝(� ↑�)↑�. (8) the whole molecule displacements T2 and rotations T1 (see [6]). Tus, one obtains total set of vibration modes of CH4 Te decomposition of induced representation in the lef- � molecule �1 +�+2�2. Hence, it follows that as opposed to hand side of (8) into IRs of is written as follows: standard methods [4–6], where a multidimensional character � � � � ↑�=∑�� Γ . of total vibration representation is constructed explicitly, the � (9) induced representation method makes it possible to obtain the same results by the character’s analysis on local subgroup Consider an alternative route of induction. Afer inducing into the intermediate subgroup �,wemaydecomposethe only. � Te method of construction of SALCs of atomic wave- result into IRs � of �: functions in molecules is very similar to the above discussed �� ↑�=∑����. � (10) technique. Consider �-orbitals in octahedral complex, in � which six atoms are in coordinate directions (see Figure 2). Directions of the axis of the frst atom in the six-atom orbit Similar decomposition is carried out afer inducing each �� � are the same as those of central atom. Te axes of the second one of IRs into : and third centers are obtained by counterclockwise rotation �� ↑�=∑��Γ�. ∘ ∘ � (11) by 120 and 240 , respectively, about axis (111) and the rest � of basis set is obtained by the space inversion. To convert basis functions on the centers 4, 5, and 6 into standard form, According to the transitivity of induction theorem (8), the frequencies are connected by the relation one can rename �� and �� basis functions, but it is not essential in our general consideration. Functions ��1 and �� =∑����. � � � (12) {��1,��1} belong to IRs �1 and �, respectively, of local group � 4 Advances in Mathematical Physics

Table 2: Characters of group �ℎ on its subgroup �4V (top part, characters of �-IRs are shown in parenthesis, when they difer from �-IRs) and characters of IRs � 1 and � of group �4V.(bottompart).

Group/subgroup IR Element (number in class)/character

�(1) �2�(1) �4�(2) �V(2) �V� (2)

��(�) 11 11(-1)1(-1)

� 2�(�) 1 1 -1 1 (-1) -1 (1) � ℎ ��(�) 22 02(-2)0

�1�(�) 3-1 1-1(1) -1 (1)

�2(�) 3-1 -1-1(1) 1(-1)

�1 11 1 1 1 �4V � 2-2 0 0 0

If frequencies in two-step induction in (10) and (11) do where �∈��. For each self-inverse double coset (i.e., −1 � not exceed unity, every repeating IR in (9) acquires unique ����=��� �), there are two extensions of �� into � ∗ additional quantum number — index of IR of intermediate extended intersection subgroup �� =�� +����,whose subgroup. Tis technique will be used in the next sections for characters are as follows: additional quantum numbers of Cooper pairs. �+ � �(�� (���)) = � (� (������)) (16) 5. Space-Group Approach to the Wavefunction �− � �(�� (���)) = −� (� (������)) , (17) of a Cooper Pair where �∈��. Ten, the symmetrized and antisym- Following Ginzburg and Landau [28], we consider SOP metrized parts of the Kronecker square of the IR of the space (superconducting order parameter) to be identical with a group are given by the two following formulae, respectively: Cooper pair wavefunction. Hereby, we can take advantage � � � � �+ of general quantum mechanical rule for construction of [(� ↑�)×(� ↑�)]=[� ×� ]↑�+∑�� � two-electron state. Tus, the wavefunction of two-electron (18) state is constructed as a direct (Kronecker) product of � ↑�+∑�� ↑� one-electron wavefunctions. If the electrons are equivalent, � the requirement of antisymmetry of the total wavefunction � � � � �− makes partial reduction of the total two-electron space into {(� ↑�)×(� ↑�)}={� ×� }↑�+∑�� singlet and triplet subspaces possible. Tus, making use of � (19) the Pauli exclusion principle, we obtain that spatial part of � ↑�+∑�� ↑�. a singlet (triplet) pair belongs to symmetrized (antisym- � metrized) Kronecker square of one-electron wavefunction. Irreducible representations of space groups are induced from Te frst items on the right-hand sides of (18) and (19) wavevector group [1–3, 26] and are defned by formula (3). correspond to the double coset defned by the identity ele- Te method of the decomposition of Kronecker squares of ment. Te summations in the second items in the right-hand induced representations of fnite group into symmetrized and side of (18) and (19) run over all self-inverse double cosets. antisymmetrized parts was developed by Mackey [12] and Te last summations in the right-hand side of (18) and (19) �� � ̸= applied to space groups by Bradley and Davies [13]. correspond to non-self-inverse double cosets (i.e., � −1 TestructureoftheKroneckersquareofIRsofaspace ��� �). According to the Pauli exclusion principle, the group depends on the structure of the space group � relative symmetrized Kronecker square (equation (18)) defnes the to wavevector group �.Tedoublecosetdecompositionof� spatial part of the singlet state (antisymmetrized with respect relative to � is written as to spin coordinates) and the antisymmetrized Kronecker �=∑�� �. square (equation (19)) defnes the spatial part of the triplet � (13) state, which is symmetrical with respect to spin coordinates. � It follows from the Mackey-Bradley theorem that possible For each double coset representative in (13), the intersection symmetries of singlet and triplet states are diferent for subgroup is considered: identity and self-inverse double cosets and are the same for non-self-inverse double cosets. Since total momentum of a −1 �� =����� ∩�. (14) Cooper pair equals zero, the following relation should be fulflled for the k-vector and double coset representative: Te representation of group �� is defned by the following �→ �→ �→ formula: �+�� �=�, (20) �± � � −1 �→ �� (�) =� (�) ×� (�� ���), (15) where � is a vector of reciprocal lattice. Advances in Mathematical Physics 5

1 1 1 It is clear from (20) that if � is inside the BZ, one can take state consists of three components: �0, �1,and�−1,wherethe the space inversion � or �2 rotation about the axis normal subscript denotes spin projection on �-axis. In the commonly �→ �→ 1 � �=�/2 accepted notation, �̂ stands for �0 and �̂ and �̂ are linear to .AtthesurfaceofaBZwhere ,doublecoset 1 1 1 defned by identity element also results =in two-electron combinations of �1 and �−1.Inthe�0 component, the spins state with zero total momentum. Pairing of electrons with of electron in a pair are opposite and this state is called �1 �1 equal moments was proposed for pseudogap states in cuprate OSP (opposite spin pairing). In the 1 and −1 components, superconductors and was called Amperean pairing [29]. Tus the electron spins in a pair are equal and these states (or double coset defned by identity element corresponds to their linear combinations �̂ and �̂) are called ESP (equal spin Amperean pairing. pairing). In the case of D4ℎ symmetry, OSP spin belongs to For �, a general point of a Brillouin zone relation (20) is IR �2� and ESP total spin belongs to IR ��.Teoremonthe � fulflled for double coset defned by the space inversion and direct product of IR Γ and induced representation � ↑� �∗ � group � consists of the identity element and the space [1, 2] is written as inversion I only. It is seen from formulas (15)-(17) that in � � this case symmetrized square belongs to even IR of group Γ×(� ↑�)=[(Γ↓�) ×� ]↑�. (21) �� and antisymmetrized square belongs to odd IR of ��.Te decomposition of corresponding induced representations can Hence, it follows that the symmetry of OSP pair on the (� ↓� )×� be easily done making use of Frobenius reciprocity theorem plane is 2� 2ℎ � and the symmetry of ESP pair (� ↓� )×� (formula (7)). Tus, one obtains that each even (odd) IR of on the plane is � 2ℎ �. Tese characters are also the whole group appears in symmetrized (antisymmetrized) presented in Table 3. Since these IRs are induced into the square with the frequency equal to its dimension. whole group (see formulas (18) and (19)), the fnal results Te space-group approach is an alternative to phe- may be easily obtained making use of Frobenius reciprocity � nomenological or point group approach [30–33] in which the theorem. Te absence of any IR of 2ℎ in the right column pair function is expressed in terms of spherical functions, results in the absence of some IRs of the whole group for the similar to spherical harmonics in crystal feld theory [7]. In k-vector on this plane. Te intersection of symmetry plane point group approach, spatial part of a singlet pair is even and with Fermi surface results in a line. Tus, the absence of any spatial part of triplet pair is odd [16, 30, 31]. It can be easily IR on the plane results in a symmetry-protected line of nodes. shown that this relation is violated for two-dimensional small Some symmetry protected lines of nodes can be identifed in IRs on the lines of symmetry [14, 15, 21]. Indeed, it follows a singlet case and for spatial part in a triplet case making use from formulas (15)-(17) that for two-dimensional small IRs of Table 3 and Frobenius reciprocity theorem. It is seen from �± �± Table 3 that if in a triplet case both ESP and OSP pairs are �[�� (�)] = 4 and �[�� (�)] = ±2. Hence, it follows that �+ �− taken into account, all odd IRs of �2ℎ,namely,�� and ��,are neither � is decomposed into even IRs only nor � is � � possible on the planes and there are no symmetry-protected decomposed into odd IRs only. It is also clear from formulas lines of nodes. Tis conclusion is in agreement with Blount (15)-(17) than on lines of symmetry the set of possible IRs theorem, according to which in a triplet case, symmetry- for a pair depends on the IR of one-electron state. Te direct protected lines of nodes are “vanishingly improbable” [32]. relation between multiplicity and parity is also violated for On the other hand, in axial symmetry, OSP and ESP pairs due nonsymmorphic space groups on the surface of a BZ [14, 18– to interaction with crystal feld will have diferent energies 20]. and only one type of pairing, namely, ESP or OSP, will take It follows from the Blount theorem [32] that there are place. In the case of Sr2RuO4, it is assumed that pairing is no symmetry requirements for lines of nodes in triplet case. of OSP type [34–36]. Hence, it follows from Table 3 that if Also phenomenological functions for �4ℎ group in a triplet only one type of pairing (ESP or OSP) takes place, symmetry- case are nodeless [33]. It was pointed out [34–36] that a protected lines of nodes are possible. contradiction exists between experimental lines of nodes in In order to consider two-dimensional IRs �� and �� in Sr2RuO4 and nodeless model functions [33]. It will be shown more detail, their characters on symmetry groups of planes in the next section that nodal and nodeless basis sets for two- are also presented in Table 3. It is immediately verifed from dimensional IRs can be distinguished by additional quantum the data of Table 3 that in the basal plane (001) IR �� is numbers. completely forbidden for singlet pairs and �� appears twice for spatial part of triplet pair. Hence, it follows that on basal 6. Nodes on Symmetry Planes plane in a singlet case there is a symmetry-protected line �� �� ∗ of nodes of IR .Inverticalplanes,IR appears ones In the symmetry plane, the group �� of a pair is �2ℎ and it in a triplet case and IR �� appears ones in a singlet case. follows from formulas (15)-(17) that spatial part of a singlet Note that according to Mackey-Bradley theorem, at general pair belongs to IR �� and that of triplet pair belongs to IR �� point of a BZ two-dimensional, IRs appear twice in the of �2ℎ. decomposition of the complete basis set. Tus, on vertical In L-S coupling scheme, total pair wavefunction is a direct planes, one two-dimensional IR �� is permitted and the other product of its spatial and spin parts. Symmetrized Kronecker IR �� is forbidden. Hence, it follows that such lines of nodes 0 square of spatial part is multiplied by singlet spin function � , are not symmetry protected in terms of IRs of the whole which belongs to IR �1�, and the nodal structure of a singlet group. Tese lines of nodes can be classifed in terms of IRs of pair is the same as that of its spatial part. Te triplet spin an intermediate group, namely, symmetry group of the plane 6 Advances in Mathematical Physics

Table 3: Characters for singlet and triplet pairs on symmetry planes in a BZ of the space group with point group �4ℎ (top part) and characters of IR �� of �4ℎ on these two types of planes (bottom part).

Type of function E �ℎ ��2 IRs of �2ℎ

Singlet Spatial part 1 1 1 1 � �

Triplet Spatial part 1 1 -1 -1 �� (a) Triplet OSP 1 1 -1 -1 �� (a) Triplet ESP 2-2-2 2 2�� (b) Triplet OSP 1-1-1 1 � � (b) Triplet ESP 20-20 � � +��

IR of �4ℎ (a) �� 222 2 (a) �� 2 2 -2 -2 (b) �� 202 0 (b) �� 20-20

(a) Basal plane �2 =�2� (b) Vertical coordinate plane �2 =�2� and vertical diagonal plane �2 =�2��.

�2ℎ. It follows from formulae (15) and (17) that the symmetry Table 4: Spatial parts of pairs in point �(�1/2, �2/2, 0),spacegroup 7 (∗) of spatial part of a triplet pair on the plane corresponds to �� P4/nmm (�4ℎ) . of group �2ℎ. It is immediately verifed that when inducing Small IRs [26] Possible pair symmetry �� from �2ℎ into �, one obtains the set of IRs, which also � � � +� +� includes ��. Hence, it follows that in general point of a BZ, Singlet 1, 2 1� 2� 2� onecanlabeltwobasissetsofIR�� by IR of intermediate pairs �3, �4 �1� +�2� +�2� � � � group 2ℎ.Tebasisset �( �) is nodeless on the plane and Triplet �1, �2 �1� the basis set ��(��)isnodalontheplane. pairs �3, �4 �1� ∗ ( )Notethat� and central extension of � coincide. IRs of � are projective 7. Nonsymmorphic Space Groups and andaredenotedaccordingto[26]. Symmetry Lines � In the center of a BZ and in some points on its faces, space for each one of four projective IRs of group 4ℎ. In a triplet inversion belongs to the wavevector group �.Inthiscase, case, only one odd IR is possible for each IR of wavevector thedoublecosetdefnedbyidentityelementresultsinzero group. total momentum of two-electron state. In the center of a BZ, Te space-group approach to the wavefunction of a the frst items in right-hand side (18), (19) are calculated by Cooper pair can be generalized for strong spin-orbit coupling standard methods [37] and one obtains even spatial part only case [15]. In this case, possible pair functions belong to for triplet and singlet pair. antisymmetrized squares of double-valued representations. � Consider as an example point �(�1/2, �2/2, 0) in a space For , a general point of a BZ and on the planes of sym- 7 group P4/nmm (�4ℎ) of Fe-pnictide superconductors [38]. In metry, where double-valued small IRs are one-dimensional, corepresentations of magnetic groups [1, 26] should be used these materials, there two sheets of Fermi surface: one near 4 point Γ and one near point � [38, 39]. In point �,space [15]. Consider two planes (001) in a BZ of �6ℎ group, namely, 4 inversion belongs to the wavevector group and two electrons �� =0plane and �� =�3/2 plane. In �6ℎ group, space group with equal nonzero moments combine into a Cooper pair and elements �� and �2� are connected with translation �,which its zero total momentum is due to translation periodicity of equals to a half of Bravais lattice translation t3.[26].Double- a crystal. Since IRs are projective at this BZ point of non- valued IRs [26] are presented in Table 5. It should be noted 7 symmorphic space group �4ℎ,projectivefactorsappearin thatsincetherearetwophasefactorsfor�� =�3/2,namely, standard formulas for symmetrization (antisymmetrization) one due to nonsymmorphic structure of group and one due [37] for fnite group IRs: to the double-valued IRs of spin rotation, double valued IRs are real [26]. �2 (�) + � (�2)�(�,�) [�2 (�)] = (22) Consider modifcations of formulas (15)-(17) due to 2 nonsymmorphic structure of group. Multiplication rule for space-group elements is as follows [1, 2, 26]: �2 (�) − � (�2)�(�,�) {�2 (�)} = . (23) 2 {� | ��}×{�|��}={��|�� +���}, (24)

It is seen from the results, presented in Table 4, that in a where � stands for a point group element and �� for its singlet pairing case, two even and one odd IRs are possible nonprimitive translation. Making use of this rule, one obtains Advances in Mathematical Physics 7

Table 5: Double-valued IRs on the (001) planes of symmetry of BZ Table 7: Possible IRs of pair in symmetry directions in BZ for �6ℎ 4 for space group �6ℎ [26]. group in strong spin-orbit coupling case.

��ℎ � SmallIR[26] IRof�6ℎ � =0 � � � � � +� +� � 1 1 1 or 2 1� 1� u �b3 �2 1-� �3 �1� +�1� +�1� +�2� � � �� � � � +� � =�/2 1 111/2 1 + 3 1 or 2 1� 2� � 3 (∗) �2 1-1 �(�1,�2) �1� +�2� +2�1� +�2� +�1� +2�2�

�3 �1� +�2� +�1� +�2� +�1� +�2� − � (∗) Table 6: Characters of representation � forCooperpairson Corepresentation. symmetry planes (001) in the case of strong spin-orbit coupling 4 (space-group �6ℎ). � � 8. Basis Functions E h I 2� Decomposition � =0 � +2� +� � 40-22 � � � In a general point of a BZ, the wavevector group � consists of � = /2 � +3� � b3 4-4-2-2 g � an identity element only and the expressions for spatial parts of singlet and triplet pairs which follow from Mackey-Bradley theorem [13] may be written as that the transformation by inversion results in point �=�3/2 (−�� /2, � ) � � aphasefactorexp 3 3 in formula (15) for z: � =� (� )� (� )+� (� )� (� ) 1 �1 1 ��1 2 �1 2 ��1 1 (27) {�|0} {� |�}{�|0} ={� |��}{�|0} � 2� �� =� (� )� (� )−� (� )� (� ), (25) �1 �1 1 ��1 2 �1 2 ��1 1 (28) ={�� |−�}{�� |�+�3}. where �1 denotes a general point in a representation domain Also, there is a phase factor in formulas (16) and (17) for of a BZ [1]. All one-electron wavefunctions are obtained �2�,since when �1 runs over representation domain of a BZ and all |�|̂ �̂ {�2� |�}{�2� |�}={�|2�} ={�|�3}. (26) elements of the central expansion of the space group � (�) G act on one-electron wavefunction �1 [1,2,26].Te One more phase factor appears, since small representa- results of action of these elements on �1 form a star {�1} of the � ×� tions are double-valued and total phase factor for 2� 2� wavevector. Tus, the number of one-electron basis functions ̂ equals one. Possible IRs of Cooper pairs for these planes are of Cooper pair for �1 a general point of a BZ equals |�|.Pair � =0 � presented in Table 6. For � , only one even IR � and all functions may be constructed for all �1 in the representation � � � odd IRs � and � of group 2ℎ are possible. Hence, it follows domain of a BZ making use of formulas (27) and (28). Since � that not all even IRs of 6ℎ appear in induced representation the space inversion is included in formulas (27) and (28), � ↑� � � 6ℎ.SincealloddIRsof 2ℎ are possible, all odd the number of elements in two-electron basis set of defnite � IRs of 6ℎ are also possible. Tus, in agreement with Blount multiplicity reduces to |�|/2̂ . theorem, there are no symmetry-protected lines of nodes for We take a space group with central extension �4ℎ (i.e., � = /2 odd IRs. It is also seen from Table 6 that for � b3 only symmetry group of unconventional superconductor S2RuO4 � one odd IR � is possible and there are symmetry-protected [32–35]) as an example of application of the space-group lines of nodes of odd IRs. Similar results were obtained by technique for construction of pair’s function. Te basis set for Micklitz and Norman [18–20] �1 a general point in a BZ of a space group with �4ℎ point �� � In direction � in a BZ for 6ℎ group, the wavevector group is shown in Figure 3. Notations in [26] are used for � group is 6V and there are three two-dimensional double- the cubic point group elements. Tis makes the direct use of � +�� valued IRs [26]. On the BZ face in direction 1/2 1 3, group multiplication table possible [26]. In these notations, symmetry group is C3� and there are two one-dimensional ℎ1 is an identity element, ℎ14, ℎ4,andℎ15 are rotations at IRs and one two-dimensional double-valued IR. Te results �/2, �,and3�/2 about the axis (001), ℎ26, ℎ27, ℎ37,and for these two symmetrical directions presented in Table 7 ℎ40 are refections on planes (100), (010), (-110), and (110), show that in highly symmetrical directions possible pair respectively. Notations in [26] for the remaining elements of symmetries depend on the internal quantum number — �4ℎ group may be obtained by making use of the following labelofIRoflittlegroup.Duetolargenumberofrotational multiplication rules for ℎ25 (space inversion): elements in group C6�, there is an analogy between IRs of this group and angular momentum j� and it was called ℎ25 ×ℎ� =ℎ�+24, if 1 < � < 24, j� dependence [21]. Present results for p3 in 1/2 b1 + �b3 (29) direction are the same as that for �� =1/2[21]. IRs p1 and p2 ℎ25 ×ℎ� =ℎ�−24, if 25 < � < 48. are one-dimensional and one can construct antisymmetrized square of p1 or p2 separately. Tese two states are connected Te labels of group elements h� are used as subscripts of basis by time-reversal and form a corepresentation �(�1,�2). functions in Figure 3; for example, Antisymmetrized square of �(�1,�2) consists of the same IRs � � as the result [21] for �� =3/2. �2 =ℎ2�1 =�2 (�1) �26 (�2) −�2 (�2) �26 (�1) . (30) 8 Advances in Mathematical Physics

k z   the signs of two mirror counterparts are changed under the 14 37 ky action of plane refection, they will cancel at the plane. Te 26 intersection of a nodal plane with Fermi surface results in 1 symmetry-protected nodal line of SOP. 4  27 Te action of refection in (010) plane on the frst line of k x �� results in 15 40 ℎ27 (�1 +�27 −�4 −�26)=�27 +�1 −�26 −�4. (32)

Figure 3: Basis set for Cooper in a general point of a BZ (�4ℎ Te sign is unchanged and there are no nodes in this symmetry). Basis function �1 is defned by formula (28), where one- plane. It is seen from Figure 3 that when �1 runs over the basis � electron wavevector 1 runs over basis domain of BZ and dashed domain of a BZ, the functions of the frst line of �� are around �� � lines correspond to wavevectors �.Basisfunction � is obtained �- axis and the functions of the second line are around �-axis. ℎ � by the action of point group element � on 1 (notations in [26] are Hence, we need to investigate nodes of the frst function in used; also see text). plane (010) only and nodes of the second function in the plane (100) only. Te action of refection in (100) on the function of

Table 8: Spatial parts of triplet wavefunctions for IR �� of �4ℎ the second row results in (a) group in �-� form .Basisfunction�1 corresponds to arbitrary ℎ (� +� −� −� ) =� +� −� −� . wave vector �1 in a Brillouin zone in one-electron space and is built 26 14 37 15 40 37 14 40 15 (33) according to formula (28) (superscript � is dropped). Other basis functions are obtained by the action of half of the elements of group Hence, it follows that there are no nodes in plane (100) �4ℎ on �1 (see Figure 3). also. We will call the nodes due to cancelation of functions of one row as nodes of the frst type. IR Wavefunction Nodes In the case of two-dimensional IRs, there is one more � +� −� −� (b) 1 27 4 26 No (c) possibility for lines of nodes, namely, cancelation between �� (110) �14 +�37 −�15 −�40 No functions of the frst and second rows. Refection in plane ℎ �2� �1 −�27 −�4 +�26 (010) (-110) 37 does not change the sign of the sum of two basis �� (-110) � �14 −�37 −�15 +�40 (100) functions of � basis set:

�1� �1 +�27 −�4 −�26 No � (-110) ℎ (� +� −� −� +� +� −� −� ) � −� −� +� +� 37 1 27 4 26 14 37 15 40 14 37 15 40 No (34) �2� �1 −�27 −�4 +�26 (010) =�37 +�14 −�40 −�15 +�27 +�1 −�26 −�4. �� (110) −�14 +�37 +�15 −�40 (100) Tus, there are no nodes on plane (-110). Te action of (a) Basis function for one-dimensional IRs and for �� in chiral form are presented elsewhere [25]. refection in (110) plane ℎ40 changes the sign of basis set: (b) Nodes of the frst type. (c) Nodes of the second type. ℎ40 (�1 +�27 −�4 −�26 +�14 +�37 −�15 −�40) (35) =�40 +�15 −�37 −�14 +�26 +�4 −�27 −�1. In the case of induced representations, standard projection � operator technique [37] may be reduced to the action of lef Hence, it follows that the basis set of IR � has a coset representatives on the basis functions of subgroup [9] nodal plane (110). We will call the nodes, resulting from the and one obtains the following formula for construction the cancellation of basis functions belonging to diferent rows of � wave function of a Cooper pair belonging to any IR Γ of �̂: two-dimensional IR, nodes of the second type. It is immediately verifed that multiplication of �� by � �� � Γ� �Γ � � all even one-dimensional IRs results also in �.Tus,the Ψ = � � ∑ Γ (ℎ�)ℎ��1. � � � �̂�� �� (31) other � basis sets can be obtained by projection on IRs � �� � ̂� � � ℎ�∈� transformed by �2�, �1�,and�2�.Basissetsof�� symmetry, labeled by these one-dimensional IRs, written as superscripts, Te prime in (31) means that the sum includes only one of the arealsopresentedinTable8.Tesebasissetsandtheresults two elements, connected by inversion. of their nodal structure analysis are presented in Table 8. It � � Spatial parts of wavefunction of triplet Cooper pairs for 2� 2� is seen from Table 8 that basis sets �� and �� have nodes IR �� of �4ℎ group in �-� representation are presented in of the frst type on two vertical coordinate planes, but basis Table 8, where it is assumed that wavevector �1 runs over � � � 1� basis domain of a BZ. Te results for one-dimensional IRs and sets � and � have no nodes of the frst type. It is also for �� in chiral representation are presented elsewhere [25]. seen from Table 8 that all IRs have one vertical nodal plane of Nodal structure on the symmetry planes may be estimated the second type. Recall that it follows from Mackey-Bradley � as follows. Since pair function on the plane is invariant theorem that � appears twice in complete basis set, but under plane refection, the nodeless linear combination is once of each vertical plane (see Table 3 and discussions in �2� unchanged under the action of the plane refection. At the the text). Consider �� and �� , difering by nodal quantum symmetry plane, mirror counterparts can merge or cancel. If number on the plane (100). It is seen from Table 8, that in Advances in Mathematical Physics 9 agreement with the results of Table 3, there is one nodeless IR, [4] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non- and the other is nodal on each vertical plane. Te same result Relativistic Teory, Pergamon Press, Oxford, UK, 1965. �1� �2� is valid for other pair of IRs �� and �� .Teauthorsof [5] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Teory experimental works [34–36] on topological superconductor Application to the Physics of Condensed Matter, Springer, Berlin, Germany, 2008. Sr2RuO4 obtained vertical lines of nodes of odd triplet SOP [6] F.A.Cotton,Chemical Application of Group Teory, Wiley, 1990. and concluded that OSP pairing is the most probable. In [7] C. J. Ballhausen, Introduction to Ligand Field Teory,McGraw- �4ℎ group, the symmetry of OSP spin is �2�.Sincewe Hill, 1962. investigated products of �� basis set by all one-dimensional [8]V.P.SmirnovandR.A.Evarestov,“Applicationofthesite IRs, our results include symmetry of OSP case and confrm symmetry for the simplifcation of the group theoretical anal- that vertical lines of nodes are possible for OSP pairs of �� � ysis of electronic and vibrational states of crystals,” Vestnik symmetry. Note that there are two diferent forms of IR �, Leningradskogo Universiteta Seriya Fizika Khimiya,vol.4,pp. namely, real and complex. Real form, considered in present 107–111, 1981 (Russian). work, is usually used in quantum chemistry [7, 8]. Basis [9] V. G. Yarzhemskii and E. N. Murav’ev, “Induced representation functions of complex form [26] have defnite momentum method in the theory of molecular orbitals,” Doklady Akademii projection on �-axis and are considered elsewhere [25]. Nauk SSSR,vol.278,no.4,pp.945–948,1984(Russian). [10] V.G. Yarzhemskii and E. N. Muravev, “Orbits and induced rep- 9. Conclusion resentations in quantum chemistry of nanoparticles,” Russian Journal of Inorganic Chemistry,vol.54,article1341,pp.1273– Te advantages of applications of induced representation 1276, 2009. method to symmetrical molecules and Cooper pairs in solids [11] V.G. Yarzhemsky and V.I. Nefedov, “Group theoretical descrip- are reviewed and some results for pairing in topological tion of two-electron wave functions in systems with subgroups of symmetry,” International Journal of Quantum Chemistry,vol. superconductors are obtained. It is shown that induced 100, no. 4, pp. 519–527, 2004. representation method makes it possible to simplify standard [12] G. W. Mackey, “Symmetric and anti symmetric Kronecker methods for normal vibrations and SALCs construction (i.e., squares and intertwining numbers of induced representations list of permitted IRs is obtained by the analysis of character of fnite groups,” American Journal of Mathematics,vol.75,no. tables on local subgroup only). Te applications of Mackey- 2, pp. 387–405, 1953. Bradley theorem in the case of nonsymmorphic space groups [13] C. J. Bradley and B. L. Davies, “Kronecker products and and projective representations are further investigated and symmetrized squares of irreducible representations of space examples of UPt3 and Fe-pnictide superconductors are con- groups,” Journal of Mathematical Physics, vol. 11, pp. 1536–1552, sidered. Nodal structure �� SOP for �4ℎ symmetry in x-y rep- 1970. resentation is obtained and its dependence upon the addition [14] V. G. Yarzhemsky, E. N. Murav’ev, and J. Phys, “Space group quantum numbers is envisaged. Tese nodal structures are in approach to the wavefunction of a Cooper pair,” Journal of qualitative agreement with experimental data for topological Physics: Condensed Matter,vol.4,no.13,pp.3525–3532,1992. superconductor Sr2RuO4. [15] V.G. Yarzhemsky, “Space-group approach to the nodal structure of the superconducting order parameter in UPt3,” Physica Status Solidi (b) – Basic Solid State Physics,vol.209,no.1,pp.101–107, Data Availability 1998. [16] P. W. Anderson, “Structure of ’triplet’ superconducting energy Te references to the sources of the data are in the paper, and gaps,” Physical Review B,vol.30,pp.4000–4002,1984. at present there are no other places where they are available. [17] V.G. Yarzhemsky, “Space-group approach to the nodal structure of superconducting order parameter in ferromagnetic and Conflicts of Interest antiferromagnetic materials,” International Journal of Quantum Chemistry,vol.80,pp.133–140,2000. Te author declares that there are no conficts of interest. [18] T. Micklitz and M. Norman, “Odd parity and line nodes in nonsymmorphic superconductors,” Physical Review B,vol.80, article 100506, 2009. Acknowledgments [19] T. Micklitz and M. R. Norman, “Symmetry-enforced line nodes Te work was carried out within the State Assignment on in unconventional superconductors,” Physical Review Letters, vol. 118, articel 207001, 2017. Fundamental Research to the Kurnakov Institute of General [20] T. Micklitz and M. R. Norman, “Nodal lines and nodal loops in and Inorganic Chemistry. nonsymmorphic odd-parity superconductors,” Physical Review B,vol.95,article024508,2017. References [21] S. Sumita and Y. Yanase, “Unconventional superconducting gap structure protected by space group symmetry,” Physical Review [1]C.J.BradleyandA.P.Cracknell,Te Mathematical Teory of B,vol.97,article134512,2018. Symmetry in Solids, Oxford University Press, Oxford, UK, 1972. [22] V. G. Yarzhemsky, Nanoscale Systems: Mathematical Modeling, [2]S.L.Altmann,Induced Representations for Crystals and Teory and Applications,vol.2,pp.96–106,2013. Molecules, Academic Press, Massachusetts, Mass, USA, 1977. [23] V. G. Yarzhemsky, “Nodal quantum numbers for two-electron [3] A. P. Cracknell, Applied Group Teory,Pergamon,Oxford,UK, states in solids,” Few-Body Systems,vol.53,no.3-4,pp.499–504, 1968. 2012. 10 Advances in Mathematical Physics

[24] V. G. Yarzhemsky and J. Adv, “Space-group approach to the wavefunction of a Cooper pair: nodal structure and additional quantum numbers for Sr2RuO4 and UPt3,” International Journal of Advanced Applied Physics Research,vol.1,pp.48–56,2016. [25] V. G. Yarzhemsky, “Group theoretical lines of nodes in triplet chiral superconductor Sr2RuO4,” Journal of the Physical Society of Japan, vol. 87, article 114711, 2018. [26] O. V.Kovalev, Irreducible Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representa- tions and Corepresentations,GordonandBreach,1993. [27] W. Lederman, Introduction to Group Characters,Cambridge University Press, 1987. [28] V. L. Ginzburg and L. D. Landay, “On the theory of supercon- ductivity,” Journal of Experimental and Teoretical Physics,vol. 20, pp. 1064–1082, 1950 (Russian). [29] P. A. Lee, “Amperean pairing and the pseudogap phase of cuprate superconductors,” Physical Review X,vol.4,article 031017, 2014. [30] G. E. Volovik and L. P. Gorkov, “Superconducting classes in heavy-fermion systems,” Journal of Experimental and Teoret- ical Physics,vol.61,no.4,pp.843–854,1985. [31] M. Sigrist and K. Ueda, “Phenomenological theory of uncon- ventional superconductivity,” Reviews of Modern Physics,vol. 63, pp. 239–311, 1991. [32] E. I. Blount, “Symmetry properties of triplet superconductors,” Physical Review B,vol.32,pp.2935–2944,1985.

[33]T.M.Rice,M.Sigrist,andJ.Phys,“Sr2RuO4:anelectronic 3 analogue of He?” Journal of Physics: Condensed Matter,vol.7, no.47,pp.L643–L648,1995. [34]C.Lupien,W.A.MacFarlane,C.Proust,L.Taillefer,Z.Q.Mao, and Y. Maeno, “Ultrasound attenuation in Sr2RuO4:anangle- resolved study of the superconducting gap function,” Physical Review Letters,vol.86,pp.5986–5989,2001. [35] K. Deguchi, Z. Q. Mao, and Y. Maeno, “Determination of the superconducting gap structure in all bands of the spin-triplet superconductor Sr2RuO4,” Journal of the Physical Society of Japan,vol.73,pp.1313–1321,2004. [36] E. Hassinger, P. Bourgeois-Hope, H. Taniguchi et al., “Vertical line nodes in the superconducting gap structure of Sr2RuO4,” Physical Review X,vol.7,article011032,2017. [37] M. Hamermesh, Group Teory and Its Application to Physical Problems,Adison-Wesley,1964. [38]V.G.Yarzhemsky,A.D.Izotov,andV.O.Izotova,“Structure of the order parameter in iron pnictide-based superconducting materials,” Inorganic Materials,vol.53,no.9,pp.923–929,2017. [39] S. V.Borisenko, D. V.Evtushinsky, Z.-H. Liu et al., “Direct obser- vation of spin-orbit coupling in iron-based superconductors,” Nature Physics,vol.12,no.4,pp.311–317,2016. Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences

Journal of Optimization Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Submit your manuscripts at www.hindawi.com

International Journal of Engineering International Journal of Mathematics Analysis Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Journal of Advances in Mathematical Problems International Journal of Discrete Dynamics in Complex Analysis Numerical Analysis in Engineering Dierential Equations Nature and Society Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

International Journal of Journal of Journal of Abstract and Advances in Stochastic Analysis Mathematics Function Spaces Applied Analysis Mathematical Physics Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018