Shuichi Murakami 1 Department of Physics, Tokyo Institute of Technology 2 TIES, Tokyo Institute of Technology 3 CREST, JST

Weyl semimetals and topological phase transitions

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)

•  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

k-space (bulk)

k-space (surface)

Wan et al., PRB (2011)

Weyl semimetal

•  Surface Fermi arc – connecting between Weyl nodes

Weyl node

Physics 4, 36 (2011)

• 　pyrochlore iridates (Wan et al., PRB (2011)…) •  TI multilayer (Burkov, Balents, PRL 107, 127205 (2011))　　　….

Dirac semimetal = Bulk 3D Dirac cones with degeneracy

Weyl semimetal = Bulk 3D Dirac cones without degeneracy

3D Weyl nodes = monopole or antimonopole for Berry curvature

Antimonopole at

0( ) ( )l k k kρ δ= −r r rMonopole at

0k k=r r

kr

0kr

Weyl node

( ) nk nkn

u uB k ik kｶ ｶ= ｴｶ ｶ

r

1( ) ( )2n nkk B kρπ

= ∇ ⋅rr rr

: Berry curvature

: monopole density

0( ) ( )l k k kρ δ= − −r r r

or

0k k=r r

•  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).

Weyl nodes in 2D and 3D 2D Weyl node :

( , ) x x y y zH k m k k ms s s= + +r

parameter opens a gap. m

( )( , ) x x y y z z z

x x y y z z

H k m k k k m

k k k m

s s s s

s s s

= + + +

= + + +

r3D Weyl node :

Weyl point moves but gap does not open. (0,0,0)à(0,0,-m)

ß Monopole charge (in 3D k-space) is conserved

0m = 0m ｹ

3D Weyl node is topological.

Bulk band structure Bulk + surface

Surface Fermi arc : effective model calc.

Weyl nodes

Fermi arcs

Okugawa, Murakami, Phys. Rev. B 89, 235315 (2014) 　　　　

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

Effective model : surface states

top surface

bottom surface

TI

( )

Fermi arcs

Unitary transf. z

Fermi arc • connect between bulk Dirac cones • tangential to Dirac cones

Okugawa, Murakami, Phys. Rev. B 89, 235315 (2014) Haldane, arXiv: 1401.0529 　　　　

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

(A) systems without inversion symmetry

Z2 topological number ß Calculated from bulk Bloch wf.

( )[ ][ ]

det ( )1

Pf ( )i

i i

ww

ν Γ− =

Γ∏

Fu,Kane, PRB(2006)

( ) ( )21

1N

m ii m

νξ

=

− = Γ∏∏

Fu,Kane, PRB(2007)

nkmkmn uukw ,,)( Θ= −

!

Parity eigenvalue +1: symmetric -1: asymmetric

　(B) systems with inversion symmetry

à How does the TI-NI topological phase transition occur in (A) & (B)?

iΓ : TRIM

Universal phase diagram in 3D and in 2D

•  In 3D, Weyl nodes are monopoles and antimonopoles. conservation of monopole charge

δ( : inversion symmetry breaking) ( : external parameter ) m

3D 2D

SM, Iso, Avishai, Onoda,Nagaosa PRB (07) SM, New J. Phys. (‘07). SM. Kuga, PRB (’08) SM, Physica E43, 748 (‘11)

Evolution of Weyl points by parameter change

•  Time-reversal symmetry

à Monopoles are symmetric w.r.t. k=0

Pair annihilation

Pair creation

Pair creation

Pair annihilation

Monopole- antimonopole pair creation

Monopole- antimonopole pair annihilation

2 monopoles and 2 antimonopoles

Weyl semimetal

SM, New J. Phys. 9, 356 (‘07); Phase transition between TI and NI phases

3D

•  No I-symmetry :

•  I-symmetry : Phase diagram

WTI (or NI) STI

WTI (or NI) TI Dirac semimetal

Lattice model: Fu-Kane-Mele model + staggered on-site energy

Weyl semimetal appears! (Murakami,Kuga, PRB78, 165313(2008)

Fu-Kane-Mele model (PRL98, 106803 (2007))

•  Diamond lattice •  nearest neighbor: spin-indep. hopping •  next nearest neighbor: spin-orbit coupling

On-site staggered potenial à breaks inversion

Inversion- symmetric

Weyl semimetal

Weyl semimetal

Dirac semimetal

k-space trajectory of the monopoles

Fu-Kane-Mele model + inversion-symmetry breaking

Weyl semimetal

Weyl semimetal

Evolution of surface states Weyl semimetal : Fermi arc (between monopole and antimonopole)

Weyl semimetal : Fermi arc

Topological insulator : Dirac cone

top surface

bottom surface

Okugawa,  Murakami,  Phys.  Rev.  B  89,  235315  (2014)  　　　　

Change of surface terminations

Weyl semimetal

Topological insulator

No dangling bonds With dangling bonds

Surface Fermion parity à surface termination changes the inside and outside of surface FS. (for inversion symmetric systems) Teo, Fu, Kane, PRB78, 045426 (2008)

Surface termination change the pairing of Weyl nodes

Okugawa,  Murakami,  Phys.  Rev.  B  89,  235315  (2014)  　　　　

Universal phase diagram in 3D

Only the inversion and time-reversal symmetries are considered.

SM, Iso, Avishai, Onoda,Nagaosa PRB (07) SM, New J. Phys. (‘07). SM. Kuga, PRB (’08) SM, Physica E43, 748 (‘11)

Question: Does it hold for crystals with additional crystallographic symmetries?

Bahramy, Nat. Commun. 3, 679 (2012) Yang et al. PRL 110, 086402 (2013)

BiTeI at high pressure: ab initio calc.

Normal insulator

Phase transition at p=pc

•  Gap closes between A and H •  No Weyl semimetal phase

Inversion symmetry breaking

Weyl semimetal should intervene between the two phases Murakami, Okugawa, preprint (2014)

　 also Liu, Vanderbilt (2014)

Topological insulator Weyl nodes move with increasing pressure

Systems with inversion symmetry

e.g. TlBi(S1-xSex)2

Sato et al., Nature Phys.7, 840 (‘11)

•  Gap closes at TRIM inversions between two bands with opposite parities. •  Insulator-to-insulator transition

WTI (or NI) STI Dirac semimetal

Systems without inversion symmetry

Monopole- antimonopole pair creation

Monopole- antimonopole pair annihilation

2 monopoles and 2 antimonopoles

Weyl semimetal

WTI (or NI) STI

•  Gap closes at non-TRIM •  Weyl semimetal appears within a finite region

Question: Does it hold for crystals with additional crystallographic symmetries?

In 3D, it holds for any STI-WTI (or STI-NI) phase transitions

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230

No inversion sym.

230 space groups 138 space groups without inversion sym.

“The Mathematical Theory of Symmetry in Solids”, Bradley, Cracknell

high-symmetry points (TRIM)

high-symmetry points (non TRIM)

high-symmetry lines

Each k point à k group

Parametric gap closing in systems without inversion symmetry.

Start with an insulator à the gap closes by changing a parameter m m

?

* * * * ** * * * *

( , ) * * * * ** * * * ** * * * *

H k m

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

r

k0

Rc

Rv

Lowest conduction band irrep.:

Highest valence band irrep.:

* *( , ) ( , )

* * i ii

H k m a k m σ⎛ ⎞

= =⎜ ⎟⎝ ⎠

∑r r

Rc and Rv should be one-dimensional (Otherwise the gap does not close at k0)

k0

effective model

Systems without inversion symmetry à Classification of parametric gap-closing

(b) Weyl semimetal

(a) Metal （gap closes along a loop) – mirror symmetric

Only two possibilities. No insulator-to-insulator transition happens.

(Example #1): mirror symmetry (i.e. k : invariant under M )

k

M eigenvalue = +i or -i

(ii)  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 .)

(i)  Same signs of M gap cannot close at k – level repulsion

(Example #2): C2 symmetry (i.e. k : invariant under C2 )

k

C2 eigenvalue = +1 or -1

(ii)  Different signs of C2 gap closing à Weyl semimetal monopole-antimonopole pair creation à move along a symmetry line

(i)  Same signs of C2 gap cannot close at k – level repulsion

monopole

anti- monopole

(Example #3): C2 and ΘC2 symmetries

k

C2 eigenvalue = +1 or -1

(ii)  Different signs of C2 Weyl semimetal one pair of Weyl nodes along C2 axis

(i)  Same signs of C2 Weyl semimetal two pairs of Weyl nodes à

ΘC2

C2

monopole

anti- monopole

C2

Systems without inversion symmetry à Classification of parametric gap-closing

Weyl semimetal Metal （gap closes along a loop)

No

Bahramy, Nat. Commun. 3, 679 (2012) Yang et al. PRL 110, 086402 (2013)

BiTeI at high pressure: ab initio calc.

Normal insulator

Phase transition at p=pc

•  Gp closes between A and H •  No Weyl semimetal phase

Inversion symmetry breaking

Weyl semimetal should intervene between the two phases Murakami, Okugawa, preprint (2014)

　 also Liu, Vanderbilt (2014)

Topological insulator Weyl nodes move with increasing pressure

Te: Weyl semimetal at high pressure

M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, arXiv:1409.7517 (2014).

Te : lattice with helical chains

P3121 P3221

•  Lattice with helical chains •  No inversion symmetry •  No mirror symmetry

M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, 　　　　　　arXiv. (2014)

Te: Weyl semimetal at high pressure

Spin-orbit coupling

Evolution of Weyl nodes

cf;: topological insulator under shear strain. A. Agapito et al., PRL 110, 176401 (2013)

M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, T. Miyake, 　　　　　　arXiv. (2014)

Conduction band hedgehog spin structure : unique to systems without mirror symmetry

Te: spin structure at ambient pressure

Valence band spin // z axis : opposite spins for H and H’ points valley degree of freedom similar to MoS2 “valleytronics”

• 　similar to a solenoid à magnetization induced by a current (Example-1) chiral nanotubes

Right-handed

Left-handed

(Example-2) chiral lattice structure (e.g.: Te)

Lattice structure with helix structure without mirror & inversion symmetry.

Miyamoto, Rubio, Cohen, Louie, PRB50, 4976 (’94) Tsuji, Takajo, Aoki, PRB75, 153406 (‘07) Wnag, Chu, Wang, Guo, PRB80, 235430 (’09)

Current-induced magnetization – model calculation with 3D crystal with chiral lattice structure

-0.05

0

0.05

0.1

0.15

0.2

0.25

-4 -3 -2 -1 0 1 2 3 4

Δ=0

Δ= 1

Δ= 2

Δ= 3

Δ= 0Δ= 1Δ=2Δ= 3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

-4 -3 -2 -1 0 1 2 3 4

Δ=0

Δ= 1

Δ= 2

Δ= 3

Δ= 0Δ= 1

Δ= 2

Δ=3

Orbital magnetization 左巻きのらせんの軌道磁化

•  Current along the helix à magnetization is induced •  Induced magnetization is opposite for opposite handedness

Spin magnetization

EF EF

Spin structure aroung H point

Brillouin zone

K

H

M

L

Γ

A

Spin structure around H

-4

-3

-2

-1

0

1

2

3

4

A Γ H A L H K Γ M K

-1

0

1

3.4 4.2 5.2

H

Radial spin texture (like Te) unlike Rashba SOC

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

Te: unique lattice structure with low symmetry

•  Weyl semimetal at high pressure •  hedgehog spin structure around H and H’.

Murakami, New J. Phys. 9, 356 (2007) Murakami, Kuga, PRB78, 165313 (2008) Okugawa, Murakami, Phys. Rev. B 89, 235315 (2014)

Hirayama et al., arXiv:1409.7517 (2014).