Symmetry Protected Line Nodes in Non-Symmorphic Magnetic Space Groups: Applications to Ucoge and Upd2al3

Symmetry protected line nodes in non-symmorphic magnetic space groups: Applications to UCoGe and UPd2Al3

Takuya Nomoto1, ∗ and Hiroaki Ikeda2 1Department of Physics, Kyoto University, Kyoto, 606-8502, Japan 2Department of Physics, Ritsumeikan University, Kusatsu, 525-8577, Japan (Dated: October 18, 2016) We present the group-thoretical classiﬁcation of gap functions in superconductors coexisting with some magnetic order in non-symmorphic magnetic space groups. Based on the weak-coupling BCS theory, we show that UCoGe-type ferromagnetic superconductors must have horizontal line nodes on either kz = 0 or ±π/c plane. Moreover, it is likely that additional Weyl point nodes exist at the axial point. On the other hand, in UPd2Al3-type antiferromagnetic superconductors, gap functions with Ag symmetry possess horizontal line nodes in antiferromagnetic Brillouin zone boundary perpendicular to c-axis. In other words, the conventional fully-gapped s-wave superconductivity is forbidden in this type of antiferromagnetic superconductors, irrelevant to the pairing mechanism, as long as the Fermi surface crosses a zone boundary. UCoGe and UPd2Al3 are candidates for unconventional superconductors possessing hidden symmetry-protected line nodes, peculiar to non-symmorphic magnetic space groups.

In the research field of superconductivity, its coexis- symmetry-protected nodes can appear at the BZ bound- tence of magnetism is a very interesting topic. Such co- ary. As for the coexisting phase, there is little progress of existence between superconductivity and magnetism is gap classification considering the ordered moment [2, 28]. often discovered in the U-based heavy-fermion supercon- In this Letter, we extend the gap classification into ductors. For example, UPd2Al3 shows an antiferromag- non-symmorphic magnetic space groups [25, 26]. The re- netic transition at TN = 14K, and then coexists with sults can be applied to the analysis of superconductivity unconventional superconductivity below Tc = 2K [1, 2]. in UCoGe or UPd2Al3. In these compounds, the mag- UGe2 [3], URhGe [4], and UCoGe [5] encounter a su- netic ordered phase belongs to the non-symmorphic mag- perconducting transition in the ferromagnetic phase. A netic (type III or IV Shubnikov) space groups due to the rare reentrant superconductivity has been discovered un- presence of time-reversal symmetry with a point group der the magnetic field [6]. Theoretically, in UPd2Al3, it operation and/or a non-primitive translation. Here, we was discussed that a spin-singlet superconductivity with show the following nontrivial consequences within the horizontal line nodes occurs via the virtual exchange of weak-coupling BCS theory. The UCoGe-type ferromag- magnetic excitons [7, 8]. In the ferromagnetic supercon- netic superconductors have horizontal line nodes (gap ze- ductors, it has been considered that the Ising-like fer- ros) on either kz = 0 or kz = ±π/c plane. In UPd2Al3- romagnetic fluctuation can lead to a spin-triplet pair- type antiferromagnetic superconductors, Ag gap func- ing state [9, 10], and many fascinating phenomena in- tions always have line nodes on kz = ±π/c plane (i.e. the cluding the odd H − T phase diagram have been stud- magnetic BZ face), in other words, the conventional fully- ied [11–16]. However, in spite of the growing interest, gapped s-wave superconductivity is forbidden. (Needless the properties characteristic of the coexisting phase is to say, in both cases, it is necessary that the Fermi sur- less well understood systematically. In this situation, face crosses nodal planes.) Thus, UCoGe and UPd2Al3 the group-theoreical classification, which provides defi- are a candidate of unconventional superconductors pos- nite statements independent of the details of materials, sessing hidden symmetry-protected line nodes, peculiar plays an important role. to non-symmorphic magnetic space groups. It is well-known that the superconducting states are Set up — First, we focus on a space group G0 that is classified into the irreducible representations (IRs) un- given as a coset decomposition G0 = {E|0}T +{2z|tz}T + der a given point group symmetry [17–19]. Such clas- {I|0}T +{σh|tz}T , where the translation group T defines c sification provides useful information in analyzing the a Bravais Lattice, and tz = 2 ec is a non-primitive trans- nodal structure of various unconventional superconduc- lation along the c-axis. The notation {p|a} is a conven-

arXiv:1610.04679v1 [cond-mat.supr-con] 15 Oct 2016 tors [20]. Also, another development of gap classiﬁcation tional Seitz space group symbol with a point-group oper- based on the space group symmetry [21–24] gives us the ation p and a translation a. E denotes an identity opera- correct way to take into account small representations at tion, 2z a π-rotation around c-axis, I a spatial-inversion, Brillouin zone (BZ) boundary in non-symmorphic space and σh a mirror about ab-plane. In this letter, we con- groups [25, 26]. T. Micklitz and M. R. Norman [24] sider the three types of systems given in Fig. 1. PP, FP, demonstrated in the pioneering work that new types of and AP correspond to paramagnetic, ferromagnetic, and antiferromagnetic phase, respectively. Unless otherwise assigned, the spin-orbit coupling is included in all systems. Note that the space group G of PP is the same as ∗ [email protected] discussed in Ref. [24]. 2

TABLE I. The characters ofγ ¯k in the case of PP, FP, and

PP: G = G0 + {θ | 0 }G0 AP. Upper and lower expressions in PP and FP correspond to the two non-equivalent IRs. FP: G = G0 Basal plane Zone face c AP: G = G0 + {θ | t z }G0 K¯k {E|0}{σh|tz} K¯k {E|0}{σh|tz} a(b) PP FP AP PP 2 0 PP 2 ±2i FP 1 ±i FP 1 ±i AP 2 0 AP 2 0 FIG. 1. (Color online) Three types of space groups considered in this letter. θ denotes a time-reversal operation. Left panel is a side view of the unit cell in the typical structures. PP and ¯ FP contain a zigzag structure, and AP possesses staggered TABLE II. The characters of Pk in PP, FP, and AP. ordered moments along the c-axis, in which the orientations Basal plane are in the ab-plane. Mk/T {E|0}{2z|tz}{I|0}{σh|tz} PP 4 2 −2 0 FP 1 1 −1 −1 Method — Let γk be a small representation [29] of a lit- AP 4 2 −2 0 tle group Kk, which represents the Bloch state with the crystal momentum k. We should note that the (zero- Zone face momentum) Cooper pairs have to be formed between Mk/T {E|0}{2z|tz}{I|0}{σh|tz} the degenerate states present at k and −k within the PP 4 −2 −2 4 BCS theory. Therefore, these two states should be con- FP 1 −1 −1 1 nected by some symmetry operations except for an acci- AP 4 −2 −2 0 dentally degenerate case. As a result, the representation of Cooper pair wave functions Pk can be constructed from γk as summarized in Refs. [22–24]. Here, we do not re- pairs Pk. In the space group operation d connecting two peat the details of the prescription, and instead, indicate states of the paired electrons, its rotation/inversion part the practical procedure step by step. pd meets pdk = −k modulo a reciprocal lattice vector. In Results — Through the present letter, we only consider the present cases, {I|0} and {2z|tz} are the candidates the Cooper pairs in the basal plane (kz = 0) and the zone for the operator d in FP, while {θ|0} ({θ|tz}) is also in face (kz = ±π/c). In both planes, the little groups Kk PP (AP). Regardless of the choice of d, Mk = Kk + dKk are given by the following coset decompositions, is identical to the space group G. Taking into account the antisymmetry of the Cooper pairs and the degen-  {E|0}T +{σ |t }T +{θI|0}T +{θ2 |t }T : PP  h z z z eracy of the two states, we can regard Pk as an anti- Kk = {E|0}T +{σh|tz}T : FP symmetrized Kronecker square [25], with zero total mo-  mentum, of the induced representation γ ↑ M . In  {E|0}T +{σh|tz}T +{θI|tz}T +{θ2z|0}T : AP k k (1) the systematic way, this is obtained by using the dou- ble coset decomposition and the corresponding Mackey- To obtain the small representations, it is sufficient to see Bradley theorem [25, 31, 32]. The obtained results are ¯ the (projective) IRs of the corresponding little co-groups summarized in Table II. Here, Pk is the representation of ¯ K¯k = Kk/T with the appropriate factor systems [25, 26]. Mk/T to meet Pk(g) = Pk(r) where g = rt for g ∈ Mk, We denote them byγ ¯k. Here, we specifies the elements r ∈ Mk/T , and t ∈ T . In the case of AP, the character ¯ of K¯k as the representatives r = {p|a} of the decompo- of Pk for {σh|tz} is equal to zero even in the zone face, sitions (1).γ ¯k can be obtained by calculating the IRs different from the case of PP. This comes from the differ- for the unitary part of K¯k, and then inducing them by ence ofγ ¯k({σh|tz}) in Table I. On the other hand, since ¯ an anti-unitary operation [30]. In Table I, we summa- Pk in FP is one-dimensional representation, only one IR rize the characters ofγ ¯k for the unitary operations in is allowed both in the basal plane and the zone face. K¯k. Now, the corresponding small representations are Finally, we reduce the representation P¯k into the IRs. given by γk(g) =γ ¯k(r)Fk(t) where g = rt for g ∈ Kk In both planes, there are four IRs, Ag,Bg,Au, and Bu −ik·t and t ∈ T . Fk is the IR of T defined by Fk(t) = e since the coset group Mk/T is isomorphic to the point for t = {E|t}. From Table I, we can see that the IRs of group C2h [33]. The results are summarized in Table III. K¯k in PP and AP become two-dimensional, which reflect Note that the gap functions should be zero, which means the Kramers degeneracy for the anti-unitary operations the appearance of gap nodes, if the corresponding IRs {θI|0} and {θI|tz}. In the case of AP, since the non- do not exist in the reduction of P¯k [21–23]. The absence primitive translation included in {θI|tz} cancels out the of Au in PP corresponds to the emergent horizontal line phase factor arising from that in {σh|tz},γ ¯k({σh|tz}) in node of the recently proposed E2u state in UPt3 super- the zone face is the same in the basal plane. This situa- conductors [24, 34–36]. On the other hand, in the case tion is in sharp contrast to the case of PP. of AP, we find that Ag does not appear in the zone face, Next, we consider the representation of the Cooper in other words, Ag possesses horizontal line nodes in the 3

face, we can ignore the anti-unitary part because this is ¯ TABLE III. The reduction of P (k) to the IRs of C2h in the not the element of the little groups. Therefore, the only case of PP, FP, and AP. difference is that the non-primitive translation becomes Basal plane Zone face tn instead of tz. As a result,γ ¯k({σh|tz}) in the zone face −ikx/2 PP Ag + 2Au + Bu PP Ag + 3Bu changes fromγ ¯k({σh|tz}) = ±i to ±ie . We can FP Au FP Bu easily confirm that this change does not affect the final AP Ag + 2Au + Bu AP Bg + Au + 2Bu results given in Table II and III. Therefore, any superconductivity in these materials have line nodes in either the basal plane or the zone face, at least, in the weak zone face (i.e. the magnetic BZ boundary). This means coupling limit. that the conventional fully-gapped s-wave superconduc- Now, let us consider the nodal structure of supercon- tivity is forbidden, if the Fermi surfaces cross kz = ±π/c ductivity in UCoGe in details. Unfortunately, the Fermi plane. Emergence of line nodes in the zone boundary surfaces of this compound have not been established in in antiferromagnetic superconductors was studied based experiments [50–52], however, the first-principles calcu- on the microscopic theory in Refs. [37, 38]. This pecu- lations show the existence of many complicated Fermi liar example can be realized in the superconductivity in surfaces, some of which cross the kz = 0 and kz = ±π/c UPd2Al3 as discussed below. In the case of FP, only planes [50]. Therefore, our results suggest the existence odd-parity pairing is allowed in both planes, due to the of horizontal line nodes in the coexistent phase of UCoGe. absence of Kramers degeneracy. Au is forbidden in the Both power law behaviors of the spin-lattice relaxation zone face, and B is forbidden in the basal plane. There- 3 u rate 1/T1 ∼ T [53, 54] and the thermal conductivity fore, the line nodes always appear, as for as the Fermi 2 κS/κN ∼ T [55] are consistent with our prediction. surface crosses kz = 0 and kz = ±π/c planes. It should Note that in the pressure-temperature phase diagram, be noted that the emergence of such nodal structure does the ferromagnetic transition seems to have little affect on not depend on the pairing mechanism. These are the the superconductivity [56, 57]. In the absence of mag- main results of this Letter. Note that these results are netism, the straightforward calculation shows that the applicable to not only conventional magnetic-dipole or- space group P nma10 leads to the same nodal structure dered states, but also magnetic multipole ordered states. as the case of PP in Table III, by regarding the IRs of D2h Discussion — Now we discuss several U-based mate- as those of C2h with the compatibility relation. There- rials in non-symmorphic magnetic space groups. First, fore, the line nodes in the basal plane of Bu IR (B2u we focus on the case of AP. The space group G = and B3u IRs of D2h group) are forbidden in the para- G0 + {θ|tz}G0 corresponds to Pb21/m (the unique axis magnetic phase, which is consistent with the Blount’s is chosen to be c-axis). Its typical example is the an- theorem of the triplet superconductors [58]. Therefore, tiferromagnetic phase of UPd2Al3, in which the order- we may expect that the realized gap function belongs to ing vector is Q = (0, 0, π/c) and the orientations of mo- Au IR (A1u or B1u in the paramagnetic phase) and has ments are in the basal plane. Many experimental obser- the line nodes at kz = ±π/c plane. As for the Au gap vations [7, 39–42] imply the presence of horizontal line functions in the coexistent phase, there should be an ad- nodes at the magnetic BZ boundary perpendicular to the ditional point node at kx = ky = 0, which is regarded as c-axis. Moreover, it has been expected that the gap func- Weyl nodes [2]. Thus, the hybrid gap structure of line tion belongs to Ag IR [8, 43–45] and the Fermi surfaces and point nodes would be realized such as proposed in cross kz = ±π/c plane [46–49]. Following our results, the URu2Si2 [59] and UPt3 [60]. expected horizontal line nodes are symmetry-protected Finally, we demonstrate the above-mentioned group nodes in the non-symmorphic magnetic space groups. theoretical arguments by using a specific model, and Next, let us consider the case of FP. Its candidates are discuss the stability of horizontal line nodes on the hotly-debated ferromagnetic superconductors, UCoGe, zone face. We consider a Bogoliubov-de Gennes (BdG) URhGe and UGe2. In the paramagnetic phase, the space Hamiltonian of an AP (FP) superconductivity with Ag 0 group of UCoGe and URhGe is P nma1 , while UGe2 (Au) gap structure as a minimal model of UPd2Al3 possesses symmorphic Cmmm10. In the ferromagnetic (UCoGe) [61]. Fig. 2(a) depicts the band structure phase, the former two belong to FP as shown below, while along the high-symmetry line (0, 0, π)-(0, π, π) in the AP the latter does not meet the condition. In the former model of UPd2Al3, and its inset shows the Fermi surface. two, since the ordered moments align parallel to c-axis in Fig. 2(b) is the case of UCoGe. In both Figs. 2(a) and the ferromagnetic phase, the space group is reduced from (b), we can find that the superconducting gap remains P nma10 into P n0m0a. This group is given explicitly by closed at the Fermi level in the particle-hole symmetric 0 0 0 G = G0 + {θ2y|ty}G0 where G0 = {E|0}T + {2z|tn}T + Bogoliubov band (red arrows), while a gap is open at b a c {I|0}T + {σh|tn}T with ty = 2 ey and tn = 2 ex + 2 ez. the inter-band crossing point far from the Fermi level Here, a, b, and c are the lattice parameters and 2y repre- (blue arrows). The emergence of gap zero on the zone sents the π-rotation around b-axis. At first glance, this face is fully consistent with the group theoretical argu- seems rather different from that of FP. However, con- ments. On the other hand, the group theory does not sidering a general point in the basal plane and the zone say anything about the inter-band gap opening, since the 4

(a) (b) gap nodes is controlled by three parameters, ∆, δM , and 3 0.4 SC SC Normal Normal δS, which correspond to respectively the gap amplitude, 2 0.2 the strength of the magnetic order and the spin-orbit 1 coupling [61]. δM and/or δS lift the band degeneracy on

0 0 the zone face [62]. Figs. 2(c) and (d) are the Bogoliubov -1 k k z -0.2 z band structure for several ∆. As expected, we can find -2 k the vanishing of gap nodes for larger ∆ via a kind of Lif- -3 ky -0.4 y k k (0,0,π) x (0,π,π) (0,0,π) x (0,π,π) shitz transition. Figs. 2(e) and (f) show kz dependence (c) (d) of the excitation gap amplitude |∆(kz)| on the Fermi sur- SC SC face at k = 0 for several δ and δ . For relatively large Normal Normal x M S δM (δS), ∆(kz) behaves like ∆(kz) ∼ cos(kz/2). On the other hand, for relatively small δM (δS), the gap ampli- arb. unit arb. unit tude sharply changes around the nodes on the zone face, δ /∆ = δ /∆ = M S which was also discussed in Ref. [38]. In the limit of 4.0 2.0 1.0 0.5 10.0 5.0 1.0 0.5 δM (δS) = 0, the nodal structure is completely lost. (e) (f) In our model, such k dependence just comes from 0.12 z k |∆( z)| 0.02 the unitary matrix diagonalizing the BdG Hamiltonian. It implies that even the BCS approximation of purely 0.08 |∆(kz)| local (on-site) interactions, such as the conventional 0.01 electron-phonon interactions, can induce the anisotropic 0.04 δM/∆ = 4.0 δ /∆ = 10.0 2.0 S 5.0 gap structure in the non-symmorphic magnetic super- 1.0 1.0 0.5 0.5 0.0 0.0 conductors. In the realistic situations, the band split- 0 0 −π 0 −π 0 kz kz tings on the zone face will be sufficiently larger than the gap amplitude. Therefore, it is expected that the FIG. 2. (Color online) Typical band structure of the BdG present nodal structure can be observed as the usual Hamiltonian assuming (a) the Ag gap function in the AP power-law behavior in thermodynamic and/or transport model of UPd2Al3 and (b) the Au gap function in the FP properties at low temperatures. Consequently, nontriv- model of UCoGe [61]. The lattice constants are set to be ial symmetry-protected line nodes in the non-symmophic unity. Measure of energy is unit of the nearest neighbor hop- magnetic space groups will be observed in magnetic su- ping integral. Blue dashed lines correspond to the original perconductors UCoGe or UPd2Al3. band in the normal state. The inset shows the corresponding Fermi surface. (c) and (d) are the enlarged figure of the Bogoliubov band for several ∆. (e) and (f) show the kz dependence of the excitation gap amplitude |∆(kz)| on the Fermi ACKNOWLEDGMENTS surface at kx = 0 for several δM and δS .

We acknowledge K. Hattori, Y. Yanase, S. Fuji- above-mentioned arguments are based on the intra-band moto and K. Shiozaki for valuable discussions. This pairs. As readily understood, if the inter-band gap is suf- work was partly supported by JSPS KAKENHI Grant ﬁciently large, then the symmetry-protected intra-band No.15H05745, 15H02014, 15J01476, 16H01081, and gap nodes can be lost. In our models, the emergence of 16H04021.

[1] C. Geibel, C. Sehank, S. Thies, H. Kitazawa, C.D. Bredl, [5] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. A. Biihm, M. Rau, A. Grauel, R. Caspary, R. Helfrich, Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, T. U. Ahlheim, G. Weber, and F. Steglieh, Z. Phys. B 84, 1 G¨orlach, and H. v. L¨ohneysen,Phys. Rev. Lett 99, 067006 (1991) (2007). [2] A. Krimmel, P. Fischer, B. Roessli, H. Maletta, C. Geibel, [6] F. Levy, I. Sheikin, B. Grenier, and A. D. Huxley, Science C. Schank, A. Grauel, A. Loidl, and F. Stegiich, Z Phys. 309, 1343 (2005). B 86, 161 (1992). [7] N. K. Sato, N. Aso, K. Miyake, R. Shiina, P. Thalmeier, [3] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. G. Varelogiannis, C. Geibel, F. Steglich, P. Fulde, and T. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, Komatsubara, Nature 410, 340 (2001). S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, [8] K. Miyake and N. K. Sato, Phys. Rev. B 63, 052508 I. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406, (2001). 587 (2000). [9] T. Hattori, Y. Ihara, Y. Nakai, K. Ishida, Y. Tada, S. [4] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flou- Fujimoto, N. Kawakami, E. Osaki, K. Deguchi, N. K. Sato, quet, J.-P. Brison, E. Lhotel, and C. Paulsen, Nature 413 and I. Satoh, Phys. Rev. Lett. 108, 066403 (2012). 613 (2001). [10] Y. Tada, S. Fujimoto, N. Kawakami, T. Hattori, Y. Ihara, 5

K. Ishida, K. Deguchi, N. K. Sato, and I. Satoh, J. Phys.: Geibel, F. Steglich, N. Sato, and T. Komatsubara, Phys. Conf. Ser. 449, 012029 (2013). Rev. Lett. 73, 1849 (1994). [11] V. P. Mineev and T. Champel, Phys. Rev. B 69, 144521 [45] K. Matsuda, Y. Kohori, and T. Kohara, Phys. Rev. B (2004). 55, 15223 (1997). [12] V. P. Mineev, Phys. Rev. B 81, 180504(R) (2010). [46] Y. Inada, H. Aono, A. Ishiguro, J. Kimura, N. Sato, A. [13] V. P. Mineev, Phys. Rev. B 83, 064515 (2011). Sawada, and T. Komatsubara, Physica B 199-200, 119 [14] Y. Tada, N. Kawakami, and S. Fujimoto, J. Phys. Soc. (1994). Jpn. 80 (2011) SA006. [47] Y. Inada, A. Ishiguro, J. Kimura, N. Sato, A. Sawada, [15] K. Hattori and H. Tsunetsugu, Phys. Rev. B 87 064501 T. Komatsubara, and H. Yamagami, Physica B 206-207, (2013). 33 (1995). [16] Y. Tada, S. Takayoshi, and S. Fujimoto, Phys. Rev B 93, [48] L. M. Sandratskii, J. K¨ubler, P. Zahn and I. Mertig, 174512 (2016). Phys. Rev. B 50, 15834 (1994). [17] G. E. Volovik and L. P. Gor’kov, JETP Lett. 39, 674 [49] Y. Inada, H. Yamagami, Y. Haga, K. Sakurai, Y. Tokiwa, (1984). T. Honma, E. Yamamoto, Y. Onuki,¯ and T. Yanagi- [18] G. E. Volovik and L. P. Gor’kov, Sov. Phys. JETP 61, sawa, J. Phys. Soc. Jpn. 68, 3643 (1999). Y. Tsutsumi, 843 (1985). T. Nomoto, H. Ikeda, and K. Machida, J. Phys. Soc. Jpn. [19] K. Ueda and T. M. Rice, Phys. Rev. B 31, 7114 (1985). 85, 033704 (2016). [20] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). [50] M. Samsel-Czeka la, S. Elgazzar, P. M. Oppeneer, E. [21] Y. A. Izyumov, V. M. Laptev, and V. N. Syromyatnikov, Talik, W. Walerczyk, and R. Troc, J. Phys.: Condens. Int. J. Mod. Phys. B 3, 1377 (1989). Matter 22, 015503 (2010). [22] V. G. Yarzhemsky and E. N. Murav’ev, J. Phys.: Con- [51] D. Aoki, I. Sheikin, T. D. Matsuda, V. Taufour, G. dens. Matter 4, 3525 (1992). Knebel, and J. Flouquet, J. Phys. Soc. Jpn. 80, 013705 [23] V. G. Yarzhemsky, Phys. Status Solidi B 209, 101 (1998). (2011). [24] T. Micklitz and M. R. Norman, Phys. Rev. B 80, [52] S. Fujimori, T. Ohkochi, I. Kawasaki, A. Yasui, Y. 100506(R) (2009). Takeda, T. Okane, Y. Saitoh, A. Fujimori, H. Yamagami, [25] C. J. Bradley and A. P. Cracknell, The Mathematical Y. Haga, E. Yamamoto, and Y. Onuki,¯ Phys. Rev. B 91, Theory of Symmetry in Solids (Oxford University Press, 174503 (2015). Oxford, 1972). [53] T. Ohta, Y. Nakai, Y. Ihara, K. Ishida, K. Deguchi, N. K. [26] C. J. Bradley and B. L. Davies, Rev. Mod. Phys. 40, 359 Sato, and I. Satoh, J. Phys. Soc. Jpn. 77, 023707 (2008). (1968). [54] T. Ohta, T. Hattori, K. Ishida, Y. Nakai, E. Osaki, K. [27] V. P. Mineev, Phys. Rev. B 66, 134504 (2002). Deguchi, N. K. Sato, and I. Satoh, J. Phys. Soc. Jpn. 79, [28] K. V. Samokhin and M. B. Walker, Phys. Rev. B 66, 023707 (2010). 174501 (2002). [55] M. Taupin, L. Howald, D. Aoki, and J. -P. Brison, Phys. [29] In this paper, we use the word “representation” both as Rev. B 90, 180501(R) (2014). a representation of a unitary group and a corepresentation [56] E. Hassinger, D. Aoki, G. Knebel, and J. Flouquet, J. of a non-unitary group. Phys. Soc. Jpn. 77, 073703 (2008). [30] M. V. Murthy, J. Math. Phys. 7, 853 (1966). [57] E. Hassinger, D. Aoki, G. Knebel, and J. Flouquet, Jour- [31] G. W. Mackey, Am. J. Math. 75, 387 (1953). nal of Physics: Conference Series 200, 012055 (2010). [32] C. J. Bradley and B. L. Davies, J. Math. Phys. 11, 1536 [58] E. I. Blount, Phys. Rev. B 32, 2935 (1985). (1970). [59] T. Shibauchi, H. Ikeda, and Y. Matsuda, Philo. Mag. 94, [33] Strictly, Mk/T is isomorphic to the gray point group of 3747 (2014). C2h in PP and AP. However, the corepresentations are [60] R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235 trivial because the anti-unitary operations do not cause (2002). the extra degeneracy. [61] For details, see Supplemental Material. [34] T. Nomoto and H. Ikeda, arXiv:1607.02708. [62] Note that, in order to lift the band degeneracy on the [35] Y. Yanase, arXiv:1606.08563. zone face, we require both the spin-orbit coupling and [36] S. Kobayashi, Y. Yanase, and M. Sato, arXiv:1607.01862. the exchange interaction with the magnetic moments in [37] S. Fujimoto, J. Phys. Soc. Jpn. 75, 083704 (2006). UCoGe, while only latter is necessary in UPd2Al3. This [38] Y. Yanase and M. Sigrist, J. Phys. Soc. Jpn. 77, 124711 comes from the following reasons. Without the spin-orbit (2008). coupling, the direction of the magnetic moments can be [39] M. Kyogaku, Y. Kitaoka, K. Asayama, C. Geibel, C. freely chosen, and then, the extended time-reversal sym- Schank, and F. Steglich, J. Phys. Soc. Jpn. 62, 4016 metry {θ2z|tn} is restored in UCoGe. As a result, there (1993). remains two-fold degeneracy on the zone face due to the [40] N. Bernhoeft, Eur. Phys. J. B 13, 685 (2000). corresponding Kramers theorem, even in the presence of [41] T. Watanabe, K. Izawa, Y. Kasahara, Y. Haga, Y. Onuki, ferromagnetism. Note also that, in UCoGe, the spin-orbit P. Thalmeier, K. Maki, and Y. Matsuda, Phys. Rev. B 70, coupling vanishes along ky = 0 line on the zone face, which 184502 (2004). is also protected by the Kramers theorem for the anti- [42] Y. Shimizu, S. Kittaka, T. Sakakibara, Y. Tsutsumi, T. unitary operation {θ2yσh|ty − tn}. This implies that the Nomoto, H. Ikeda, K. Machida, Y. Homma, and D. Aoki, present line nodes also disappear along ky = 0 line, which Phys. Rev. Lett. 117, 037001 (2016). can be directly conﬁrmed by the present group-theoretical [43] M. Kyogaku, Y. Kitaoka, K. Asayama, N. Sato, T. arguments, and become arc line nodes similar to ones dis- Sakon, T. Komatsubara, C. Geibel, C. Schank, and F. cussed in Ref. [35]. Steglich, Physica B 186-188, 285 (1993). [44] R. Feyerherm, A. Amato, F.N. Gygax, A. Schenck, C. 1

Supplemental Materials: Symmetry protected line nodes in non-symmorphic magnetic space groups: Applications to UCoGe and UPd2Al3

I. MODEL HAMILTONIAN

In this supplementary material, we show the details of our model Hamiltonians, which mimic UPd2Al3 or UCoGe superconductors. For simplicity, we set the lattice constant to unity, and consider a single orbital on each U site.

A. UPd2Al3

The antiferromagnetic phase of UPd2Al3 belongs to the magnetic space group Pb21/m (the unique axis is chosen 1 to be c-axis). In the unit cell, two U atoms are placed at x1 = (0, 0, 0) and x2 = (0, 0, 2 ). A minimal tight-binding Hamiltonian contains four orbitals, corresponding to two atoms (sublattice) and the spin-1/2 degrees of freedom. The Hamiltonian H is deﬁned by,

X X X † H = hασ,βσ0 (k)cασ(k)cβσ0 (k), (S1) k αβ σσ0

† 0 where cασ(k)(cβσ0 (k)) is a creation (annihilation) operator of electrons with spin σ (σ ) =↑, ↓ on an atom α (β) = 1, 2. Note that, through this paper, we have used the site dependent Fourier transformation deﬁned as, 1 † X ik·(R+xα) † cασ(k) = √ e cασ(R), (S2) N R where N is a total number of unit cell, R is a lattice vector, and xα is a relative position for the site α in the unit cell. In this case, hασ,βσ0 (k) in the matrix form is given by,

0 0 x 0 z x h(k) = ε0(k)τ ⊗ σ + ε1(k)τ ⊗ σ + δM τ ⊗ σ , (S3)

0 kz µ µ where ε0(k) = −2txy(cos kx + cos ky) − 2tz cos kz − µ and ε1(k) = −2tz cos 2 . τ and σ (µ = 0, x, y, and z) are the Pauli matrices acting on the sublattice and the spin degrees of freedom, respectively. The third term of Eq. (S3) tunes the magnitude of the staggered magnetic moment along a-axis in the antiferromagnetic phase. In the superconducting state, the anomalous part ΨΓ given by

Γ X X Γ † † Ψ = ϕασ,βσ0 (k)cασ(k)cβσ0 (−k) (S4) k αβ

Γ should be added in the Hamiltonian H. Here, ϕασ,βσ0 (k) is the corresponding order parameter. The superscript Γ denotes an IR of the point group C2h. For the Ag spin-singlet pairing state discussed in Fig. 1, we can take momentum-independent (constant) order parameter,

ϕAg (k) = ∆τ 0 ⊗ (iσy), (S5) where ∆ is the gap amplitude. 0 The band structure in Fig. 1(a) is the result for the parameters (txy, tz, tz, δM , ∆, µ) = (1.0, 0.4, 0.1, 0.4, 0.1, −2.0). Figs. 1(c) and (e) were obtained by changing ∆ and δM , respectively.

B. UCoGe

0 0 The ferromagnetic phase of UCoGe belongs to the magnetic space group P n m a. U atoms are placed at x1 = 1 1 3 1 3 1 1 3 (x, 4 , z), x2 = ( 2 − x, 4 , z − 2 ), x3 = (1 − x, 4 , 1 − z), and x4 = ( 2 + x, 4 , 2 − z) in the unit cell, where x = 0.0101, z = 0.7075 [S1]. The ferromagnetic moments are aligned along c-axis. In this case, h(k) in Eq. (S1) is given by,

h(k) = h0(k) + hS(k) + hM , (S6) 2

TABLE I. The character table of point group P in UCoGe. ω is an arbitrary phase factor.

IRs E 2z θ2y θ2x I σh θI2y θI2x Ag 1 1 ω ω 1 1 ω ω Bg 1 −1 ω −ω 1 −1 ω −ω Au 1 1 ω ω −1 −1 −ω −ω Bu 1 −1 ω −ω −1 1 −ω ω

where h0(k), hS(k), and hM represent the hopping integral, the spin-orbit coupling, and the exchange interaction with the magnetic moments, respectively. These are given by,   ε0(k) ε12(k) ε13,−(k) ε14(k) ∗  ε12(k) ε0(k) ε14(k) ε13,+(k) 0 h0(k) =  ∗ ∗ ∗  ⊗ σ , (S7a) ε13,−(k) ε14(k) ε0(k) ε12(k)  ∗ ∗ ε14(k) ε13,+(k) ε12(k) ε0(k) z hS(k) = δS sin ky diag(1, −1, −1, 1) ⊗ σ , (S7b) z hM = δM diag(1, 1, 1, 1) ⊗ σ . (S7c)

Here, we only consider a simple spin-orbit coupling term, which lifts the band degeneracy on the zone face. Each element of the hopping matrix ε0(k), ε12(k), ε13,±(k), and ε14(k) is given by,

ε0(k) = −2ty cos ky − µ (S8a)

ky kz ε (k) = 4t cos cos (e−(4x−1)kx/2i + λ e−(4x+1)kx/2i), (S8b) 12 12 2 2 1 ky ε (k) = 2t cos (e−(2z−1)kz i + λ e−(2z−2)kz i)e±2xkxi, (S8c) 13,± 13 2 2 kx ε (k) = 2t cos e−(4z−3)kz /2i. (S8d) 14 14 2

In the superconducting phase, we set a typical Au-type gap function,

Au x y y y ϕ (k) = ∆ diag(1, 1, 1, 1) ⊗ (sin kxiσ σ + sin kyiσ σ ), (S9) where ∆ is the gap amplitude. Note that each element of the (magnetic) point group P is given by P = {E, 2z, θ2y, θ2x, I, σh, θI2y, θI2x}, whose IRs are summarized in TABLE I [S2]. Since the unitary transformation to the basis functions can connect the representations with diﬀerent ω in TABLE I, these representations are equivalent in the sense of corepresentations. Eq. (S9) corresponds to the case of ω = 1. Gap functions with ω 6= 1 can be Γ − 1 Γ simply obtained by the transformation ϕ (k) 7→ ω 2 ϕ (k) without any change of the nodal structure. The band structure in Fig. 1(b) is the result for the parameters,

(ty, t12, t13, t14, λ1, λ2, δM , δS, ∆, µ) = (0.1, 0.4, 1.0, 0.8, 0.8, 0.8, 0.2, 0.2, 0.02, −1.9).

Figs. 1(d) and (f) were obtained by changing ∆ and δS, respectively.

[S1] F. Canepa, P. Manfrinetti, M. Pani, and A. Palenzona, J. Alloy. Compd. 234, 225 (1996). [S2] V. P. Mineev, Phys. Rev. B 66, 134504 (2002).