Quantum Spin Liquids and Majorana Metals

Transcription

1 Quantum Spin Liquids and Majorana Metals Maria Hermanns University of Cologne M.H., S. Trebst, PRB 89, (2014) M.H., K. O Brien, S. Trebst, PRL 114, (2015) M.H., S. Trebst, A. Rosch, arxiv: (2015)

2 Collaborators Kevin O Brien Simon Trebst Achim Rosch University of Cologne

3 Fractionalization in strongly correlated systems (Strong) interactions can lead to emergent quasiparticles with fractional quantum numbers Spin-charge separation Fractional electric charge Magnetic monopoles e Fermi gas in 1D! q e charge spin e Electron gas in 2D e/3 e/3 e/3 Spin ice q J. Schlappa et al., Nature 485, 82 (2012) 1D Tsui, Störmer, Gossard, PRL 48, 1559 (1982) 2D Castelnovo et al., Nature 451, (2007) 3D 3

4 Fractionalization in strongly correlated systems (Strong) interactions can lead to emergent quasiparticles with fractional quantum numbers Spin-charge separation Fractional electric charge Magnetic monopoles e Fermi gas in 1D! q e charge spin e Electron gas in 2D e/3 e/3 e/3 Skyrmions in chiral magnets q J. Schlappa et al., Nature 485, 82 (2012) 1D Tsui, Störmer, Gossard, PRL 48, 1559 (1982) 2D Milde et al., Science 340, 1076 (2013) 3D 3

5 Towards spin liquids frustration geometric frustration exchange frustration? z i z j? x i x j y i y j? 4

6 Towards spin liquids frustration geometric frustration exchange frustration? z i z j? x i x j y i y j? hyperkagome lattice Na4Ir3O8 Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, PRL 99, (2007) 4

7 Towards spin liquids frustration geometric frustration exchange frustration? z i z j? x i x j y i y j? hyperkagome lattice Na4Ir3O8 Na2IrO3 Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, PRL 99, (2007) Chun, S. H. et al. Nature Physics 11, (2015) Trebst, Gegenwart, Nature Physics 11, (2015) 4

8 Towards spin liquids frustration geometric frustration exchange frustration? z i z j? x i x j y i y j? hyperkagome lattice Na4Ir3O8 hyperhoneycomb lattice β-li2iro3 Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, PRL 99, (2007) Takayama et al., PRL 114, (2015) 4

9 Exchange frustration in Iridates G. Jackeli and G. Khaliullin, PRL 102, (2009) J. Chaloupka, G. Jackeli, and G. Khaliullin, PRL 105, (2010) Interplay of octahedral crystal field spin-orbit coupling correlations spin-orbit-entangled spin-1/2 moment on Ir atom IrO6 Heisenberg exchange dominant Kitaev exchange corner-sharing edge-sharing parallel edge-sharing Sr2IrO4 superconductivity Na2IrO3 / Li2IrO3 quantum spin liquids Ba3IrTi2O9 Z2 vortex lattice M. Becker, M. H., B. Bauer, M. Garst, and S. Trebst, PRB 91, (2015) 5

10 A family of Li2IrO3 compounds Li 2 IrO 3 Li 2 IrO 3 Li 2 IrO 3 crystals Singh et al., PRL 108, (2008) Takayama et al., PRL 114, (2015) Modic et al., Nat. Comm. 5, 4203 (2014) Kitaev models Na2IrO3: Singh, Gegenwart, PRB 82, (2010) RuCl3: Majumder et al. PRB 91, (R) (2015) 6

11 3D Kitaev models 7

12 Spin fractionalization and Majorana fermions A. Kitaev, Annals of Physics 321, 2 (2006) splitting spins = ia c a x a y a z C X H = bond J j k represent spins by four Majorana fermions emergent Z2 gauge field on bonds joining Majoranas û jk = ia j a k Hilbert space split into two separate sectors: 2 N =2 N/2 2 N/2 ia x a x ia y a y Majorana fermions cj spinons flux loops visons (static and gapped) ia z a z C 8

13 Spin fractionalization and Majorana fermions A. Kitaev, Annals of Physics 321, 2 (2006) splitting spins = ia c a x a y a z C X H = i J c j c k bond represent spins by four Majorana fermions emergent Z2 gauge field on bonds joining Majoranas û jk = ia j a k Hilbert space split into two separate sectors: 2 N =2 N/2 2 N/2 ia x a x ia y a y Majorana fermions cj spinons flux loops visons (static and gapped) ia z a z C 8

14 Kitaev spin liquids in 2D Kitaev, Annals of Physics 06 J z gapped spin liquid Toric code J x + J y + J z = const. H = i X bond J c j c k J y gapless spin liquid Dirac cones J x 9

15 A family of Kitaev models in 3D A. F. Wells, 3D nets and polyhedra Tri-coordinated lattices with equal bond length 120 bond angles 2D 3D descritpion # (6,3) 1 (10,3) 3 (9,3) 2 (8,3) 4 (n, p) connectivity: p nearest neighbors polygonality: shortest circuit including any pair of neighboring links is an n-gon 10

16 (10,3)b Fermi line space group # 70 (Fddd) inversion symmetric edge-sharing oxygen octahedra 11

17 (10,3)b Fermi line Mandal, Surendran, PRB 79, (2009) Lee etal., PRB 89, (2014) Kimchi, Analytis, Vishwanath, PRB 90, (2014) Nasu, Udagawa, Motome, PRL 113, (2014 ) J z gapped spin liquid H = i X J c j c k J x + J y + J z = const. bond J x J y gapless spin liquid Fermi line 12

18 (10,3)a Majorana Fermi surface space group # 214 (I 4132) cubic symmetry chiral edge-sharing oxygen octahedra 90 screw-rotation 120 rotation rotation

19 Majorana Fermi surface M.H., S. Trebst, PRB 89, (2014) J z gapped spin liquid J x + J y + J z = const. energy E(K) J x -0.8 J y gapless spin liquid with Majorana Fermi surface Γ P N H Γ N P H momentum K P N H 14

20 Majorana Fermi surface in the gapless region J z energy E(K) J x = 1/2 J x = 0.46 J x = 0.42 J x = 0.37 J x = 1/3-0.8 J x Γ P N H Γ N P H momentum K J y 15

21 Majorana Fermi surface in the gapless region J z energy E(K) J x = 0 J x = 0.1 J x = 0.17 J x = 0.26 J x = 1/3-0.8 J x Γ P N H Γ N P H momentum K J y 16

22 Symmetries in Majorana systems Particle-hole symmetry (k) = ( k) honeycomb lattice Sublattice symmetry (k) = (k k 0 ) Time-reversal symmetry (k) = ( k + k 0 ) ~a 2 ~a 1 k 0 =0 hyperoctagon lattice k 0 is the reciprocal lattice vector of the translation vector of the sublattice ~a 2 ~a 3 ~a 1 k 0 =(q 2 + q 3 )/2 17

23 Symmetries Fermi surface Particle-hole symmetry Time-reversal symmetry (k) = ( k) (k) = ( k k 0 ) k 0 =0: honeycomb / hyperhoneycomb k 0 6=0: hyperoctagon particle-hole symmetry at every k point doubly degenerate zero-modes at same k 0 A h(k) = A 0 protected by time-reversal stable zero-mode manifolds are separated points (2D) and lines (3D) generic band hamiltonian at given k zero-modes are at different k h(k) = 0 0 A... A 0 stable zero-mode manifolds are lines (2D) and surfaces (3D) 1 C A 18

24 Spin-Peierls BCS instability M.H., S. Trebst, A. Rosch, arxiv: (2015) natural BCS-type instability for time-reversal invariant systems Majorana fermions ck perfect nesting k 0 complex fermion fk BCS pairing Fermi surface centered around k0/2 order parameter has finite momentum k0: hf k 0 +q f k 0 q i spontanous breaking of translational symmetry spin-peierls transition additional breaking of rotation symmetry possible order parameter distribution 19

25 Stabilizing the Fermi surface BCS instability cut off at low T by breaking of time-reversal external magnetic field in (1,1,1)-direction H eff = J X hj,ki j k apple X hj,k,li j k l j l k apple > apple =0.1 apple =0.5 apple =

26 Weyl spin liquids 21

27 Weyl nodes and Weyl semi-metals Touching of two bands in 3D is generically linear 3X Ĥ = ~v 0 ~q 1 + ~v j ~q j Weyl nodes j=1 Weyl nodes are sources/sinks of Berry flux ~B n ( ~ k)=r ~k ihn( ~ k) r ~k n( ~ k)i with chirality sign[~v 1 (~v 2 ~v 3 )] Surface BZ Fermi arc if the Weyl nodes are pinned at zero energy unusal surface states: Fermi arcs topological semi-metal with protected surface states (metallic cousin of the topological insulator) bulk BZ ky k z k x Wan et al., PRB 83, (2011) 22

28 Gapping the Majorana Fermi line external magnetic field in (1,1,1)-direction H eff = J X hj,ki j k apple X hj,k,li j k l j l k apple > 0 23

29 Gapping the Majorana Fermi line external magnetic field in (1,1,1)-direction H eff = J X hj,ki j k apple X hj,k,li j k l j l k ε(k) k z k y apple > 0 ε(k) k z apple > 0 k y 23

30 Gapping the Majorana Fermi line external magnetic field in (1,1,1)-direction H eff = J X hj,ki j k apple X hj,k,li j k l j l k apple > 0 Breaking time-reversal reduces line to a pair of gapless Weyl nodes Weyl nodes are pinned to zero energy due to inversion symmetry 23

31 Topology of Weyl superconductors Sources/sinks of Berry flux with chirality sign[~v 1 (~v 2 ~v 3 )] non-zero Chern number for surface surrounding a Weyl node sink 1 chern number N(K) 0.5 apple =0.05 source momentum K 24

32 Fermi arcs slab geometry topologically protected gapless surface states apple =0 apple =0.01 surface Brillouin zone 25

33 Evolution of Weyl nodes and Fermi arcs apple =0 apple =0.01 apple = 1 2 r

34 Evolution of Weyl nodes and Fermi arcs apple =0 apple =0.01 apple =0.4 26

35 Evolution of Weyl nodes and Fermi arcs apple =0 apple =0.01 apple = apple =0.45 apple =0.5 apple =0.55

36 More Majorana metals 27

37 3D Kitaev lattices Name Chirality Gapless modes TR broken Lieb theorem Peierls instability (10,3)a (hyperoctagon) (10,3)b (hyperhoneycomb) chiral Fermi surface Fermi surface - inversion Fermi line Weyl nodes - - (10,3)c chiral Fermi line topological Fermi surface - - (9,3)a inversion Weyl nodes Weyl nodes - - (9,3)b inversion (8,3)a chiral Fermi surface Fermi surface - (8,3)b inversion Weyl nodes Weyl nodes (8,3)c inversion Fermi line Weyl nodes - - (8,3)d inversion gapped gapped - - (6,3)a honeycomb inversion Dirac points gapped NA phase - 28

38 3D Kitaev lattices Name Chirality Gapless modes TR broken Lieb theorem Peierls instability (10,3)a (hyperoctagon) (10,3)b (hyperhoneycomb) chiral Fermi surface Fermi surface - inversion Fermi line Weyl nodes - - (10,3)c chiral Fermi line topological Fermi surface - - (9,3)a inversion Weyl nodes Weyl nodes - - (9,3)b inversion (8,3)a chiral Fermi surface Fermi surface - (8,3)b inversion Weyl nodes Weyl nodes (8,3)c inversion Fermi line Weyl nodes - - (8,3)d inversion gapped gapped - - (6,3)a honeycomb inversion Dirac points gapped NA phase - 29

39 (8,3)b Weyl SL with Inversion & TR K. O Brien, M.H., S. Trebst, in preparation space group # 166 (R-3m) inversion symmetric edge-sharing oxygen octahedra Lieb theorem 30

40 Kitaev interaction on the (8,3)b lattice effect of symmetries: PHS: k = k TR: k = k+k 0 Inv: k = k source Weyl nodes with both time-reversal and inversion possible perfect nesting with vector k 0 =(2, 2 p 3, 5 p 6 ) k 0 sink 31

41 Topology of Weyl spin liquids non-zero Chern number for surface surrounding a Weyl node Fermi arcs for open boundary conditions along 001-direction k 0 32

42 Breaking time-reversal symmetry apple =0 apple =0.1 k 0 k 0 33

43 Breaking time-reversal symmetry breaking time-reversal destroys perfect nesting condition apple =0 apple =0.1 k 0 k 0 33

44 3D Kitaev lattices Name Chirality Gapless modes TR broken Lieb theorem Peierls instability (10,3)a (hyperoctagon) (10,3)b (hyperhoneycomb) chiral Fermi surface Fermi surface - inversion Fermi line Weyl nodes - - (10,3)c chiral Fermi line topological Fermi surface - - (9,3)a inversion Weyl nodes Weyl nodes - - (9,3)b inversion (8,3)a chiral Fermi surface Fermi surface - (8,3)b inversion Weyl nodes Weyl nodes (8,3)c inversion Fermi line Weyl nodes - - (8,3)d inversion gapped gapped - - (6,3)a honeycomb inversion Dirac points gapped NA phase - 34

45 Summary (Strong) interactions can lead to emergent quasiparticles with fractional quantum numbers Spin-charge separation Electron fractionalization Magnetic monopoles e Fermi gas in 1D e! q charge spin e Quantum Hall liquids e/3 e/3 e/3 Skyrmions in chiral metals q Tsui, Störmer, Gossard, PRL 48, 1559 (1982) Milde et al., Science 340, 1076 (2013) 35

46 Summary (Strong) interactions can lead to emergent quasiparticles with fractional quantum numbers Spin liquid with emergent spinon Fermi surface Spin liquid with emergent spinon Fermi line Weyl spin liquid Majorana Metal Zoo of Majorana metals Weyl superconductor M.H., S. Trebst, PRB 89, (2014) M.H., K. O Brien, S. Trebst, PRL 114, (2015) M.H., S. Trebst, A. Rosch, arxiv: (2015) K. O Brien, M.H., S. Trebst, in preparation 35