Multipole Superconductivity and Unusual Gap Closing Application to - PowerPoint PPT Presentation

Novel Quantum States in Condensed Matter 2017 Nov. 17, 2017 Multipole Superconductivity and Unusual Gap Closing ― Application to Sr 2 IrO 4 and UPt 3 ― Department of Physics, Kyoto University Shuntaro Sumita SS, T. Nomoto, & Y. Yanase, Phys. Rev. Lett. 119 , 027001 (2017). S. Kobayashi, SS, Y. Yanase, & M. Sato, in preparation. SS & Y. Yanase, in preparation.

Outline 1. Introduction 2. Method: Superconducting gap classification based on space group symmetry 3. Result 1: Condition for nonsymmorphic line nodes (a) Complete classification (b) Application: Sr 2 IrO 4 4. Result 2: j z -dependent point nodes in UPt 3 5. Conclusion

Outline 1. Introduction 2. Method: Superconducting gap classification based on space group symmetry 3. Result 1: Condition for nonsymmorphic line nodes (a) Complete classification (b) Application: Sr 2 IrO 4 4. Result 2: j z -dependent point nodes in UPt 3 5. Conclusion

Superconducting gap classification (point group) ‣ Most studies: gap classification based on 
 crystal point group symmetry ‣ Classification theory by Sigrist & Ueda (1991) (1) Most simple basis of OP 
 Classification for D 6h for each point group (2) Gap or node @ specific k ‣ Ex.) Point group D 6h Line node @ k z = 0 Γ 1 − ( A 1u ): k x ˆ x + k y ˆ y, k z ˆ z → No line node with SOC Blount's theorem (1985): "No line node for odd-parity SC w/ SOC" M. Sigrist & K. Ueda, RMP (1991)

Superconducting gap classification (space group) ‣ Recent study: importance of space group symmetry (Point group) + (Translation) ‣ Classification theory based on (magnetic) space group (1) Choosing specific k at first (2) The presence or absence of Cooper pair w.f. ‣ Ex.) Space group P6 3 /mmc ( D 6h4 ) + TRS M. R. Norman (1995) T. Micklitz & M. R. Norman (2009) A 1u : → Gap, → Line node 
 k z = 0 , k z = π Incompatible w/ Blount's theorem due to nonsymmorphic symmetry ‣ Unusual gap structures beyond Sigrist-Ueda method !

Results by space group & Unsolved questions ‣ Nontrivial line nodes by nonsymmorphic symmetry • UPt 3 , UCoGe, UPd 2 Al 3 T. Micklitz & M. R. Norman (2009, 2017) T. Nomoto & H. Ikeda (2017) k z Nonsymmorphic symmetry Zone face (ZF) ↓ Basal plane (BP) Difference in reps. of gap k y between BP and ZF k x → What is the condition for nontrivial line nodes ? ‣ Only a few & less-known studies of point nodes → Do point nodes peculiar to crystal group exist ?

Abstract of this study Aim 1 Aim 2 Investigate the condition for Consider crystal symmetry- nonsymmorphic line nodes protected point nodes Method Gap classification based on space group symmetry Result 1 Result 2 Complete classification j z -dependent point nodes ‣ Application: Sr 2 IrO 4 ‣ Application: UPt 3

Outline 1. Introduction 2. Method: Superconducting gap classification based on space group symmetry 3. Result 1: Condition for nonsymmorphic line nodes (a) Complete classification (b) Application: Sr 2 IrO 4 4. Result 2: j z -dependent point nodes in UPt 3 5. Conclusion

Gap classification (Flow) High-symmetry Little group k -point Magnetic space group ‣ Small representation ‣ Centrosymmetric = Bloch state at k I k = − k Bloch state at − k Bloch state at k Antisymmetric direct product ( Mackey-Bradley theorem ) Representation of Cooper pair wave function

Outline 1. Introduction 2. Method: Superconducting gap classification based on space group symmetry 3. Result 1: Condition for nonsymmorphic line nodes (a) Complete classification (b) Application: Sr 2 IrO 4 4. Result 2: j z -dependent point nodes in UPt 3 5. Conclusion

System for classification of line nodes ‣ Gap classification: inversion ‣ On high-sym. k -plane: mirror (glide) & primitive lattice → Space group contains C 2h :  { E | 0 } T + { I | 0 } T + { C 2 ? | 0 } T + { σ ? | 0 } T (i) Rotation + Mirror    { E | 0 } T + { I | 0 } T + { C 2 ? | τ k } T + { σ ? | τ k } T (ii) Rotation + Glide  G = { E | 0 } T + { I | 0 } T + { C 2 ? | τ ? } T + { σ ? | τ ? } T (iii) Screw + Mirror    { E | 0 } T + { I | 0 } T + { C 2 ? | τ k + τ ? } T + { σ ? | τ k + τ ? } T (iv) Screw + Glide  twofold axis Note) • : space group operator { p | a } r = p r + a • T : translation group mirror plane • : non-primitive translation τ � , τ � → Nonsymmorphic

Magnetic space group M ‣ Ferromagnetic (FM) (Unitary space group) M = G ‣ Paramagnetic (PM) or Antiferromagnetic (AFM) M = G + ˜ θ G (b) ‣ Anti-unitary operator: twofold  { θ | 0 } (a) PM axis    { θ | τ � } (b) AFM 1  ˜ θ = { θ | τ � } (c) AFM 2    { θ | τ � + τ � } (d) AFM 3 

Classification results by IRs of C 2h  { θ | 0 } (a)  { E | 0 } T + { I | 0 } T + { C 2 ? | 0 } T + { σ ? | 0 } T (i) R      { θ | τ � } (b)  { E | 0 } T + { I | 0 } T + { C 2 ? | τ k } T + { σ ? | τ k } T (ii)   ˜ θ = G = { θ | τ � } (c) { E | 0 } T + { I | 0 } T + { C 2 ? | τ ? } T + { σ ? | τ ? } T (iii)       { θ | τ � + τ � } (d)  { E | 0 } T + { I | 0 } T + { C 2 ? | τ k + τ ? } T + { σ ? | τ k + τ ? } T (iv)  ˜ BP ( k ⊥ = 0) ZF ( k ⊥ = π ) G θ (i), (ii) - A u FM A u ← UCoGe [1] (iii), (iv) - B u (i), (ii) (a), (b) A g + 2 A u + B u ← Sr 2 IrO 4 PM (i), (ii) (c), (d) B g + 3 A u A g + 2 A u + B u AFM ← UPt 3 [2] (iii), (iv) (a), (b) A g + 3 B u ← UPd 2 Al 3 [1], Sr 2 IrO 4 (iii), (iv) (c), (d) B g + A u + 2 B u [1] T. Nomoto & H. Ikeda (2017) / [2] T. Micklitz & M. R. Norman (2009) Non-primitive translation mirror (glide) plane ⊥ → Nonsymmorphic line node (gap opening) ! T. Micklitz & M. R. Norman (2017)

Application to centrosymmetric space groups Tetragonal  { E | 0 } T + { I | 0 } T + { C 2 ? | 0 } T + { σ ? | 0 } T (i) R    { E | 0 } T + { I | 0 } T + { C 2 ? | τ k } T + { σ ? | τ k } T (ii)  No. SG ⊥ = z ⊥ = x, y G = 83 P 4 /m (i) - { E | 0 } T + { I | 0 } T + { C 2 ? | τ ? } T + { σ ? | τ ? } T (iii)  84 P 4 2 /m (i) -   { E | 0 } T + { I | 0 } T + { C 2 ? | τ k + τ ? } T + { σ ? | τ k + τ ? } T (iv)  85 P 4 /n (ii) - 86 P 4 2 /n (ii) - Monoclinic Orthorhombic 123 P 4 /mmm (i) (i) 124 P 4 /mcc (i) (ii) 125 P 4 /nbm (ii) (ii) No. SG ⊥ = y No. SG ⊥ = x ⊥ = y ⊥ = z 126 P 4 /nnc (ii) (ii) 10 P 2 /m (i) 47 Pmmm (i) (i) (i) UPd 2 Al 3 127 P 4 /mbm (i) (iv) 11 P 2 1 /m (iii) 48 (ii) (ii) (ii) Pnnn 128 P 4 /mnc (i) (iv) 49 (ii) (ii) (i) 13 P 2 /c (ii) Pccm UCoGe 129 P 4 /nmm (ii) (iii) 50 Pban (ii) (ii) (ii) 14 P 2 1 /c (iv) 130 P 4 /ncc (ii) (iv) 51 (iii) (i) (ii) Pmma 131 P 4 2 /mmc (i) (i) 52 (ii) (iv) (ii) Pnna 132 P 4 2 /mcm (i) (ii) 53 Pmna (i) (ii) (iv) Sr 2 IrO 4 133 P 4 2 /nbc (ii) (ii) 54 (iv) (ii) (ii) Pcca 134 P 4 2 /nnm (ii) (ii) 55 (iv) (iv) (i) Pbam (This talk) 135 P 4 2 /mbc (i) (iv) 56 Pccn (iv) (iv) (ii) 136 P 4 2 /mnm (i) (iv) 57 (ii) (iv) (iii) Pbcm 137 P 4 2 /nmc (ii) (iii) 58 (iv) (iv) (i) Pnnm Cubic 138 P 4 2 /ncm (ii) (iv) 59 Pmmn (iii) (iii) (ii) Hexagonal 60 (iv) (ii) (iv) Pbcn No. SG ⊥ = x, y, z 61 (iv) (iv) (iv) Pbca Pm ¯ 200 3 (i) No. SG ⊥ = z ⊥ = [1 − 10] , [120] , [210] 62 Pnma (iv) (iii) (iv) Pn ¯ 201 3 (ii) 63 Cmcm - - (iii) 175 P 6 /m (i) - Pa ¯ 205 3 (iv) 64 - - (iv) 176 P 6 3 /m (iii) - Cmca Pm ¯ 221 3 m (i) 65 - - (i) Cmmm 191 P 6 /mmm (i) (i) Pn ¯ 222 3 n (ii) 66 Cccm - - (i) 192 P 6 /mcc (i) (ii) Pm ¯ 223 3 n (i) 67 - - (ii) 193 P 6 3 /mcm (iii) (i) Cmma UPt 3 Pn ¯ 224 3 m (ii) 68 - - (ii) 194 P 6 3 /mmc (iii) (ii) Ccca

Outline 1. Introduction 2. Method: Superconducting gap classification based on space group symmetry 3. Result 1: Condition for nonsymmorphic line nodes (a) Complete classification (b) Application: Sr 2 IrO 4 4. Result 2: j z -dependent point nodes in UPt 3 5. Conclusion

What is Sr 2 IrO 4 ? ‣ A layered perovskite insulator ‣ Nonsymmorphic lattice: I4 1 /acd or I4 1 /a ‣ Locally parity violation on Ir sites 
 → Sublattice-dependent ASOC ‣ Many similarities to high- T c cuprate • Pseudogap & d -wave gap under doping Y. K. Kim et al. (2014, 2016) / Y. J. Yan et al. (2015) • Theory of d -wave superconductivity F. Ye et al. (2013) H. Watanabe et al. (2013) → Expectation for (high- T c ) superconductivity ! • J eff = 1/2 antiferromagnet

Magnetic structures on Ir site ‣ Canted moments: AFM along a axis & FM along b axis − ++ − pattern B. J. Kim et al. (2009) Symmetry analysis ‣ B 1g representation of D 4h ‣ Even-parity magnetic octupole Unusual gap structures b − + − + pattern L. Zhao et al. (2016) Symmetry analysis a ‣ E u representation of D 4h SS, T. Nomoto, & Y. Yanase ‣ Odd-parity magnetic quadrupole PRL 119 , 027001 (2017) FFLO superconductivity