Multipole Superconductivity and Unusual Gap Closing

Transcription

1 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, (2017). S. Kobayashi, SS, Y. Yanase, & M. Sato, in preparation. SS & Y. Yanase, in preparation.

2 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

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

4 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 specific k Ex.) Point group D 6h Γ 1 (A 1u ): k xˆx + k y ŷ, k z ẑ Line k z = 0 No line node with SOC Blount's theorem (1985): "No line node for odd-parity SC w/ SOC" M. Sigrist & K. Ueda, RMP (1991)

5 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 6h 4) + TRS M. R. Norman (1995) T. Micklitz & M. R. Norman (2009) A 1u : k z =0, Gap, k z = Line node Incompatible w/ Blount's theorem due to nonsymmorphic symmetry Unusual gap structures beyond Sigrist-Ueda method!

6 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) Nonsymmorphic symmetry Difference in reps. of gap k z Zone face (ZF) Basal plane (BP) between BP and ZF k x k y 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?

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

8 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

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

10 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

11 System for classification of line nodes Gap classification: inversion On high-sym. k-plane: mirror (glide) & primitive lattice Space group contains C 2h : G = Note) 8 {E 0}T + {I 0}T + {C 2? 0}T + {? 0}T >< {E 0}T + {I 0}T + {C 2? k }T + {? k }T {E 0}T + {I 0}T + {C 2?? }T + {?? }T >: {E 0}T + {I 0}T + {C 2? k +? }T + {? k +? }T {p a}r = pr + a : space group operator T : translation group, : non-primitive translation Nonsymmorphic (i) Rotation + Mirror (ii) Rotation + Glide (iii) Screw + Mirror (iv) Screw + Glide twofold axis mirror plane

12 Magnetic space group M Ferromagnetic (FM) M = G (Unitary space group) Paramagnetic (PM) or Antiferromagnetic (AFM) M = G + G Anti-unitary operator: { 0} (a) PM { } (b) AFM 1 = { } (c) AFM 2 { + } (d) AFM 3 (b) twofold axis

13 Classification results by IRs of C 2h G = 8 {E 0}T + {I 0}T + {C 2? 0}T + {? 0}T >< {E 0}T + {I 0}T + {C 2? k }T + {? k }T {E 0}T + {I 0}T + {C 2?? }T + {?? }T >: {E 0}T + {I 0}T + {C 2? k +? }T + {? k +? }T (i) R (ii) (iii) (iv) = { 0} (a) { } (b) { } (c) { + } (d) FM PM AFM G BP (k? = 0) ZF (k? = ) (i), (ii) - A u A u (iii), (iv) - B u (i), (ii) (a), (b) A g +2A u + B u (i), (ii) (c), (d) B g +3A u A g +2A u + B u (iii), (iv) (a), (b) A g +3B u (iii), (iv) (c), (d) B g + A u +2B u UCoGe [1] Sr 2 IrO 4 UPt 3 [2] UPd 2 Al 3 [1], Sr 2 IrO 4 [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)

14 Application to centrosymmetric space groups 8 {E 0}T + {I 0}T + {C 2? 0}T + {? 0}T >< {E 0}T + {I 0}T + {C 2? k }T + {? k }T G = {E 0}T + {I 0}T + {C 2?? }T + {?? }T >: {E 0}T + {I 0}T + {C 2? k +? }T + {? k +? }T Monoclinic Orthorhombic No. SG?= y 10 P 2/m (i) 11 P 2 1 /m (iii) 13 P 2/c (ii) 14 P 2 1 /c (iv) Cubic No. SG?= x, y, z 200 Pm 3 (i) 201 Pn 3 (ii) 205 Pa 3 (iv) 221 Pm 3m (i) 222 Pn 3n (ii) 223 Pm 3n (i) 224 Pn 3m (ii) UPd 2 Al 3 UCoGe No. SG?= x?= y?= z 47 P mmm (i) (i) (i) 48 P nnn (ii) (ii) (ii) 49 Pccm (ii) (ii) (i) 50 P ban (ii) (ii) (ii) 51 P mma (iii) (i) (ii) 52 P nna (ii) (iv) (ii) 53 P mna (i) (ii) (iv) 54 Pcca (iv) (ii) (ii) 55 P bam (iv) (iv) (i) 56 Pccn (iv) (iv) (ii) 57 P bcm (ii) (iv) (iii) 58 P nnm (iv) (iv) (i) 59 P mmn (iii) (iii) (ii) 60 P bcn (iv) (ii) (iv) 61 P bca (iv) (iv) (iv) 62 P nma (iv) (iii) (iv) 63 Cmcm - - (iii) 64 Cmca - - (iv) 65 Cmmm - - (i) 66 Cccm - - (i) 67 Cmma - - (ii) 68 Ccca - - (ii) (i) R (ii) (iii) (iv) Sr 2 IrO 4 (This talk) Tetragonal No. SG?= z?= x, y 83 P 4/m (i) - 84 P 4 2 /m (i) - 85 P 4/n (ii) - 86 P 4 2 /n (ii) P 4/mmm (i) (i) 124 P 4/mcc (i) (ii) 125 P 4/nbm (ii) (ii) 126 P 4/nnc (ii) (ii) 127 P 4/mbm (i) (iv) 128 P 4/mnc (i) (iv) 129 P 4/nmm (ii) (iii) 130 P 4/ncc (ii) (iv) 131 P 4 2 /mmc (i) (i) 132 P 4 2 /mcm (i) (ii) 133 P 4 2 /nbc (ii) (ii) 134 P 4 2 /nnm (ii) (ii) 135 P 4 2 /mbc (i) (iv) 136 P 4 2 /mnm (i) (iv) 137 P 4 2 /nmc (ii) (iii) 138 P 4 2 /ncm (ii) (iv) Hexagonal No. SG?= z?= [1 10], [120], [210] 175 P 6/m (i) P 6 3 /m (iii) P 6/mmm (i) (i) 192 P 6/mcc (i) (ii) 193 P 6 3 /mcm (iii) (i) 194 P 6 3 /mmc (iii) (ii) UPt 3

15 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

16 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 Theory of d-wave superconductivity Expectation for (high-t c ) superconductivity! J eff = 1/2 antiferromagnet Y. K. Kim et al. (2014, 2016) / Y. J. Yan et al. (2015) H. Watanabe et al. (2013) F. Ye et al. (2013)

17 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) a SS, T. Nomoto, & Y. Yanase PRL 119, (2017) Symmetry analysis E u representation of D 4h Odd-parity magnetic quadrupole FFLO superconductivity

18 Gap classification (1) Magnetic space group in ++ state (P I cca) M ++ = G ++ + { x + y + z }G ++, G ++ = {E 0}T + {I 0}T + {C 2z x + z }T + { z x + z }T + {C 2x z }T + { x z }T + {C 2y x }T + { y x }T Comparison with general classification k z =0, /c : (iv) Screw + Glide, (d) AFM 3 k x,y =0, /a : (ii) Rotation + Glide, (d) AFM 3 G = 8 {E 0}T + {I 0}T + {C 2? 0}T + {? 0}T >< {E 0}T + {I 0}T + {C 2? k }T + {? k }T {E 0}T + {I 0}T + {C 2?? }T + {?? }T >: {E 0}T + {I 0}T + {C 2? k +? }T + {? k +? }T (i) R (ii) (iii) (iv) = { 0} (a) { } (b) { } (c) { + } (d)

19 Gap classification (2) Representations decomposed by IRs of C 2h G BP (k? = 0) ZF (k? = ) (i), (ii) - A u A u (iii), (iv) - B u (i), (ii) (a), (b) A g +2A u + B u (i), (ii) (c), (d) B g +3A u A g +2A u + B u (iii), (iv) (a), (b) A g +3B u (iii), (iv) (c), (d) B g + A u +2B u k x,y =0, /a k z =0, /c Induction to D 4h space k z =0, /c A 1g + A 2g + B 1g + B 2g +2A 1u +2A 2u +2B 1u +2B 2u +2E u BP 2E g + A 1u + A 2u + B 1u + B 2u +4E u ZF k x,y =0, /a A 1g + B 1g + E g +2A 1u + A 2u +2B 1u + B 2u +3E u BP A 2g + B 2g + E g +3A 1u +3B 1u +3E u ZF

20 Gap classification (3) k z =0, /c : horizontal s-wave d-wave A 1g + A 2g + B 1g + B 2g +2A 1u +2A 2u +2B 1u +2B 2u +2E u BP 2E g + A 1u + A 2u + B 1u + B 2u +4E u k x,y =0, /a : vertical ZF Zone face (ZF) Basal plane (BP) A 1g + B 1g + E g +2A 1u + A 2u +2B 1u + B 2u +3E u BP A 2g + B 2g + E g +3A 1u +3B 1u +3E u ZF Nontrivial gap structures! k z =0 k z = /c k x,y =0 k x,y = /a A 1g (s-wave) gap node gap node B 2g (d-wave) gap node node gap

21 Numerical calculation Gap structure obtained by group theory k z =0 k z = /c k x,y =0 k x,y = /a A 1g (s-wave) gap node gap node B 2g (d-wave) gap node node gap Demonstration by numerical calculation Effective J eff = 1/2 model 3D single-orbital tight-binding model 8 Ir atoms / unit cell & 3 types of ASOC Mean-field theory: space ˆ (s) (k) = ˆ (d) (k) = 0 iˆ(spin) y ˆ1 2 0 sin k xa 2 sin k ya iˆ(spin) y 2 ˆ(sl) x ˆ1 2

22 Numerical results (s-wave & horizontal plane) Calculation using effective Jeff = 1/2 model kz = 0 kz = /c kx,y = 0 kx,y = /a A1g (s-wave) gap node gap node B2g (d-wave) gap node node gap Nontrivial line nodes protected by nonsymmophic symmetry!

23 Numerical results (s-wave & vertical plane) Calculation using effective Jeff = 1/2 model kz = 0 kz = /c kx,y = 0 kx,y = /a A1g (s-wave) gap node gap node B2g (d-wave) gap node node gap Nontrivial line nodes protected by nonsymmophic symmetry!

24 Numerical results (d-wave & horizontal plane) Calculation using effective Jeff = 1/2 model kz = 0 kz = /c kx,y = 0 kx,y = /a A1g (s-wave) gap node gap node B2g (d-wave) gap node node gap Nontrivial line nodes protected by nonsymmophic symmetry!

25 Numerical results (d-wave & vertical plane) Calculation using effective Jeff = 1/2 model kz = 0 kz = /c kx,y = 0 kx,y = /a A1g (s-wave) gap node gap node B2g (d-wave) gap node node gap Nontrivial gap opening protected by nonsymmophic symmetry! Usual d-wave OP: vanishes on ZF and BP

26 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

27 Symmetry-protected point nodes Previous subject: Complete classification of line nodes Gap classification on high-symmetry plane This subject: unusual "j z -dependent" point nodes Gap classification on high-symmetry line Consider n-fold axis in BZ (n = 2, 3, 4, 6) A example of unconventional result: 3-fold line of UPt 3

28 Crystal symmetry of UPt 3 Point group: D 6h Globally centrosymmetric Site symmetry: D 3h Local parity violation Sublattice-dependent ASOC (Zeeman-type) r 1 e 1 e 2 r 3 r 2 e 3 a : z = 0 b : z = 1/2 Two uranium sublattices (a & b)

29 Symmetry of superconductivity in UPt 3 Multiple SC phases R. A. Fisher et al. (1989) S. Adenwalla et al. (1990) Multi-component E 2u order parameter: ˆ (k) = 1ˆE2u 1 + 2ˆE2u 2 Consistency with experiments: Phase diagram Broken TRS in B phase Hybrid gap structure R. Joynt & L. Taillefer, RMP (2002) Weak in-plane anisotropy in H c2 (θ) H C (η 1, η 2 ) = (0, 1) B (1, i) (1, 0) A T Suppression in H c2 c

30 Fermi surfaces of UPt 3 First-principle study Γ-FSs, A-FSs, & K-FSs K-FSs: NOT sufficiently studied (cf. Weyl nodes on Γ- & A-FSs based on E 2u ) H K T. Nomoto & H. Ikeda (2016) K-H line: 3-fold rotation symmetry Investigate superconducting gap structure on the K-H line of UPt 3!

31 Gap classification on K-H line Little group on K-H line: Gap classification Bloch state [ k (m)] M k /T C 3v + { I 0}C 3v Cooper pair [P k (m)] C 3v E C 3 C v E 1/ E 3/ D 3d E C 3,C 2 3 3C 0 2 I IC 3,IC v P k P k j z = ±1/2, ±3/2 P k 1 = A 1g + A 1u + E u P k 2 = A 1g +2A 1u + A 2u Two nonequivalent representations of Cooper pair depending on Bloch-state angular momentum j z!

32 Gap structure Order parameter: E 2u representation of D 6h Compatibility relations between D 6h and D 3d (IR of D 6h ) A 1u A 2u B 1u B 2u E 1u E 2u (IR of D 6h ) D 3d A 1u A 2u A 2u A 1u E u E u R. Joynt & L. Taillefer, RMP (2002) Irreducible decomposition P k 1 P k 2 = A 1g + A 1u + E u = A 1g +2A 1u + A 2u Gap (normal: E 1/2 ) Point node (normal: E 3/2 ) Next problem: Which normal state is realized on K-H line?

33 Effective model of UPt 3 3D single-orbital tight-binding model by Yanase Two uranium sublattices (a & b) Zeeman-type ASOC Y. Yanase, PRB (2016, 2017) r 1 e 1 e 2 r 3 r 2 e a : z = 0 3 b : z = 1/2

34 j z -dependent gap structure K-H line K -H line ASOC α > 0 K', H' Ik K, H orbital spin perm. j z =?? j z = l z + s z + λ z α < 0 = 0 ± 1/2 +?? Exchange j z =??

35 Effective orbital angular momentum Bloch function on each site has a phase factor e ik r Phase factor by 3-fold rotation at K point... Note) Opposite phase factor at K' point Effective angular momentum by site permutation! a : e +i2 /3 z =+1 b : e i2 /3 z = 1

36 j z -dependent gap structure K-H line K -H line ASOC α > 0 orbital spin perm. j z = l z + s z + λ z = 0 ± 1/2 ± 1 Point node α < 0 j z = 0 ± 1/2 1 Gap

37 Numerical results E 2u order parameter by Yanase Y. Yanase, PRB (2016, 2017) Intra-sublattice p-wave Inter-sublattice d+f-wave Disappear on K-H line... α > 0: E 3/2 Point node + Inter-sublattice p-wave ab initio study by J. Ishizuka α < 0: E 1/2 Gap

38 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

39 Conclusions Gap structures beyond the Sigrist-Ueda method Classification on high-symmetry plane in BZ: Line nodes protected by nonsymmorphic symmetry due to non-primitive translation? mirror plane Application to magnetic octupole state in Sr 2 IrO 4 Classification on high-symmetry line in BZ: j z -dependent point nodes (gap opening) on K-H line of UPt 3 This result is general on 3- or 6-fold axis