Weyl semimetals and topological phase transitions

Transcription

1 Weyl semimetals and topological phase transitions Shuichi Murakami 1 Department of Physics, Tokyo Institute of Technology 2 TIES, Tokyo Institute of Technology 3 CREST, JST Collaborators: R. Okugawa (Tokyo Tech) T. Yoda (Tokyo Tech) J. Tanaka (Tokyo Tech) M. Noro (Tokyo Tech) T. Yokoyama (Tokyo Tech) M. Hirayama (AIST) T. Miyake (AIST) S. Ishibashi (AIST) S. Kuga (U. Tokyo) N. Nagaosa (RIKEN, U. Tokyo) Y. Avishai (Ben Gurion) S. Iso (KEK) M. Onoda (Akita)

2 Introduction Weyl semimetals, Dirac semimetals Topological phase transitions topological insulator (TI) normal insulator (NI) phase transition Dirac/Weyl semimetals Universal phase diagram Application to any space groups without inversion symm. Crystals with helical lattice structure Weyl semimetal Current induced magnetization

3 Weyl semimetal Weyl semimetal = Bulk 3D Dirac cones without degeneracy Dirac semimetal = Bulk 3D Dirac cones with degeneracy Weyl node Surface Fermi arc connecting between Weyl nodes k-space (surface) k-space (bulk) Physics 4, 36 (2011) pyrochlore iridates (Wan et al., PRB (2011) ) TI multilayer (Burkov, Balents, PRL 107, (2011)). Wan et al., PRB (2011)

4 3D Weyl nodes = monopole or antimonopole for Berry curvature r カunk カunk Bn ( k) = i エカk カk r 1 ρn( k) = Bn( k) k 2π r r r : Berry curvature : monopole density k r or k r 0 Weyl node r r k = k 0 Monopole at r r r ρl ( k) = δ( k k0) r r k = k 0 Antimonopole at r r r ρl ( k) = δ( k k0) Weyl nodes are either monopole or antimonopole Quantized monopole charge They can appear/disappear only by pair creation/annihilation. C. Herring, Phys. Rev. 52, 365 (1937). G. E. Volovik, The Universe in a Helium Droplet (2007). S. Murakami, New J. Phys. 9, 356 (2007).

5 Weyl nodes in 2D and 3D m = 0 m ケ 0 2D Weyl node : r Hkm (, ) = kxs x+ kys y+ ms z parameter m opens a gap. 3D Weyl node : r Hkm (, ) = kxs x+ kys y + kzs z + ms z ( ) = k s + k s + k + m s x x y y z z Weyl point moves but gap does not open. (0,0,0)à (0,0,-m) ß Monopole charge (in 3D k-space) is conserved 3D Weyl node is topological.

6 Surface Fermi arc : effective model calc. Bulk band structure Bulk + surface Fermi arcs Weyl nodes Okugawa, Murakami, Phys. Rev. B 89 (2014)

7 Effective model: bulk Bulk dispersion m<0: bulk gap = = topological or normal insulator m>0: bulk is gapless gap closed at = Weyl semimetal Weyl points

8 Effective model : surface states Okugawa, Murakami, Phys. Rev. B 89, (2014) Haldane, arxiv: Unitary transf. z top surface bottom surface TI Fermi arcs ( ) Fermi arc connect between bulk Dirac cones tangential to Dirac cones

9 NI-TI phase transitions and Weyl/Dirac semimetals NI: normal insulator TI: topological insulator

10 Z 2 topological number ß Calculated from bulk Bloch wf. (A) systems without inversion symmetry (B) systems with inversion symmetry ( 1) ν =! i [ w Γi ] [ w Γ ] det ( ) Pf ( ) w mn( k) = u k, m Θ uk, n i ν ( 1) = ξ ( Γ ) i N m= 1 2m i Parity eigenvalue +1: symmetric -1: asymmetric Γ i : TRIM Fu,Kane, PRB(2007) Fu,Kane, PRB(2006) à How does the TI-NI topological phase transition occur in (A) & (B)?

11 Universal phase diagram in 3D and in 2D 3D 2D SM, Iso, Avishai, Onoda,Nagaosa PRB (07) SM, New J. Phys. ( 07). SM. Kuga, PRB ( 08) SM, Physica E43, 748 ( 11) δ m In 3D, Weyl nodes are monopoles and antimonopoles. conservation of monopole charge ( : inversion symmetry breaking) ( : external parameter )

12 Evolution of Weyl points by parameter change Time-reversal symmetry à Monopoles are symmetric w.r.t. k=0 Pair creation Pair annihilation Pair annihilation Pair creation

13 Phase transition between TI and NI phases SM, New J. Phys. 9, 356 ( 07); 3D No I-symmetry : WTI (or NI) Weyl semimetal STI I-symmetry : Monopoleantimonopole pair creation 2 monopoles and 2 antimonopoles Monopoleantimonopole pair annihilation Phase diagram WTI (or NI) Dirac semimetal TI

14 Lattice model: Fu-Kane-Mele model + staggered on-site energy Fu-Kane-Mele model (PRL98, (2007)) Diamond lattice nearest neighbor: spin-indep. hopping next nearest neighbor: spin-orbit coupling On-site staggered potenial à breaks inversion Weyl semimetal appears! (Murakami,Kuga, PRB78, (2008) Weyl semimetal Dirac semimetal Inversionsymmetric Weyl semimetal

15 Fu-Kane-Mele model + inversion-symmetry breaking Weyl semimetal Weyl semimetal k-space trajectory of the monopoles

16 top surface Evolution of surface states Weyl semimetal : Fermi arc (between monopole and antimonopole) bottom surface Weyl semimetal : Fermi arc Topological insulator : Dirac cone Okugawa, Murakami, Phys. Rev. B 89, (2014)

17 Change of surface terminations No dangling bonds Okugawa, Murakami, Phys. Rev. B 89, (2014) With dangling bonds Surface termination change the pairing of Weyl nodes Weyl semimetal Topological insulator Surface Fermion parity à surface termination changes the inside and outside of surface FS. (for inversion symmetric systems) Teo, Fu, Kane, PRB78, (2008)

18 Universal phase diagram in 3D SM, Iso, Avishai, Onoda,Nagaosa PRB (07) SM, New J. Phys. ( 07). SM. Kuga, PRB ( 08) SM, Physica E43, 748 ( 11) Only the inversion and time-reversal symmetries are considered. Question: Does it hold for crystals with additional crystallographic symmetries?

19 BiTeI at high pressure: ab initio calc. Bahramy, Nat. Commun. 3, 679 (2012) Yang et al. PRL 110, (2013) Normal insulator Topological insulator Weyl nodes move with increasing pressure Inversion symmetry breaking Phase transition at p=p c Gap closes between A and H No Weyl semimetal phase Weyl semimetal should intervene between the two phases Murakami, Okugawa, preprint (2014) also Liu, Vanderbilt (2014)

20 Systems with inversion symmetry Gap closes at TRIM inversions between two bands with opposite parities. Insulator-to-insulator transition e.g. TlBi(S 1-x Se x )2 WTI (or NI) Dirac semimetal STI Sato et al., Nature Phys.7, 840 ( 11)

21 Systems without inversion symmetry Gap closes at non-trim Weyl semimetal appears within a finite region WTI (or NI) Monopoleantimonopole pair creation Weyl semimetal 2 monopoles and 2 antimonopoles STI Monopoleantimonopole pair annihilation Question: Does it hold for crystals with additional crystallographic symmetries? In 3D, it holds for any STI-WTI (or STI-NI) phase transitions

22 Parametric gap closing in systems without inversion symmetry. Start from an insulator à suppose a gap closes by changing a parameter m m? Classification by space groups & k-points.

23 230 space groups 138 space groups without inversion sym No inversion sym. high-symmetry points (TRIM) high-symmetry points (non TRIM) high-symmetry lines The Mathematical Theory of Symmetry in Solids, Bradley, Cracknell Each k point à k group

24 Parametric gap closing in systems without inversion symmetry. Start with an insulator à the gap closes by changing a parameter m R c and R v should be one-dimensional m (Otherwise the gap does not close at k 0 ) r * * r Hkm (, ) = = ai( kmσ, ) i * * i k 0? k 0 effective model * * * * * * * * * * r Hkm (, ) = * * * * * * * * * * * * * * * Lowest conduction band irrep.: R c Highest valence band irrep.: R v

25 Systems without inversion symmetry à Classification of parametric gap-closing (a) Metal (gap closes along a loop) mirror symmetric (b) Weyl semimetal Only two possibilities. No insulator-to-insulator transition happens.

26 (Example #1): mirror symmetry (i.e. k : invariant under M ) M eigenvalue = +i or -i k (i) (ii) Same signs of M gap cannot close at k level repulsion Different signs of M metal with gap closing along a loop on a mirror plane (ß on the mirror plane the two bands are totally decoupled because of sign difference of M.)

27 (Example #2): C 2 symmetry (i.e. k : invariant under C 2 ) C 2 eigenvalue = +1 or -1 k (i) Same signs of C 2 gap cannot close at k level repulsion (ii) Different signs of C 2 gap closing à Weyl semimetal monopole-antimonopole pair creation à move along a symmetry line monopole antimonopole

28 (Example #3): C 2 and ΘC 2 symmetries C 2 eigenvalue = +1 or -1 k ΘC 2 (i) Same signs of C 2 Weyl semimetal two pairs of Weyl nodes à C 2 C 2 (ii) Different signs of C 2 Weyl semimetal one pair of Weyl nodes along C 2 axis monopole antimonopole

29 Systems without inversion symmetry à Classification of parametric gap-closing Metal (gap closes along a loop) Weyl semimetal No

30 BiTeI at high pressure: ab initio calc. Bahramy, Nat. Commun. 3, 679 (2012) Yang et al. PRL 110, (2013) Normal insulator Topological insulator Inversion symmetry breaking Weyl nodes move with increasing pressure Gp closes between A and H Phase transition at p=p c No Weyl semimetal phase Weyl semimetal should intervene between the two phases Murakami, Okugawa, preprint (2014) also Liu, Vanderbilt (2014)

31 Te: Weyl semimetal at high pressure M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, arxiv: (2014).

32 Te : lattice with helical chains M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, arxiv. (2014) P P Lattice with helical chains No inversion symmetry No mirror symmetry

33 Te: Weyl semimetal at high pressure M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, arxiv. (2014) Evolution of Weyl nodes Spin-orbit coupling cf;: topological insulator under shear strain. A. Agapito et al., PRL 110, (2013)

34 Te: spin structure at ambient pressure Valence band spin // z axis : opposite spins for H and H points valley degree of freedom similar to MoS 2 valleytronics Conduction band hedgehog spin structure : unique to systems without mirror symmetry

35 Lattice structure with helix structure without mirror & inversion symmetry. similar to a solenoid à magnetization induced by a current (Example-1) chiral nanotubes (Example-2) chiral lattice structure (e.g.: Te) Miyamoto, Rubio, Cohen, Louie, PRB50, 4976 ( 94) Tsuji, Takajo, Aoki, PRB75, ( 07) Wnag, Chu, Wang, Guo, PRB80, ( 09) Righthanded Lefthanded

36 Current-induced magnetization model calculation with 3D crystal with chiral lattice structure Orbital magnetization Δ= 0 Δ= 0 Δ= 1 Δ= 2 Δ= Spin 左巻きのらせんの軌道磁化 magnetization Δ= 3 Δ= Δ= 1 Δ= 2 Δ= Δ= 1 Δ= 0 Δ= 0 Δ= 1 Δ= 2 Δ= E F E F Current along the helix à magnetization is induced Induced magnetization is opposite for opposite handedness

37 Spin structure aroung H point Spin structure around H H -3-4 A Γ H A L H K Γ M K Brillouin zone A H L K Γ M Radial spin texture (like Te) unlike Rashba SOC

38 Summary Universal phase diagram for TI/NI/Weyl/Dirac semimetals Weyl semimetal phase in 3D, but not in 2D Checked by model calculation (Fu-Kane-Mele model) Surface state evolution Holds for any space groups Murakami, New J. Phys. 9, 356 (2007) Murakami, Kuga, PRB78, (2008) Okugawa, Murakami, Phys. Rev. B 89, (2014) Te: unique lattice structure with low symmetry Weyl semimetal at high pressure hedgehog spin structure around H and H. Hirayama et al., arxiv: (2014).